text: MOTIFS ET ADJOINTS 5 Théorème 3. — Soit K/k un corps de fonctions d’une variable, et soit ∈ A une vari abenne sur K. Alors il existe un motif x(A, K/k) Choweﬀ(k, Q), dont l’image dans Moteﬀ(k, Q) est de poids 2 pour toute cohomologie de H Weil classique (§), et tel que H (cid:39) ∗ − x X H ( (A, K/k)) V ( (A , Kk ))( 1) l l s pour tout nombre premier (cid:54) , o X ∗ est le groupe de Tate- l = car k (A , Kk ) s Šafarevič “gique” de la duale de et est la risation -adique. Ce motif A H l l est fonctoriel en pour les homomorphismes de varis abennes. A Voir cor. 8.4 et th. 8.6. Le dernier rltat a trait aux “correspondances au point grique” di par Bloch, Rovinsky et Beilinson. Pour toute -vari projective lisse , on k X note End (X) la Q-alge de ces correspondances [4]. gen Théorème 4. — Soient K/k une extension de type ﬁni, X une K-vari X projective lisse et un -mod projectif lisse de . Alors il existe un k X homomorphisme canonique d’alges → X End (X) End ( ) gen gen dont on peut expliciter le conoyau. Voir th. 9.3. X En particulier, si et si est une courbe (donc ) degtr(K/k) = 1 X est une surface), est une -alge semi-simple de dimension End (X) Q gen ﬁnie; ceci donne une approche possible de la conjecture de Bloch sur les surfaces (rem. 9.5). Dans la premi section, on fait des “rappels” sur les thies motiviques et on ﬁxe quelques notations. La seconde section est une liste d’exemples qui interviennent dans la suite de l’article. La troisi est une collection de sorites sur les adjonctions. Dans la section 3.8, on traite le cas d’une ex- tension ﬁnie: ces rltats ne sont ni diﬃciles ni surprenants (pour certaines thies motiviques, ils sont des cas extrment particuliers de ceux de [3]), mais ils ont d une application non triviale rltat d’eﬀectivitur les motifs purs: le th 4.8. La section est consacrux motifs numques. Vu le th de semi-simplicit Jannsen [17], l’existence des adjoints est ici un cas particulier de leur existence pour tout foncteur pleinement ﬁd entre catries semi-simples (proposition 5.3).

text: MOTIFS

text: ET

text: ADJOINTS

text: Théorème

text: Soit

text: K/k

text: un

text: corps

text: de

text: fonctions

text: d’une

text: variable,

text: et

text: soit

text: A

text: une

text: vari

text: abenne

text: sur

text: K.

text: Alors

text: il

text: existe

text: un

text: motif

text: x(A,

text: K/k)

text: Choweﬀ(k,

text: Q),

text: dont

text: l’image

text: dans

text: Moteﬀ(k,

text: Q)

text: est

text: de

text: poids

text: pour

text: toute

text: cohomologie

text: de

text: H

text: Weil

text: classique

text: et

text: tel

text: que

text: H

text: x

text: X

text: H

text: (A,

text: K/k))

text: V

text: (A

text: Kk

text: l

text: l

text: s

text: pour

text: tout

text: nombre

text: premier

text: o

text: X

text: est

text: le

text: groupe

text: de

text: Tate-

text: l

text: car

text: k

text: (A

text: Kk

text: s

text: Šafarevič

text: “gique”

text: de

text: la

text: duale

text: de

text: et

text: est

text: la

text: risation

text: -adique.

text: Ce

text: motif

text: A

text: H

text: l

text: l

text: est

text: fonctoriel

text: en

text: pour

text: les

text: homomorphismes

text: de

text: varis

text: abennes.

text: A

text: Voir

text: cor.

text: et

text: th.

text: Le

text: dernier

text: rltat

text: a

text: trait

text: aux

text: “correspondances

text: au

text: point

text: grique”

text: di

text: par

text: Bloch,

text: Rovinsky

text: et

text: Beilinson.

text: Pour

text: toute

text: -vari

text: projective

text: lisse

text: on

text: k

text: X

text: note

text: End

text: (X)

text: la

text: Q-alge

text: de

text: ces

text: correspondances

text: gen

text: Théorème

text: Soient

text: K/k

text: une

text: extension

text: de

text: type

text: ﬁni,

text: X

text: une

text: K-vari

text: X

text: projective

text: lisse

text: et

text: un

text: -mod

text: projectif

text: lisse

text: de

text: Alors

text: il

text: existe

text: un

text: k

text: X

text: homomorphisme

text: canonique

text: d’alges

text: X

text: End

text: (X)

text: End

text: gen

text: gen

text: dont

text: on

text: peut

text: expliciter

text: le

text: conoyau.

text: Voir

text: th.

text: X

text: En

text: particulier,

text: si

text: et

text: si

text: est

text: une

text: courbe

text: (donc

text: degtr(K/k)

text: X

text: est

text: une

text: surface),

text: est

text: une

text: -alge

text: semi-simple

text: de

text: dimension

text: End

text: (X)

text: Q

text: gen

text: ﬁnie;

text: ceci

text: donne

text: une

text: approche

text: possible

text: de

text: la

text: conjecture

text: de

text: Bloch

text: sur

text: les

text: surfaces

text: (rem.

text: Dans

text: la

text: premi

text: section,

text: on

text: fait

text: des

text: “rappels”

text: sur

text: les

text: thies

text: motiviques

text: et

text: on

text: ﬁxe

text: quelques

text: notations.

text: La

text: seconde

text: section

text: est

text: une

text: liste

text: d’exemples

text: qui

text: interviennent

text: dans

text: la

text: suite

text: de

text: l’article.

text: La

text: troisi

text: est

text: une

text: collection

text: de

text: sorites

text: sur

text: les

text: adjonctions.

text: Dans

text: la

text: section

text: on

text: traite

text: le

text: cas

text: d’une

text: ex-

text: tension

text: ﬁnie:

text: ces

text: rltats

text: ne

text: sont

text: ni

text: diﬃciles

text: ni

text: surprenants

text: (pour

text: certaines

text: thies

text: motiviques,

text: ils

text: sont

text: des

text: cas

text: extrment

text: particuliers

text: de

text: ceux

text: de

text: mais

text: ils

text: ont

text: d

text: une

text: application

text: non

text: triviale

text: rltat

text: d’eﬀectivitur

text: les

text: motifs

text: purs:

text: le

text: th

text: La

text: section

text: est

text: consacrux

text: motifs

text: numques.

text: Vu

text: le

text: th

text: de

text: semi-simplicit

text: Jannsen

text: l’existence

text: des

text: adjoints

text: est

text: ici

text: un

text: cas

text: particulier

text: de

text: leur

text: existence

text: pour

text: tout

text: foncteur

text: pleinement

text: ﬁd

text: entre

text: catries

text: semi-simples

text: (proposition


text: 50 BRUNO KAHN ﬁbre générique de l’immersion fermée Y × X → Y × X . S k On aun diagramme commutatif     X × Y −−−→ X × Y → X × Y − X × Y → (cid:121) (cid:121) (cid:121) CH ( ) CH ( ) CH ( ) 0 n S n k n k S     −−−→ Y → Y − → (cid:121) (cid:121) CH (Y ) CH ( X ) CH ( Y ) 0 0 K(X) 0 k( ) 0 k(X) K(X) × X × Y CH (X Y ) −−−→ CH ( ) n ≡ K n+d ≡ k où la suite supérieure est exacte et où les ﬂèches verticales sont surjec- tives. D’où le premier isomorphisme de c); le second se dit du lemme 9.2. Remarque 9.4. — Pour d = n = 1, on obtient la “description plus fonctorielle” de l’homomorphisme r de [20, Prop. 3.1 a)], promise dans la preuve de loc. cit. Ceci permet de montrer que le motif x (A, K/k) de la dnition 8.5 coide avec celui introduit dans [20, prop. 3.2] (ce qui dntre l’unicit ce dernier omorphisme unique pr et sa fonctorialit ). En A eﬀet, en appliquant le foncteur de (9.1) et en tenant compte du th S 2 9.3 b), on obtient (cid:39) f2(S (ho(A)) S (xo(A, K/k)). (cid:93) 1 1 2 Revenant dnstration de la proposition 8.3, on voit avec ses notations que S (xo(A, K/k)) est dupr S (to(S)) par le projecteur f2(S (π)) qui, 2 2 2 (cid:93) 1 vu (9.2), n’est autre que celui de [20, d de la prop. 3.2]. On conclut en observant que −→∼ End (t (S)) End (S (to(S))) Moteraﬀt(k,Q) 2 d2 Motorat(k,Q) 2 2 via , d’apr[th. 7.4.3]kmp. S P 2 2 Remarque 9.5. — On peut comparer ivalences rationnelle et numque. Avec les notations du th 9.3 et (qu’on omet dorvant), cela donne A = Q un diagramme commutatif: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: BRUNO

text: KAHN

text: ﬁbre

text: générique

text: de

text: l’immersion

text: fermée

text: Y

text: X

text: Y

text: X

text: S

text: k

text: On

text: aun

text: diagramme

text: commutatif

text: X

text: Y

text: X

text: Y

text: X

text: Y

text: X

text: Y

text: CH

text: CH

text: CH

text: n

text: S

text: n

text: k

text: n

text: k

text: S

text: Y

text: Y

text: CH

text: (Y

text: CH

text: X

text: CH

text: Y

text: K(X)

text: k(

text: k(X)

text: K(X)

text: X

text: Y

text: CH

text: (X

text: Y

text: CH

text: n

text: K

text: n+d

text: k

text: où

text: la

text: suite

text: supérieure

text: est

text: exacte

text: et

text: où

text: les

text: ﬂèches

text: verticales

text: sont

text: surjec-

text: tives.

text: D’où

text: le

text: premier

text: isomorphisme

text: de

text: c);

text: le

text: second

text: se

text: dit

text: du

text: lemme

text: Remarque

text: Pour

text: d

text: n

text: on

text: obtient

text: la

text: “description

text: plus

text: fonctorielle”

text: de

text: l’homomorphisme

text: r

text: de

text: Prop.

text: a)],

text: promise

text: dans

text: la

text: preuve

text: de

text: loc.

text: cit.

text: Ceci

text: permet

text: de

text: montrer

text: que

text: le

text: motif

text: x

text: (A,

text: K/k)

text: de

text: la

text: dnition

text: coide

text: avec

text: celui

text: introduit

text: dans

text: prop.

text: (ce

text: qui

text: dntre

text: l’unicit

text: ce

text: dernier

text: omorphisme

text: unique

text: pr

text: et

text: sa

text: fonctorialit

text: En

text: A

text: eﬀet,

text: en

text: appliquant

text: le

text: foncteur

text: de

text: et

text: en

text: tenant

text: compte

text: du

text: th

text: S

text: b),

text: on

text: obtient

text: (ho(A))

text: S

text: (xo(A,

text: K/k)).

text: Revenant

text: dnstration

text: de

text: la

text: proposition

text: on

text: voit

text: avec

text: ses

text: notations

text: que

text: S

text: (xo(A,

text: K/k))

text: est

text: dupr

text: S

text: (to(S))

text: par

text: le

text: projecteur

text: (π))

text: qui,

text: vu

text: n’est

text: autre

text: que

text: celui

text: de

text: d

text: de

text: la

text: prop.

text: On

text: conclut

text: en

text: observant

text: que

text: End

text: (t

text: (S))

text: End

text: (S

text: (to(S)))

text: Moteraﬀt(k,Q)

text: Motorat(k,Q)

text: via

text: d’apr[th.

text: S

text: P

text: Remarque

text: On

text: peut

text: comparer

text: ivalences

text: rationnelle

text: et

text: numque.

text: Avec

text: les

text: notations

text: du

text: th

text: et

text: (qu’on

text: omet

text: dorvant),

text: cela

text: donne

text: A

text: Q

text: un

text: diagramme

text: commutatif:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 BRUNO KAHN Dans la section , on s’attaque aux catries birationnelles di dans [25, 26, 27]. Ici, la technique utilisour montrer l’existence d’un adjoint uche est celle de la proposition 3.7. Elle marche de mani remarquablement aist uniforme. On donne aussi d’autres dnstrations, fond sur le calcul explicite des ensembles de morphismes dans ces catries et qui fournissent peut-e une meilleure intuition de ce qui se passe; ces dnstrations ont toutefois l’inconvent d’utiliser la rlution des singularit sous la forme de Hironaka ou sous celle de de Jong. Dans la section , on donne des exemples o les adjoints n’existent pas: extensions algiques inﬁnies, extension rlis inﬁnies pour les motifs de Chow eﬀectifs. Ce dernier cas (th 7.4) n’est pas entiment trivial et repose sur le fait qu’une vari abenne non nulle a un rang positif sur un corps algiquement clos de dimension de Kronecker . Le cas o la base de > 0 l’extension est algique sur un corps ﬁni est d’ailleurs une exception, voir proposition 7.5. Dans la section , on fait quelques calculs explicites d’adjoints dans le cas des motifs de Chow birationnels. On en dit le corollaire 8.4 (et donc une partie du th 3). Enﬁn, dans la section , on applique ces rltats aux catries de “motifs au point grique” dlopp dans [22, §7.8] rtir de l’ide Bloch-Rovinsky-Beilinson. Il reste crire ce qui n’est pas fait dans cet article. J’avais initialement pr de traiter les -motifs: cela s’av beaucoup plus dcat que je ne le 1 pensais, et devra faire l’objet d’un autre travail. Dans le cas des motifs purs, l’existence ou l’inexistence d’adjoints n’est claire que pour l’ivalence rationnelle et l’ivalence numque: je ne sais rien dans le cas d’une relation intermaire (voir § pour le cas des motifs homologiques). Dans le cas des motifs eﬀectifs, un analogue du th 7.4 pour l’ivalence algique semble particuliment intssant cider: je soupçonne que le groupe de Griﬃths donne une obstruction existence d’un adjoint, mais je n’ai pas rsi briquer un argument convaincant pour le dntrer. Enﬁn, la dnstration du th 8.6 est incompl. Ces résultats ont été initialement inspirés par mon travail avec R. Su- jatha sur les motifs birationnels [23, 24, 25, 26, 27]. Une partie de cet article a été imaginée au cours d’un séjour à l’IMPA en août 2008, ﬁnancé par la Coopération France-Brésil. Je tiens à remercier cette dernière de son soutien, l’IMPA de son hospitalité, et Marc Hindry et Amílcar Pa- checo pour des échanges lors de ce séjour. Je remercie également Ah- med Abbes, Joseph Ayoub, Daniel Bertrand, Mikhail Bondarko, Antoine Chambert-Loir, Denis-Charles Cisinski, Pierre Colmez, Frédéric Déglise, Mathieu Florence, V. Srinivas, Claire Voisin et Jörg Wildeshaus pour des

text: BRUNO

text: KAHN

text: Dans

text: la

text: section

text: on

text: s’attaque

text: aux

text: catries

text: birationnelles

text: di

text: dans

text: Ici,

text: la

text: technique

text: utilisour

text: montrer

text: l’existence

text: d’un

text: adjoint

text: uche

text: est

text: celle

text: de

text: la

text: proposition

text: Elle

text: marche

text: de

text: mani

text: remarquablement

text: aist

text: uniforme.

text: On

text: donne

text: aussi

text: d’autres

text: dnstrations,

text: fond

text: sur

text: le

text: calcul

text: explicite

text: des

text: ensembles

text: de

text: morphismes

text: dans

text: ces

text: catries

text: et

text: qui

text: fournissent

text: peut-e

text: une

text: meilleure

text: intuition

text: de

text: ce

text: qui

text: se

text: passe;

text: ces

text: dnstrations

text: ont

text: toutefois

text: l’inconvent

text: d’utiliser

text: la

text: rlution

text: des

text: singularit

text: sous

text: la

text: forme

text: de

text: Hironaka

text: ou

text: sous

text: celle

text: de

text: de

text: Jong.

text: Dans

text: la

text: section

text: on

text: donne

text: des

text: exemples

text: o

text: les

text: adjoints

text: n’existent

text: pas:

text: extensions

text: algiques

text: inﬁnies,

text: extension

text: rlis

text: inﬁnies

text: pour

text: les

text: motifs

text: de

text: Chow

text: eﬀectifs.

text: Ce

text: dernier

text: cas

text: (th

text: n’est

text: pas

text: entiment

text: trivial

text: et

text: repose

text: sur

text: le

text: fait

text: qu’une

text: vari

text: abenne

text: non

text: nulle

text: a

text: un

text: rang

text: positif

text: sur

text: un

text: corps

text: algiquement

text: clos

text: de

text: dimension

text: de

text: Kronecker

text: Le

text: cas

text: o

text: la

text: base

text: de

text: l’extension

text: est

text: algique

text: sur

text: un

text: corps

text: ﬁni

text: est

text: d’ailleurs

text: une

text: exception,

text: voir

text: proposition

text: Dans

text: la

text: section

text: on

text: fait

text: quelques

text: calculs

text: explicites

text: d’adjoints

text: dans

text: le

text: cas

text: des

text: motifs

text: de

text: Chow

text: birationnels.

text: On

text: en

text: dit

text: le

text: corollaire

text: (et

text: donc

text: une

text: partie

text: du

text: th

text: Enﬁn,

text: dans

text: la

text: section

text: on

text: applique

text: ces

text: rltats

text: aux

text: catries

text: de

text: “motifs

text: au

text: point

text: grique”

text: dlopp

text: dans

text: rtir

text: de

text: l’ide

text: Bloch-Rovinsky-Beilinson.

text: Il

text: reste

text: crire

text: ce

text: qui

text: n’est

text: pas

text: fait

text: dans

text: cet

text: article.

text: J’avais

text: initialement

text: pr

text: de

text: traiter

text: les

text: -motifs:

text: cela

text: s’av

text: beaucoup

text: plus

text: dcat

text: que

text: je

text: ne

text: le

text: pensais,

text: et

text: devra

text: faire

text: l’objet

text: d’un

text: autre

text: travail.

text: Dans

text: le

text: cas

text: des

text: motifs

text: purs,

text: l’existence

text: ou

text: l’inexistence

text: d’adjoints

text: n’est

text: claire

text: que

text: pour

text: l’ivalence

text: rationnelle

text: et

text: l’ivalence

text: numque:

text: je

text: ne

text: sais

text: rien

text: dans

text: le

text: cas

text: d’une

text: relation

text: intermaire

text: (voir

text: pour

text: le

text: cas

text: des

text: motifs

text: homologiques).

text: Dans

text: le

text: cas

text: des

text: motifs

text: eﬀectifs,

text: un

text: analogue

text: du

text: th

text: pour

text: l’ivalence

text: algique

text: semble

text: particuliment

text: intssant

text: cider:

text: je

text: soupçonne

text: que

text: le

text: groupe

text: de

text: Griﬃths

text: donne

text: une

text: obstruction

text: existence

text: d’un

text: adjoint,

text: mais

text: je

text: n’ai

text: pas

text: rsi

text: briquer

text: un

text: argument

text: convaincant

text: pour

text: le

text: dntrer.

text: Enﬁn,

text: la

text: dnstration

text: du

text: th

text: est

text: incompl.

text: Ces

text: résultats

text: ont

text: été

text: initialement

text: inspirés

text: par

text: mon

text: travail

text: avec

text: R.

text: Su-

text: jatha

text: sur

text: les

text: motifs

text: birationnels

text: Une

text: partie

text: de

text: cet

text: article

text: a

text: été

text: imaginée

text: au

text: cours

text: d’un

text: séjour

text: à

text: l’IMPA

text: en

text: août

text: ﬁnancé

text: par

text: la

text: Coopération

text: France-Brésil.

text: Je

text: tiens

text: à

text: remercier

text: cette

text: dernière

text: de

text: son

text: soutien,

text: l’IMPA

text: de

text: son

text: hospitalité,

text: et

text: Marc

text: Hindry

text: et

text: Amílcar

text: Pa-

text: checo

text: pour

text: des

text: échanges

text: lors

text: de

text: ce

text: séjour.

text: Je

text: remercie

text: également

text: Ah-

text: med

text: Abbes,

text: Joseph

text: Ayoub,

text: Daniel

text: Bertrand,

text: Mikhail

text: Bondarko,

text: Antoine

text: Chambert-Loir,

text: Denis-Charles

text: Cisinski,

text: Pierre

text: Colmez,

text: Frédéric

text: Déglise,

text: Mathieu

text: Florence,

text: V.

text: Srinivas,

text: Claire

text: Voisin

text: et

text: Jörg

text: Wildeshaus

text: pour

text: des


text: 54 BRUNO KAHN [34] A. Neeman Triangulated categories, Ann. of Math. Studies 148, Princeton Univ. Press, 2001. [35] J.S. Milne Etale cohomology, Princeton Univ. Press, 1980. [36] H.-G. Quebbemann, W. Scharlau, M. Schulte Quadratic and Hermi- tian forms in additive and abelian categories, J. Algebra 59 (1979), 264–289. [37] N. Ramachandran Duality of Albanese and Picard 1-motives, K- Theory 22 (2001), 271–301. [38] D. Roy, M. Waldschmidt Autour du théorème du sous-groupe algé- brique, Canad. Math. Bull. 36 (1993), 358–367. [39] R. Sebastian Examples of smash nilpotent cycles on rationally connected varieties, J. of Algebra 438 (2015), 119–129. [40] A. Suslin Motivic complexes over nonperfect ﬁelds, Ann. of K-theory 2 (2017), 277–302. [41] C. Vial Chow-Künneth decomposition for 3- and 4-folds ﬁbred by va- rieties with trivial Chow group of zero cycles, J. Algebraic Geometry 24 (2015), 51–80. [42] V. Voevodsky A1-homotopy theory, Proc. ICM (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, 579–604. [43] V. Voevodsky Triangulated categories of motives over a ﬁeld, in E. Friedlander, A. Suslin, V. Voevodsky Cycles, transfers and motivic cohomology theories, Ann. Math. Studies 143, Princeton University Press, 2000, 188–238. [44] A. Weil Varis abennes et courbes algiques, Hermann, 1948. [SGA1] A. Grothendieck, Revments les et groupe fondamental (SGA1), nou- velle tion, Doc. Math. 3, SMF, 2003. [SGA4-I] E. Artin, A. Grothendieck, J.-L. Verdier Théorie des topos et co- homologie étale des schémas (SGA4), Vol. 1, Lect. Notes in Math. 269, Springer, 1972. 7 mars 2017 Bruno Kahn, Institut de Mathématiques de Jussieu - PRG, Case 247, 4 place • Jussieu, 75252 Paris Cedex 05, FRANCE E-mail : bruno.kahn@imj-prg.fr ##LTLine##

text: BRUNO

text: KAHN

label: reference

text: A.

text: Neeman

text: Triangulated

text: categories,

text: Ann.

text: of

text: Math.

text: Studies

text: Princeton

text: Univ.

text: Press,

text: J.S.

text: Milne

text: Etale

text: cohomology,

text: Princeton

text: Univ.

text: Press,

text: H.-G.

text: Quebbemann,

text: W.

text: Scharlau,

text: M.

text: Schulte

text: Quadratic

text: and

text: Hermi-

text: tian

text: forms

text: in

text: additive

text: and

text: abelian

text: categories,

text: J.

text: Algebra

text: N.

text: Ramachandran

text: Duality

text: of

text: Albanese

text: and

text: Picard

text: K-

text: Theory

text: D.

text: Roy,

text: M.

text: Waldschmidt

text: Autour

text: du

text: théorème

text: du

text: sous-groupe

text: algé-

text: brique,

text: Canad.

text: Math.

text: Bull.

text: R.

text: Sebastian

text: Examples

text: of

text: smash

text: nilpotent

text: cycles

text: on

text: rationally

text: connected

text: varieties,

text: J.

text: of

text: Algebra

text: A.

text: Suslin

text: Motivic

text: complexes

text: over

text: nonperfect

text: ﬁelds,

text: Ann.

text: of

text: K-theory

text: C.

text: Vial

text: Chow-Künneth

text: decomposition

text: for

text: and

text: ﬁbred

text: by

text: va-

text: rieties

text: with

text: trivial

text: Chow

text: group

text: of

text: zero

text: cycles,

text: J.

text: Algebraic

text: Geometry

text: V.

text: Voevodsky

text: theory,

text: Proc.

text: ICM

text: (Berlin,

text: Doc.

text: Math.

text: Extra

text: Vol.

text: I,

text: V.

text: Voevodsky

text: Triangulated

text: categories

text: of

text: motives

text: over

text: a

text: ﬁeld,

text: in

text: E.

text: Friedlander,

text: A.

text: Suslin,

text: V.

text: Voevodsky

text: Cycles,

text: transfers

text: and

text: motivic

text: cohomology

text: theories,

text: Ann.

text: Math.

text: Studies

text: Princeton

text: University

text: Press,

text: A.

text: Weil

text: Varis

text: abennes

text: et

text: courbes

text: algiques,

text: Hermann,

text: A.

text: Grothendieck,

text: Revments

text: les

text: et

text: groupe

text: fondamental

text: nou-

text: velle

text: tion,

text: Doc.

text: Math.

text: SMF,

text: E.

text: Artin,

text: A.

text: Grothendieck,

text: J.-L.

text: Verdier

text: Théorie

text: des

text: topos

text: et

text: co-

text: homologie

text: étale

text: des

text: schémas

text: Vol.

text: Lect.

text: Notes

text: in

text: Math.

text: Springer,

text: mars

text: Bruno

text: Kahn,

text: Institut

text: de

text: Mathématiques

text: de

text: Jussieu

text: PRG,

text: Case

text: place

text: Jussieu,

text: Paris

text: Cedex

text: FRANCE

text: E-mail

text: bruno.kahn@imj-prg.fr

text: ##LTLine##


text: MOTIFS ET ADJOINTS 7 discussions autour de cet article. Finalement, je remercie le rapporteur pour ses commentaires qui m’ont permis d’amorer la prntation. 1. Notations et conventions A A A Pour toute catrie , on note (resp. ) la classe des ob- Ob( ) F l( ) A ∈ A jets (resp. des morphismes) de ; si X, Y , on note HomA(X, Y ) A ou l’ensemble des morphismes de vers , en choisissant la (X, Y ) X Y notation la plus pratique selon le contexte. Soit la catégorie des corps commutatifs (morphismes: les exten- corps K ⊂ sions de corps). On se donne une sous-catégorie . Exemples: corps la sous-catégorie pleine des corps d’une caractéristique donnée, la sous- catégorie non pleine formée des extensions ﬁnies séparables, etc. C Soit une -catégorie (par exemple la -catégorie des catégories, 2 2 Cat la -catégorie des catégories additives, celle des catégories triangulées, 2 celle des catégories -linéaires tensorielles rigides. . . ). On s’intéresse Q aux pseudo-foncteurs covariants [SGA1, VI.8] M K → C : . Un tel pseudo-foncteur consiste donc en M ∈ C ∈ K — un objet pour tout ; (k) k M ∗ ∗ M → M — un -morphisme pour toute 1 (f) = fM = f : (k) (K) → K extension de ; f : k K → → — étant donné deux extensions composables et , f : k K g : K L un -isomorphisme 2 M ◦ ∗ −→∼ ∗ ◦ ∗ c = c : (g f) g f f,g f,g où les satisfont aux relations de -cocycle habituelles. (En c 2 f,g ∗ particulier, (1 ) = IdM .) M k (k) K On dira que est une théorie motivique (déﬁnie sur , à valeurs dans C C ). On prsera “thie motivique additive, triangul’, etc. si est (etc.).

text: MOTIFS

text: ET

text: ADJOINTS

text: discussions

text: autour

text: de

text: cet

text: article.

text: Finalement,

text: je

text: remercie

text: le

text: rapporteur

text: pour

text: ses

text: commentaires

text: qui

text: m’ont

text: permis

text: d’amorer

text: la

text: prntation.

label: section

text: Notations

text: et

text: conventions

text: A

text: A

text: A

text: Pour

text: toute

text: catrie

text: on

text: note

text: (resp.

text: la

text: classe

text: des

text: ob-

text: Ob(

text: F

text: l(

text: A

text: A

text: jets

text: (resp.

text: des

text: morphismes)

text: de

text: si

text: X,

text: Y

text: on

text: note

text: HomA(X,

text: Y

text: A

text: ou

text: l’ensemble

text: des

text: morphismes

text: de

text: vers

text: en

text: choisissant

text: la

text: (X,

text: Y

text: X

text: Y

text: notation

text: la

text: plus

text: pratique

text: selon

text: le

text: contexte.

text: Soit

text: la

text: catégorie

text: des

text: corps

text: commutatifs

text: (morphismes:

text: les

text: exten-

text: corps

text: K

text: sions

text: de

text: corps).

text: On

text: se

text: donne

text: une

text: sous-catégorie

text: Exemples:

text: corps

text: la

text: sous-catégorie

text: pleine

text: des

text: corps

text: d’une

text: caractéristique

text: donnée,

text: la

text: sous-

text: catégorie

text: non

text: pleine

text: formée

text: des

text: extensions

text: ﬁnies

text: séparables,

text: etc.

text: C

text: Soit

text: une

text: -catégorie

text: (par

text: exemple

text: la

text: -catégorie

text: des

text: catégories,

text: Cat

text: la

text: -catégorie

text: des

text: catégories

text: additives,

text: celle

text: des

text: catégories

text: triangulées,

text: celle

text: des

text: catégories

text: -linéaires

text: tensorielles

text: rigides.

text: On

text: s’intéresse

text: Q

text: aux

text: pseudo-foncteurs

text: covariants

text: M

text: K

text: C

text: Un

text: tel

text: pseudo-foncteur

text: consiste

text: donc

text: en

text: M

text: C

text: K

text: un

text: objet

text: pour

text: tout

text: (k)

text: k

text: M

text: M

text: M

text: un

text: -morphisme

text: pour

text: toute

text: (f)

text: fM

text: f

text: (k)

text: (K)

text: K

text: extension

text: de

text: f

text: k

text: K

text: étant

text: donné

text: deux

text: extensions

text: composables

text: et

text: f

text: k

text: K

text: g

text: K

text: L

text: un

text: -isomorphisme

text: M

text: c

text: c

text: (g

text: f)

text: g

text: f

text: f,g

text: f,g

text: où

text: les

text: satisfont

text: aux

text: relations

text: de

text: -cocycle

text: habituelles.

text: (En

text: c

text: f,g

text: particulier,

text: IdM

text: M

text: k

text: (k)

text: K

text: On

text: dira

text: que

text: est

text: une

text: théorie

text: motivique

text: (déﬁnie

text: sur

text: à

text: valeurs

text: dans

text: C

text: C

text: On

text: prsera

text: “thie

text: motivique

text: additive,

text: triangul’,

text: etc.

text: si

text: est

text: (etc.).


text: Robustness Analysis of Preconditioned Successive Projection Algorithm for General Form of Separable NMF Problem ∗ Tomohiko Mizutani May 27, 2020 Abstract The successive projection algorithm (SPA) has been known to work well for separable non- negative matrix factorization (NMF) problems arising in applications, such as topic extraction from documents and endmember detection in hyperspectral images. One of the reasons is in that the algorithm is robust to noise. Gillis and Vavasis showed in [SIAM J. Optim., 25(1), pp. 677–698, 2015] that a preconditioner can further enhance its noise robustness. The proof rested on the condition that the dimension d and factorization rank r in the separable NMF problem coincide with each other. However, it may be unrealistic to expect that the condition holds in separable NMF problems appearing in actual applications; in such problems, d is usu- ally greater than r. This paper shows, without the condition d = r, that the preconditioned SPA is robust to noise. Keywords: nonnegative matrix factorization, separability, successive projection algorithm, robustness to noise, preconditioning AMS classiﬁcation: 15A23, 15A12, 65F30, 90C25 1 Introduction A d-by-m nonnegative matrix A is said to be separable if it has a decomposition of the form ∈ R × ∈ R × A = F W for F d r and W = (I, K)Π r m (1) + + − where I is an r-by-r identity matrix, K is an r-by-(m r) nonnegative matrix, and Π is an m-by-m permutation matrix. Here, we call F the basis matrix of A and r the factorization rank. The separable nonnegative matrix factorization problem is stated as follows. (Separable NMF Problem) Let A be of the form given in (1). Find an index set I I with r elements such that A( ) coincides with the basis matrix F . ∗ Department of Industrial Engineering and Management, Tokyo Institute of Technology, 2-12-1-W9-69, Ookayama, Meguro, Tokyo, 152-8552, Japan. mizutani.t.ab@m.titech.ac.jp 1 ##LTLine##

text: Robustness

label: title

text: Analysis

text: of

text: Preconditioned

text: Successive

text: Projection

text: Algorithm

text: for

text: General

text: Form

text: of

text: Separable

text: NMF

text: Problem

label: author

text: Tomohiko

text: Mizutani

text: May

label: date

text: Abstract

text: The

label: abstract

text: successive

text: projection

text: algorithm

text: (SPA)

text: has

text: been

text: known

text: to

text: work

text: well

text: for

text: separable

text: non-

text: negative

text: matrix

text: factorization

text: (NMF)

text: problems

text: arising

text: in

text: applications,

text: such

text: as

text: topic

text: extraction

text: from

text: documents

text: and

text: endmember

text: detection

text: in

text: hyperspectral

text: images.

text: One

text: of

text: the

text: reasons

text: is

text: in

text: that

text: the

text: algorithm

text: is

text: robust

text: to

text: noise.

text: Gillis

text: and

text: Vavasis

text: showed

text: in

text: [SIAM

text: J.

text: Optim.,

text: pp.

text: that

text: a

text: preconditioner

text: can

text: further

text: enhance

text: its

text: noise

text: robustness.

text: The

text: proof

text: rested

text: on

text: the

text: condition

text: that

text: the

text: dimension

text: d

text: and

text: factorization

text: rank

text: r

text: in

text: the

text: separable

text: NMF

text: problem

text: coincide

text: with

text: each

text: other.

text: However,

text: it

text: may

text: be

text: unrealistic

text: to

text: expect

text: that

text: the

text: condition

text: holds

text: in

text: separable

text: NMF

text: problems

text: appearing

text: in

text: actual

text: applications;

text: in

text: such

text: problems,

text: d

text: is

text: usu-

text: ally

text: greater

text: than

text: r.

text: This

text: paper

text: shows,

text: without

text: the

text: condition

text: d

text: r,

text: that

text: the

text: preconditioned

text: SPA

text: is

text: robust

text: to

text: noise.

text: Keywords:

text: nonnegative

text: matrix

text: factorization,

text: separability,

text: successive

text: projection

text: algorithm,

text: robustness

text: to

text: noise,

text: preconditioning

text: AMS

text: classiﬁcation:

text: Introduction

text: A

text: d-by-m

text: nonnegative

text: matrix

text: A

text: is

text: said

text: to

text: be

text: separable

text: if

text: it

text: has

text: a

text: decomposition

text: of

text: the

text: form

text: R

text: R

text: A

text: F

text: W

text: for

text: F

text: d

text: r

text: and

text: W

text: (I,

text: K)Π

text: r

text: m

text: where

text: I

text: is

text: an

text: r-by-r

text: identity

text: matrix,

text: K

text: is

text: an

text: r-by-(m

text: r)

text: nonnegative

text: matrix,

text: and

text: Π

text: is

text: an

text: m-by-m

text: permutation

text: matrix.

text: Here,

text: we

text: call

text: F

text: the

text: basis

text: matrix

text: of

text: A

text: and

text: r

text: the

text: factorization

text: rank.

text: The

text: separable

text: nonnegative

text: matrix

text: factorization

text: problem

text: is

text: stated

text: as

text: follows.

text: (Separable

text: NMF

text: Problem)

text: Let

text: A

text: be

text: of

text: the

text: form

text: given

text: in

text: Find

text: an

text: index

text: set

text: I

text: I

text: with

text: r

text: elements

text: such

text: that

text: A(

text: coincides

text: with

text: the

text: basis

text: matrix

text: F

text: Department

text: of

text: Industrial

text: Engineering

text: and

text: Management,

text: Tokyo

text: Institute

text: of

text: Technology,

text: Ookayama,

text: Meguro,

text: Tokyo,

text: Japan.

text: mizutani.t.ab@m.titech.ac.jp

text: ##LTLine##


text: | − | ≤ (cid:107) (cid:107) Lemma 4 (Lemma 7(c) of [11]). σj((cid:98)F ) σj(G) 2 N 2 for each (cid:98)j = 1, . . . , r. (cid:98) (cid:98) (cid:62) Proof. We see from (12) that G and F has the relation (G; 0) = U F . Since U is an orthogonal matrix, the singular values of G coincide with those of F . Also, F is of the form (9). Thus, from | − | | − | ≤ (cid:107) (cid:107) ≤ (cid:107) (cid:107) Lemma 1, we have σ (F ) σ (G) = σ (F ) σ (F + N(1)) N(1) 2 N . The last j j j j 2 2 (cid:101) (cid:101) inequality follows from Lemma 2. (cid:101) Lemma 5. Let a and p be the column vectors of A and P . Also, let f and g be those of F (cid:107)k − k(cid:107) ≤ (cid:107) − (cid:107) (cid:107) (cid:107) j { j } and G. We have ak fj 2 pk (cid:101)gj 2 + 3 N 2 for any k and any j in 1, . . . , r . (cid:18) (cid:19) Proof. We see from (7) and (6) that a and p are related as k (cid:101)k (cid:101) (cid:101) (cid:101) p (cid:62) k = U ar and a = ar + ar,c (cid:101) (cid:101) 0 (cid:101)k (cid:101)k k k (cid:18) (cid:19) where ark and ark,c are the column vectors of A(cid:98)r and (cid:98)Ar,c. Also, from (12) and (9), gj and fj are related as g (cid:62) (cid:98) 0j = U f(cid:98)j and fj = fj + n(j1) where f and n(1) are the column vectors of F and N(1). Therefore, j j (cid:101) (cid:101) (cid:101) (cid:98) (cid:107)a − f (cid:107) = (cid:107)(ar +(cid:101) ar,c)(cid:98)− (f − n(cid:101)(1))(cid:107) k j 2 k k j j 2 = (cid:107)U(cid:62)(a(cid:101)r − f(cid:98) ) + U(cid:62)(ar,(cid:101)c + n(1))(cid:107) k j k j 2 ≤ (cid:107)U(cid:62)(ar − f )(cid:107)(cid:101) + (cid:107)U(cid:62)ar,c(cid:107) + (cid:107)U(cid:62)n(1)(cid:107) k j 2 k 2 j 2 (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) = p g + ar,c + n(1) . k j 2 k 2 j 2 (cid:101) U is a d-by-d orthogonal matrix in (7) that consists of the left singular vectors of A. By (cid:101) (cid:101) (cid:101) (cid:107) (cid:107) ≤ (cid:107) (cid:107) ≤ (cid:107) (cid:107) (1) (1) Le(cid:101)mma 2, we can put an upper bound on the norm of n such that n N 2 N . j (cid:107) (cid:107) ≤j (cid:107) 2 (cid:107) 2 (cid:107) (cid:107) ≤2 Also, we can put an upper bound on the norm of ar,c such that ar,c N due to ar,c (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:101) k k 2 2 k 2 Ar,c N . The last inequality is obtained in the same way as in the proof of Lemma 2. 2 2 (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) (cid:107) (cid:107) Therefore, we have a f p g + 3 N . k j 2 k j 2 2 We are now ready to prove Proposition 1. (Proof of Proposion 1). We show that Theorem 2 can apply to P in step 1 of Algorithm 2, if the (cid:107) (cid:107) amount of noise N is smaller than some level. P can be written as (11), and hence, is an 2 r-by-m near-separable matrix having an r-by-r basis matrix G. We choose some real number γ (cid:107) (cid:107) ≤ such that γ > 2. Suppose that N 1σ (F ). It follows from Lemma 4 and Assumption 1(a) ≥ − 2 γ min that σ (G) γ 2σ (F ) > 0. Therefore, Assumption 1 holds for G. This means that Theorem min γ min ∗ 2 can apply to P . Its application leads to the following statement. Let L be the optimal solution S S { } (cid:107) (cid:107) ≤ − of P( ) where =√ p , . . . , p for P . If s of S satisﬁes s (cid:15) for all i = 1, . . . , m r with ≤ 1 m i ∗ i 2 I (cid:15) O(σ (G)/r r), then, Algorithm 1 with ((L )1/2P , r) returns the output such that there min I (cid:107) − (cid:107) ≤ is an order of the elements in satisfying pI g O(κ(G)(cid:15)) for j = 1, . . . , r. (j) j 2 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: (Lemma

text: of

text: σj(G)

text: N

text: for

text: each

text: r.

text: Proof.

text: We

text: see

text: from

text: that

text: G

text: and

text: F

text: has

text: the

text: relation

text: (G;

text: U

text: F

text: Since

text: U

text: is

text: an

text: orthogonal

text: matrix,

text: the

text: singular

text: values

text: of

text: G

text: coincide

text: with

text: those

text: of

text: F

text: Also,

text: F

text: is

text: of

text: the

text: form

text: Thus,

text: from

text: Lemma

text: we

text: have

text: σ

text: (F

text: σ

text: (G)

text: σ

text: (F

text: σ

text: (F

text: N

text: The

text: last

text: j

text: j

text: j

text: j

text: inequality

text: follows

text: from

text: Lemma

text: Lemma

text: Let

text: a

text: and

text: p

text: be

text: the

text: column

text: vectors

text: of

text: A

text: and

text: P

text: Also,

text: let

text: f

text: and

text: g

text: be

text: those

text: of

text: F

text: j

text: j

text: and

text: G.

text: We

text: have

text: ak

text: fj

text: pk

text: N

text: for

text: any

text: k

text: and

text: any

text: j

text: in

text: r

text: Proof.

text: We

text: see

text: from

text: and

text: that

text: a

text: and

text: p

text: are

text: related

text: as

text: k

text: p

text: k

text: U

text: ar

text: and

text: a

text: ar

text: ar,c

text: k

text: k

text: where

text: ark

text: and

text: ark,c

text: are

text: the

text: column

text: vectors

text: of

text: and

text: Also,

text: from

text: and

text: gj

text: and

text: fj

text: are

text: related

text: as

text: g

text: U

text: and

text: fj

text: fj

text: where

text: f

text: and

text: are

text: the

text: column

text: vectors

text: of

text: F

text: and

text: Therefore,

text: j

text: j

text: f

text: (f

text: k

text: j

text: k

text: k

text: j

text: j

text: k

text: j

text: k

text: j

text: f

text: k

text: j

text: k

text: j

text: p

text: g

text: ar,c

text: k

text: j

text: k

text: j

text: U

text: is

text: a

text: d-by-d

text: orthogonal

text: matrix

text: in

text: that

text: consists

text: of

text: the

text: left

text: singular

text: vectors

text: of

text: A.

text: By

text: we

text: can

text: put

text: an

text: upper

text: bound

text: on

text: the

text: norm

text: of

text: n

text: such

text: that

text: n

text: N

text: N

text: j

text: ≤j

text: ≤2

text: Also,

text: we

text: can

text: put

text: an

text: upper

text: bound

text: on

text: the

text: norm

text: of

text: ar,c

text: such

text: that

text: ar,c

text: N

text: due

text: to

text: ar,c

text: k

text: k

text: k

text: Ar,c

text: N

text: The

text: last

text: inequality

text: is

text: obtained

text: in

text: the

text: same

text: way

text: as

text: in

text: the

text: proof

text: of

text: Lemma

text: Therefore,

text: we

text: have

text: a

text: f

text: p

text: g

text: N

text: k

text: j

text: k

text: j

text: We

text: are

text: now

text: ready

text: to

text: prove

text: Proposition

text: (Proof

text: of

text: Proposion

text: We

text: show

text: that

text: Theorem

text: can

text: apply

text: to

text: P

text: in

text: step

text: of

text: Algorithm

text: if

text: the

text: amount

text: of

text: noise

text: N

text: is

text: smaller

text: than

text: some

text: level.

text: P

text: can

text: be

text: written

text: as

text: and

text: hence,

text: is

text: an

text: r-by-m

text: near-separable

text: matrix

text: having

text: an

text: r-by-r

text: basis

text: matrix

text: G.

text: We

text: choose

text: some

text: real

text: number

text: γ

text: such

text: that

text: γ

text: Suppose

text: that

text: N

text: 1σ

text: (F

text: It

text: follows

text: from

text: Lemma

text: and

text: Assumption

text: γ

text: min

text: that

text: σ

text: (G)

text: γ

text: 2σ

text: (F

text: Therefore,

text: Assumption

text: holds

text: for

text: G.

text: This

text: means

text: that

text: Theorem

text: min

text: γ

text: min

text: can

text: apply

text: to

text: P

text: Its

text: application

text: leads

text: to

text: the

text: following

text: statement.

text: Let

text: L

text: be

text: the

text: optimal

text: solution

text: S

text: S

text: of

text: P(

text: where

text: p

text: p

text: for

text: P

text: If

text: s

text: of

text: S

text: satisﬁes

text: s

text: for

text: all

text: i

text: m

text: r

text: with

text: m

text: i

text: i

text: I

text: O(σ

text: (G)/r

text: r),

text: then,

text: Algorithm

text: with

text: ((L

text: r)

text: returns

text: the

text: output

text: such

text: that

text: there

text: min

text: I

text: is

text: an

text: order

text: of

text: the

text: elements

text: in

text: satisfying

text: pI

text: g

text: for

text: j

text: r.

text: (j)

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≤ The ﬁrst constraints hold if θ 1. For the second constraints, we have (cid:62) − (cid:62) − (cid:107) − (cid:107) (Gk + s ) (G 1) IG 1(Gk + s ) = (cid:18)k + G 1s 2(cid:19) i i i i i i 2 ≤ (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107) ( k + G 1 s )2 i 2 2 i 2 ≤ (cid:18) √4(cid:19) 2 1 + − √ α r 2 α√r + 2 2 = − . α r 2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:107) (cid:107) ≤ The second inequality follows from√Lemma 6 and also √k 1 due to Assumption 1(b). Thus, ≤ − i− 2 th√e second constraints hold if θ α√r 2 2. Let θ = α√r 2 2. It satisﬁes 0 < θ < 1 because of α r+2 α r+2 S ∗ α r > 2. Thus, θI is a feasible solution of Q( ). Accordingly, for the optimal solution C and (cid:32) (cid:33) the feasible solution θI, we have √ − 2r ∗ ≥ α√r 2 det(C ) det(θI) = θr = . α r + 2 Lemma 8 corresponds to Lemmas 2.5 and 2.6 of [8]. The lemmas of that paper need to put ∗ a condition on the amount of noise in order to derive of the upper bound on det(C ), while this lemma does not need to do so. This comes from the diﬀerence in the structures of near-separable matrices. Our lemma handles a near-separable matrix of the form (11). It has a preferable structure wherein the noise matrix contains an r-by-r zero submatrix and this zero submatrix corresponds to a basis matrix. ≥ (cid:32) (cid:33) Lemma 8. Suppose that r 2. Then, − − r 1 ∗ ≤ r λ det(C ) − j λ j r 1 for each j = 1, . . . , r. ∗ ∗ Proof. We derive an upper bound on the sum of the eigenvalues of C that is equivalent to tr(C ). ∗ S Since C is feasible for Q( ), we have (cid:62) − (cid:62) ∗ − ≤ (cid:107) ∗ (cid:107) ≤ g (G 1) C G 1g 1, equivalently, (C )1/2e 2 1 for j = 1, . . . , r. j j j 2 Thus, · · · ∗ λ + + λ = tr(C ) 1 r (cid:107)(cid:88) ∗ (cid:107) = (C )1/2 2 F r (cid:107) ∗ (cid:107) ≤ = (C )1/2e 2 r. j 2 j=1 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: ﬁrst

text: constraints

text: hold

text: if

text: θ

text: For

text: the

text: second

text: constraints,

text: we

text: have

label: equation

text: (Gk

text: s

text: (G

text: IG

text: s

text: G

text: i

text: i

text: i

text: i

text: i

text: i

text: k

text: G

text: s

text: i

text: i

text: α

text: r

text: α√r

text: α

text: r

text: The

text: second

text: inequality

text: follows

text: from√Lemma

text: and

text: also

text: √k

text: due

text: to

text: Assumption

text: Thus,

text: i−

text: th√e

text: second

text: constraints

text: hold

text: if

text: θ

text: α√r

text: Let

text: θ

text: α√r

text: It

text: satisﬁes

text: θ

text: because

text: of

text: α

text: α

text: S

text: α

text: r

text: Thus,

text: θI

text: is

text: a

text: feasible

text: solution

text: of

text: Q(

text: Accordingly,

text: for

text: the

text: optimal

text: solution

text: C

text: and

text: the

text: feasible

text: solution

text: θI,

text: we

text: have

text: α√r

text: det(C

text: det(θI)

text: θr

text: α

text: r

text: Lemma

text: corresponds

text: to

text: Lemmas

text: and

text: of

text: The

text: lemmas

text: of

text: that

text: paper

text: need

text: to

text: put

text: a

text: condition

text: on

text: the

text: amount

text: of

text: noise

text: in

text: order

text: to

text: derive

text: of

text: the

text: upper

text: bound

text: on

text: det(C

text: while

text: this

text: lemma

text: does

text: not

text: need

text: to

text: do

text: so.

text: This

text: comes

text: from

text: the

text: diﬀerence

text: in

text: the

text: structures

text: of

text: near-separable

text: matrices.

text: Our

text: lemma

text: handles

text: a

text: near-separable

text: matrix

text: of

text: the

text: form

text: It

text: has

text: a

text: preferable

text: structure

text: wherein

text: the

text: noise

text: matrix

text: contains

text: an

text: r-by-r

text: zero

text: submatrix

text: and

text: this

text: zero

text: submatrix

text: corresponds

text: to

text: a

text: basis

text: matrix.

text: Lemma

text: Suppose

text: that

text: r

text: Then,

text: r

text: r

text: λ

text: det(C

text: j

text: λ

text: j

text: r

text: for

text: each

text: j

text: r.

text: Proof.

text: We

text: derive

text: an

text: upper

text: bound

text: on

text: the

text: sum

text: of

text: the

text: eigenvalues

text: of

text: C

text: that

text: is

text: equivalent

text: to

text: tr(C

text: S

text: Since

text: C

text: is

text: feasible

text: for

text: Q(

text: we

text: have

text: g

text: (G

text: C

text: G

text: equivalently,

text: (C

text: for

text: j

text: r.

text: j

text: j

text: j

text: Thus,

text: λ

text: λ

text: tr(C

text: r

text: (C

text: F

text: r

text: (C

text: r.

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Accordingly, P can be represented as P = (G, GK + S)Π (11) by using G ∈ Rr×r and S ∈ Rr×(m(cid:18)−r) su(cid:19)ch that (cid:18) (cid:19) (cid:98) (cid:99) (cid:62) G (cid:62) S U F = and U N = . (12) 0 0 ◦ P in the step 3 of Algorithm 2 is ◦ ◦ ◦ ◦ P = (G , G K + S )Π ◦ ∈ R × ◦ ∈ R × − by using G r r and S r (m r) such that ◦ ∗ ◦ ∗ G = (L )1/2G and S = (L )1/2S. ◦ ◦ ◦ Hence, P and P are near-separable matrices, and (G, GK)Π and (G , G K)Π correspond to ◦ ◦ the separable matrices. In particular, G and G are the basis matrices, and (0, S)Π and (0, S )Π ◦ are the noise matrices. It should be noted that P and P are r-by-m near-separable matrices and these have r-by-r basis matrices. 3.2 Proof of Proposition 1 Here, we prove several lemmas that will be necessary for the subsequent discussion. Similar statements have already been proven in [11]. More precisely, Lemmas 2, 3 and 4 correspond to (a), (b) and (c) of Lemma 7 of that paper, and we have included them here to make the discussion self-contained. After that, we prove Proposition 1. The proof is obtained from Theorem 2 together (cid:101) (cid:101) with the following lemmas. ∈ R × | − | ≤ (cid:107) (cid:107) Lemma 1. Let A = A + N d m. Then, σ (A) σ (A) N for each i = 1, . . . , t where i i 2 t = min(d, m). Proof. See Corollary 8.6.2 of [9]. (cid:107) (cid:107) ≤ (cid:107) (cid:107) Lemma 2 (Lemma 7(a)(cid:101)of [11]). N 2(cid:101)N . (cid:101) 2 2 (cid:101) (cid:101) (cid:101) (cid:107) (cid:107) (cid:107) − (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) Proof. N = N Ar,c N + Ar,c since N is of the form (8). Also, Ar,c = 2 2 2 2 ≤ (cid:107) (cid:107) 2 σ (Ar,c) = σ (A). From Lemma 1 and σ (A) = 0, we have σ (A) N . Thus, (cid:107)ma(cid:107)x ≤ (cid:107) (cid:107) r+1 r+1 r+1 2 N 2 N . 2 2 Lemma 3 (Lemma 7(b) of [11]). Let s be the column vector of S in (11). Suppose that K satisﬁes (cid:107) (cid:107) ≤ (cid:107) (cid:107)i − Assumption 1(b). Then, s 4 N for each i = 1, . . . , m r. i 2 2 (cid:98) (cid:98) (cid:62) − Proof. We see from (12) and (10) that (s ; 0) = U n and n = N(1)k + n(2). Here, k and i i i (cid:107) (cid:107) (cid:107)i − i i(cid:107) ≤ n(2) are the column vectors of K and N(2), respectively. Thus, s = N(1)k + n(2) (cid:107) i (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) ≤ i 2 i (cid:107)i (cid:107)2 ≤ N(1) k + n(2) . From Lemma 2 and k 1 due to Assumption 1(b), we have s (cid:107) (cid:107) 2 i 2 i 2 i 2 i 2 4 N . 2 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Accordingly,

text: P

text: can

text: be

text: represented

text: as

text: P

text: (G,

text: GK

text: S)Π

text: by

text: using

text: G

text: Rr×r

text: and

text: S

text: that

text: G

text: S

text: U

text: F

text: and

text: U

text: N

text: P

text: in

text: the

text: step

text: of

text: Algorithm

text: is

text: P

text: (G

text: G

text: K

text: S

text: )Π

text: R

text: R

text: by

text: using

text: G

text: r

text: r

text: and

text: S

text: r

text: (m

text: r)

text: such

text: that

text: G

text: (L

text: and

text: S

text: (L

text: Hence,

text: P

text: and

text: P

text: are

text: near-separable

text: matrices,

text: and

text: (G,

text: GK)Π

text: and

text: (G

text: G

text: K)Π

text: correspond

text: to

text: the

text: separable

text: matrices.

text: In

text: particular,

text: G

text: and

text: G

text: are

text: the

text: basis

text: matrices,

text: and

text: S)Π

text: and

text: S

text: )Π

text: are

text: the

text: noise

text: matrices.

text: It

text: should

text: be

text: noted

text: that

text: P

text: and

text: P

text: are

text: r-by-m

text: near-separable

text: matrices

text: and

text: these

text: have

text: r-by-r

text: basis

text: matrices.

text: Proof

text: of

text: Proposition

text: Here,

text: we

text: prove

text: several

text: lemmas

text: that

text: will

text: be

text: necessary

text: for

text: the

text: subsequent

text: discussion.

text: Similar

text: statements

text: have

text: already

text: been

text: proven

text: in

text: More

text: precisely,

text: Lemmas

text: and

text: correspond

text: to

text: (a),

text: (b)

text: and

text: (c)

text: of

text: Lemma

text: of

text: that

text: paper,

text: and

text: we

text: have

text: included

text: them

text: here

text: to

text: make

text: the

text: discussion

text: self-contained.

text: After

text: that,

text: we

text: prove

text: Proposition

text: The

text: proof

text: is

text: obtained

text: from

text: Theorem

text: together

text: with

text: the

text: following

text: lemmas.

text: R

text: Lemma

text: Let

text: A

text: A

text: N

text: d

text: m.

text: Then,

text: σ

text: (A)

text: σ

text: (A)

text: N

text: for

text: each

text: i

text: t

text: where

text: i

text: i

text: t

text: min(d,

text: m).

text: Proof.

text: See

text: Corollary

text: of

text: Lemma

text: (Lemma

text: N

text: Proof.

text: N

text: N

text: Ar,c

text: N

text: Ar,c

text: since

text: N

text: is

text: of

text: the

text: form

text: Also,

text: Ar,c

text: σ

text: (Ar,c)

text: σ

text: (A).

text: From

text: Lemma

text: and

text: σ

text: (A)

text: we

text: have

text: σ

text: (A)

text: N

text: Thus,

text: N

text: N

text: Lemma

text: (Lemma

text: of

text: Let

text: s

text: be

text: the

text: column

text: vector

text: of

text: S

text: in

text: Suppose

text: that

text: K

text: satisﬁes

text: Assumption

text: Then,

text: s

text: N

text: for

text: each

text: i

text: m

text: r.

text: i

text: Proof.

text: We

text: see

text: from

text: and

text: that

text: (s

text: U

text: n

text: and

text: n

text: Here,

text: k

text: and

text: i

text: i

text: i

text: i

text: are

text: the

text: column

text: vectors

text: of

text: K

text: and

text: respectively.

text: Thus,

text: s

text: i

text: i

text: i

text: k

text: From

text: Lemma

text: and

text: k

text: due

text: to

text: Assumption

text: we

text: have

text: s

text: i

text: i

text: i

text: i

text: N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 AHMET BEYAZ Deﬁnition 1.2. Two contact manifolds (N , ξ ) and (N , ξ ) are said to be con- 0 0 1 1 → tactomorphic if there is an orientation preserving diﬀeomorphism f : N N → 0 1 with df(ξ ) = ξ , where df : TN TN denotes the diﬀerential of f. If ξ = 0 1 0 1 ∗ i kernel(α ), i = 0, 1, this is equivalent to saying that α and f α determine the i 0 1 same hyperplane ﬁeld, and hence equivalent to the existence of a positive function → R ∗ g : N + such that f α = gα . 0 1 0 Two contact structures ξ and ξ on a smooth manifold N are said to be homo- 0 1 topy equivalent if their respective almost contact structures are homotopy equivalent. ∈ ξ and ξ are said to be isotopic if there is a smooth isotopy ψ (t [0, 1]) of N 0 1 ∈ ∗ t such that Tψ (ξ ) = ξ for each t [0, 1]. Equivalently, ψ α = λ α , where → R t 0 t t t t 0 λ : N + is a suitable smooth family of smooth functions. This is equivalent t → to existence of a contactomorphism f : (N, ξ ) (N, ξ ) which is isotopic to the 0 1 identity. If two contact structures ξ and ξ on N are isotopic then (N, ξ ) and (N, ξ ) 0 1 0 1 are contactomorphic. Homotopy equivalence is much weaker than the isotopy and there may be many nonisotopic contact structures in a homotopy type. A simply connected 5-manifold N admits an almost contact structure if and only if its integral Stiefel-Whitney class W vanishes. Homotopy classes of almost 3 contact structures are in one to one correspondence with integral lifts of w (TN). 2 The correspondence is given by associating to an almost contact structure its ﬁrst Chern class ([8] p368). Deﬁnition 1.3. A compact symplectic manifold (M, ω) is called a strong (sym- plectic) ﬁlling of (N, ξ) if ∂M = N and there is a Liouville vector ﬁeld Y deﬁned · | near ∂M, pointing outwards along ∂M, and satisfying ξ = kernel(ω(Y, ) ) (as TM cooriented contact structure). In this case we say that (N, ξ) is the convex (or more precisely: ω-convex) boundary of (M, ω). For contact manifolds of dimensions greater than three, M is a strong ﬁlling of | N if and only if ∂M = N as oriented manifolds and ω is in the conformal class of | ξ dα (Theorem 5.1.5 of [8]). The boundary of a strong ﬁlling is said to be of contact ξ type. Let (X, ω) be a symplectic 4-manifold and e be a second cohomology class of X. Let’s denote the 2-disk bundle over X with Euler class e by M . Let π be the e ∗ ⊕ projection map of the ﬁbration. Any symplectic form on M is locally π ω ω ∈ e a where ω is the symplectic structure on the ﬁber for a X. Any two such symplectic a forms agree on the zero section, therefore M has a symplectic structure which is e unique up to symplectomorphism by the symplectic neighborhood theorem ([10]). The next lemma is deduced from [3]. Lemma 1.4. Let [ω] be the cohomology class of the symplectic form. If [ω]e < 0 then the contact structure on the boundary is of contact type. 2. Nonsymplectomorphic Fillings of a Homotopy Type ∈ Assume that X is a closed, simply connected, smooth 4-manifold and e Z H2(X; ) is a primitive, characteristic class. If X is the total space of the S1- e × bundle over X with Euler class e, then X is diﬀeomorphic to # − S2 S3 e b2(X) 1 ([2]).

text: AHMET

text: BEYAZ

text: Deﬁnition

text: Two

text: contact

text: manifolds

text: (N

text: ξ

text: and

text: (N

text: ξ

text: are

text: said

text: to

text: be

text: con-

text: tactomorphic

text: if

text: there

text: is

text: an

text: orientation

text: preserving

text: diﬀeomorphism

text: f

text: N

text: N

text: with

text: df(ξ

text: ξ

text: where

text: df

text: TN

text: TN

text: denotes

text: the

text: diﬀerential

text: of

text: f.

text: If

text: ξ

text: i

text: kernel(α

text: i

text: this

text: is

text: equivalent

text: to

text: saying

text: that

text: α

text: and

text: f

text: α

text: determine

text: the

text: i

text: same

text: hyperplane

text: ﬁeld,

text: and

text: hence

text: equivalent

text: to

text: the

text: existence

text: of

text: a

text: positive

text: function

text: R

text: g

text: N

text: such

text: that

text: f

text: α

text: gα

text: Two

text: contact

text: structures

text: ξ

text: and

text: ξ

text: on

text: a

text: smooth

text: manifold

text: N

text: are

text: said

text: to

text: be

text: homo-

text: topy

text: equivalent

text: if

text: their

text: respective

text: almost

text: contact

text: structures

text: are

text: homotopy

text: equivalent.

text: ξ

text: and

text: ξ

text: are

text: said

text: to

text: be

text: isotopic

text: if

text: there

text: is

text: a

text: smooth

text: isotopy

text: ψ

text: (t

text: of

text: N

text: t

text: such

text: that

text: Tψ

text: (ξ

text: ξ

text: for

text: each

text: t

text: Equivalently,

text: ψ

text: α

text: λ

text: α

text: where

text: R

text: t

text: t

text: t

text: t

text: t

text: λ

text: N

text: is

text: a

text: suitable

text: smooth

text: family

text: of

text: smooth

text: functions.

text: This

text: is

text: equivalent

text: t

text: to

text: existence

text: of

text: a

text: contactomorphism

text: f

text: (N,

text: ξ

text: (N,

text: ξ

text: which

text: is

text: isotopic

text: to

text: the

text: identity.

text: If

text: two

text: contact

text: structures

text: ξ

text: and

text: ξ

text: on

text: N

text: are

text: isotopic

text: then

text: (N,

text: ξ

text: and

text: (N,

text: ξ

text: are

text: contactomorphic.

text: Homotopy

text: equivalence

text: is

text: much

text: weaker

text: than

text: the

text: isotopy

text: and

text: there

text: may

text: be

text: many

text: nonisotopic

text: contact

text: structures

text: in

text: a

text: homotopy

text: type.

text: A

text: simply

text: connected

text: N

text: admits

text: an

text: almost

text: contact

text: structure

text: if

text: and

text: only

text: if

text: its

text: integral

text: Stiefel-Whitney

text: class

text: W

text: vanishes.

text: Homotopy

text: classes

text: of

text: almost

text: contact

text: structures

text: are

text: in

text: one

text: to

text: one

text: correspondence

text: with

text: integral

text: lifts

text: of

text: w

text: (TN).

text: The

text: correspondence

text: is

text: given

text: by

text: associating

text: to

text: an

text: almost

text: contact

text: structure

text: its

text: ﬁrst

text: Chern

text: class

text: Deﬁnition

text: A

text: compact

text: symplectic

text: manifold

text: (M,

text: ω)

text: is

text: called

text: a

text: strong

text: (sym-

text: plectic)

text: ﬁlling

text: of

text: (N,

text: ξ)

text: if

text: ∂M

text: N

text: and

text: there

text: is

text: a

text: Liouville

text: vector

text: ﬁeld

text: Y

text: deﬁned

text: near

text: ∂M,

text: pointing

text: outwards

text: along

text: ∂M,

text: and

text: satisfying

text: ξ

text: kernel(ω(Y,

text: (as

text: TM

text: cooriented

text: contact

text: structure).

text: In

text: this

text: case

text: we

text: say

text: that

text: (N,

text: ξ)

text: is

text: the

text: convex

text: (or

text: more

text: precisely:

text: ω-convex)

text: boundary

text: of

text: (M,

text: ω).

text: For

text: contact

text: manifolds

text: of

text: dimensions

text: greater

text: than

text: three,

text: M

text: is

text: a

text: strong

text: ﬁlling

text: of

text: N

text: if

text: and

text: only

text: if

text: ∂M

text: N

text: as

text: oriented

text: manifolds

text: and

text: ω

text: is

text: in

text: the

text: conformal

text: class

text: of

text: ξ

text: dα

text: (Theorem

text: of

text: The

text: boundary

text: of

text: a

text: strong

text: ﬁlling

text: is

text: said

text: to

text: be

text: of

text: contact

text: ξ

text: type.

text: Let

text: (X,

text: ω)

text: be

text: a

text: symplectic

text: and

text: e

text: be

text: a

text: second

text: cohomology

text: class

text: of

text: X.

text: Let’s

text: denote

text: the

text: bundle

text: over

text: X

text: with

text: Euler

text: class

text: e

text: by

text: M

text: Let

text: π

text: be

text: the

text: e

text: projection

text: map

text: of

text: the

text: ﬁbration.

text: Any

text: symplectic

text: form

text: on

text: M

text: is

text: locally

text: π

text: ω

text: ω

text: e

text: a

text: where

text: ω

text: is

text: the

text: symplectic

text: structure

text: on

text: the

text: ﬁber

text: for

text: a

text: X.

text: Any

text: two

text: such

text: symplectic

text: a

text: forms

text: agree

text: on

text: the

text: zero

text: section,

text: therefore

text: M

text: has

text: a

text: symplectic

text: structure

text: which

text: is

text: e

text: unique

text: up

text: to

text: symplectomorphism

text: by

text: the

text: symplectic

text: neighborhood

text: theorem

text: The

text: next

text: lemma

text: is

text: deduced

text: from

text: Lemma

text: Let

text: [ω]

text: be

text: the

text: cohomology

text: class

text: of

text: the

text: symplectic

text: form.

text: If

text: [ω]e

text: then

text: the

text: contact

text: structure

text: on

text: the

text: boundary

text: is

text: of

text: contact

text: type.

text: Nonsymplectomorphic

text: Fillings

text: of

text: a

text: Homotopy

text: Type

text: Assume

text: that

text: X

text: is

text: a

text: closed,

text: simply

text: connected,

text: smooth

text: and

text: e

text: Z

text: is

text: a

text: primitive,

text: characteristic

text: class.

text: If

text: X

text: is

text: the

text: total

text: space

text: of

text: the

text: e

text: bundle

text: over

text: X

text: with

text: Euler

text: class

text: e,

text: then

text: X

text: is

text: diﬀeomorphic

text: to

text: e


text: ON FILLINGS OF HOMOTOPY EQUIVALENT CONTACT STRUCTURES 3 The pullback of the almost contact structure which is compatible with dα on the 5-manifold is the pullback of the symplectic form ω on the base 4-manifold. The next lemma is relating the ﬁrst Chern class of the contact structure on the boundary and the symplectic structure on the base 4-manifold. Lemma 2.1. The ﬁrst Chern class of a compatible almost contact structure is the pullback of the ﬁrst Chern class of the symplectic structure ω on X. × Theorem 2.2. In the homotopy equivalence class of contact structures on # S2 2 S3 with the ﬁrst Chern class equal to zero, there are contact structures which have nonsymplectomorphic strong ﬁllings. Proof. According to Fintushel and Stern ([5,6]), there are inﬁnitely many mutually C C nondiﬀeomorphic smooth manifolds which are homeomorphic to P2# P2, two 2 C C of which carry symplectic structures. Let (X , ω ) be P2# P2 and let (X , ω ) 0 0 2 1 1 C C be the symplectic 4-manifold which is homeomorphic to P2# P2, but not dif- 2 feomorphic to it as given in [6]. X is not minimal and X is minimal. Both of 0 1 the manifolds have just two Seiberg-Witten basic classes which are plus and minus − − the canonical class. The ﬁrst Chern class of X is 3H E E . On the other 0 1 2 hand the canonical class of X evaluates positive with the symplectic form, because 1 it is a surface of general type ([6] page 66). The ﬁrst Chern class of X is either − − − − − 1 3H E E or (3H E E ). 1 2− −1 −2 − ∈ Z Let e be c (X ) = (3H E E ) H2(X ; ) and e be c (X ) in Z0 1 0 1 2 0 1 1 1 H2(X ; ). For j = 0, 1, the boundary of M is the circle bundle over (X , ω ) 1 ej j j with Euler class e , and the smooth structures on the boundaries of M and M j × e1 e2 are diﬀeomorphic to # S2 S3 ([2]). Since X is simply connected, a part of the 2 j Gysin sequence for this circle bundle over X is as shown below. j (1) 0 → H0(X ; Z) ∪→ej H2(X ; Z) →π∗ H2(# − S2 × S3; Z) → 0 = H1(X ; Z) j j b2(Xj) 1 j ∪ The image of the the map e is generated by e that is by plus or minus c (X ). j j × 1 j By Lemma 2.1, the pullbacks of c (ξ ) and c (ξ ) on # S2 S3 are the ﬁrst Chern 1 0 1 1 2 classes of the respective symplectic structures on X and X . These classes are in ∗ 0 1 the kernel of π , therefore the ﬁrst Chern classes of the respective contact structures are zero. · The symplectic form ω on X couples negatively with e ([9]) and ω e is less 0 0 0 1 1 than zero by the discussion above. By Lemma 1.4 the boundaries are of contact × type. The boundaries are diﬀeomorphic to # S2 S3 and the ﬁrst Chern classes 2 of the corresponding contact structures are zero. It remains to show that the symplectic structures are diﬀerent. This is done by a count of J-holomorphic curves. For j = 0, 1 the inclusion of X into M induces Z Z j ∈ ej Z an injection of H (X ; ) into H (M ; ). Let E be the image of E H (X ; ) Z 2 j · 2 −ej 2 j in H (M ; ). Remember E E is 1 in X . Let J be a generic almost complex 2 ej j j structure which is compatible with the symplectic structure on M . Since E is the ej class of an exceptional sphere in X , E has an almost complex sphere representative 0 in X . The image of this sphere under the inclusion map is a J -holomorphic sphere 0 0 representative of E in M . Assume E has an J -holomorphic sphere M that ej 1 e1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: FILLINGS

text: OF

text: HOMOTOPY

text: EQUIVALENT

text: CONTACT

text: STRUCTURES

text: The

text: pullback

text: of

text: the

text: almost

text: contact

text: structure

text: which

text: is

text: compatible

text: with

text: dα

text: on

text: the

text: is

text: the

text: pullback

text: of

text: the

text: symplectic

text: form

text: ω

text: on

text: the

text: base

text: The

text: next

text: lemma

text: is

text: relating

text: the

text: ﬁrst

text: Chern

text: class

text: of

text: the

text: contact

text: structure

text: on

text: the

text: boundary

text: and

text: the

text: symplectic

text: structure

text: on

text: the

text: base

text: Lemma

text: The

text: ﬁrst

text: Chern

text: class

text: of

text: a

text: compatible

text: almost

text: contact

text: structure

text: is

text: the

text: pullback

text: of

text: the

text: ﬁrst

text: Chern

text: class

text: of

text: the

text: symplectic

text: structure

text: ω

text: on

text: X.

text: Theorem

text: In

text: the

text: homotopy

text: equivalence

text: class

text: of

text: contact

text: structures

text: on

text: with

text: the

text: ﬁrst

text: Chern

text: class

text: equal

text: to

text: zero,

text: there

text: are

text: contact

text: structures

text: which

text: have

text: nonsymplectomorphic

text: strong

text: ﬁllings.

text: Proof.

text: According

text: to

text: Fintushel

text: and

text: Stern

text: there

text: are

text: inﬁnitely

text: many

text: mutually

text: C

text: C

text: nondiﬀeomorphic

text: smooth

text: manifolds

text: which

text: are

text: homeomorphic

text: to

text: two

text: C

text: C

text: of

text: which

text: carry

text: symplectic

text: structures.

text: Let

text: (X

text: ω

text: be

text: and

text: let

text: (X

text: ω

text: C

text: C

text: be

text: the

text: symplectic

text: which

text: is

text: homeomorphic

text: to

text: but

text: not

text: dif-

text: feomorphic

text: to

text: it

text: as

text: given

text: in

text: X

text: is

text: not

text: minimal

text: and

text: X

text: is

text: minimal.

text: Both

text: of

text: the

text: manifolds

text: have

text: just

text: two

text: Seiberg-Witten

text: basic

text: classes

text: which

text: are

text: plus

text: and

text: minus

text: the

text: canonical

text: class.

text: The

text: ﬁrst

text: Chern

text: class

text: of

text: X

text: is

text: E

text: E

text: On

text: the

text: other

text: hand

text: the

text: canonical

text: class

text: of

text: X

text: evaluates

text: positive

text: with

text: the

text: symplectic

text: form,

text: because

text: it

text: is

text: a

text: surface

text: of

text: general

text: type

text: page

text: The

text: ﬁrst

text: Chern

text: class

text: of

text: X

text: is

text: either

text: E

text: E

text: or

text: E

text: E

text: 2−

text: −1

text: −2

text: Z

text: Let

text: e

text: be

text: c

text: (X

text: E

text: E

text: and

text: e

text: be

text: c

text: (X

text: in

text: For

text: j

text: the

text: boundary

text: of

text: M

text: is

text: the

text: circle

text: bundle

text: over

text: (X

text: ω

text: ej

text: j

text: j

text: with

text: Euler

text: class

text: e

text: and

text: the

text: smooth

text: structures

text: on

text: the

text: boundaries

text: of

text: M

text: and

text: M

text: j

text: are

text: diﬀeomorphic

text: to

text: Since

text: X

text: is

text: simply

text: connected,

text: a

text: part

text: of

text: the

text: j

text: Gysin

text: sequence

text: for

text: this

text: circle

text: bundle

text: over

text: X

text: is

text: as

text: shown

text: below.

text: j

text: Z)

text: ∪→ej

text: Z)

text: →π∗

text: Z)

text: Z)

text: j

text: j

text: j

text: The

text: image

text: of

text: the

text: the

text: map

text: e

text: is

text: generated

text: by

text: e

text: that

text: is

text: by

text: plus

text: or

text: minus

text: c

text: (X

text: j

text: j

text: j

text: By

text: Lemma

text: the

text: pullbacks

text: of

text: c

text: (ξ

text: and

text: c

text: (ξ

text: on

text: are

text: the

text: ﬁrst

text: Chern

text: classes

text: of

text: the

text: respective

text: symplectic

text: structures

text: on

text: X

text: and

text: X

text: These

text: classes

text: are

text: in

text: the

text: kernel

text: of

text: π

text: therefore

text: the

text: ﬁrst

text: Chern

text: classes

text: of

text: the

text: respective

text: contact

text: structures

text: are

text: zero.

text: The

text: symplectic

text: form

text: ω

text: on

text: X

text: couples

text: negatively

text: with

text: e

text: and

text: ω

text: e

text: is

text: less

text: than

text: zero

text: by

text: the

text: discussion

text: above.

text: By

text: Lemma

text: the

text: boundaries

text: are

text: of

text: contact

text: type.

text: The

text: boundaries

text: are

text: diﬀeomorphic

text: to

text: and

text: the

text: ﬁrst

text: Chern

text: classes

text: of

text: the

text: corresponding

text: contact

text: structures

text: are

text: zero.

text: It

text: remains

text: to

text: show

text: that

text: the

text: symplectic

text: structures

text: are

text: diﬀerent.

text: This

text: is

text: done

text: by

text: a

text: count

text: of

text: J-holomorphic

text: curves.

text: For

text: j

text: the

text: inclusion

text: of

text: X

text: into

text: M

text: induces

text: Z

text: Z

text: j

text: ej

text: Z

text: an

text: injection

text: of

text: H

text: (X

text: into

text: H

text: (M

text: Let

text: E

text: be

text: the

text: image

text: of

text: E

text: H

text: (X

text: Z

text: j

text: −ej

text: j

text: in

text: H

text: (M

text: Remember

text: E

text: E

text: is

text: in

text: X

text: Let

text: J

text: be

text: a

text: generic

text: almost

text: complex

text: ej

text: j

text: j

text: structure

text: which

text: is

text: compatible

text: with

text: the

text: symplectic

text: structure

text: on

text: M

text: Since

text: E

text: is

text: the

text: ej

text: class

text: of

text: an

text: exceptional

text: sphere

text: in

text: X

text: E

text: has

text: an

text: almost

text: complex

text: sphere

text: representative

text: in

text: X

text: The

text: image

text: of

text: this

text: sphere

text: under

text: the

text: inclusion

text: map

text: is

text: a

text: J

text: -holomorphic

text: sphere

text: representative

text: of

text: E

text: in

text: M

text: Assume

text: E

text: has

text: an

text: J

text: -holomorphic

text: sphere

text: M

text: that

text: ej

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:18) (cid:19) (cid:0) (cid:1) −1 − r < R : ∂ r∂ + i∂ /r 2 ψ = 0 (2) r r r θ in (cid:18) (cid:19) −1 − − r > R : ∂ r∂ + ( i∂ /r Φ/2πr) 2 ψ = 0 (3) r r r θ out The Pauli equation (1) around at r = R gives the jump conditions between ψ and ψ . In in out a small enough neighborhood of r = R we can keep only second derivatives in r and the Stern- − − Gerlach term, while neglecting all the other terms with respect to them, to get: ( ∂2 gB)ψ = 0. r 2 Integrating ∂2 in a small neighborhood around r = R, we get the jump condition of ψ between r the two domains, r < R and r > R : − | −g Φ (∂rψout ∂rψin) r=R = 2 2πRψ (4) In order to ﬁnd the wavefunction in the two domains, we will use the separation of variables: ψ = f (r) eiθm (m ∈ Z), and get:1. − ψ = C rmeimθ + D r meimθ in 0 0 −Φ − −Φ ψout = E0rm 2π eimθ + F0r (m 2π)eimθ From the restrictions of continuity, normalization and matching of the jump condition we must ≥ take D = F = 0 and m 0. The jump condition gives us C = E , which also give us continuity. 0 0 0 0 Note that the existence of the Aharonov and Casher zero mode seems to be very sensitive to the jump condition, determined by the value of g. The remaining wavefunction is normalizable if, and only if, ψ is normalizable. Therefore we require: out (cid:18) (cid:19) ˆ ˆ ∞ ∞ | | −Φ ∞ ψ 22πrdr (cid:108)= (cid:109) r(m 2π) 2r2πdr < R R − − or m < Φ 1. This gives us the Φ 1 diﬀerent solutions predicted by Aharonov and 2π 2π Casher. 2 Bounded states with correction to the gyromagnetic factor In this section we will ﬁnd the wavefunctions and binding energies for the case of g > 2 (the case of g < 2 does not yield a nor(cid:18)malizable wavefunct(cid:19)ions). Using the same separation of variables as before: ψ = f (r) eiθm, we get: −1 r < R : ∂ r∂ + m2/r2 f (r) = Ef (r) (cid:18) r r r (cid:19)in in −1 − r > R : ∂ r∂ + (m (Φ/2π)) 2/r2 f (r) = Ef (r) r r r out out 1For m = 0, the second term of ψin is D0 ln(r), and is still not continuous at 0. A similar correction should be noted for the solutions of ψout, where the solution containing a logarithmic function is rejected for not being normalizable. 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: −1

text: r

text: R

text: r∂

text: i∂

text: /r

text: ψ

text: r

text: r

text: r

text: θ

text: in

text: −1

text: r

text: R

text: r∂

text: i∂

text: /r

text: ψ

text: r

text: r

text: r

text: θ

text: out

text: The

text: Pauli

text: equation

text: around

text: at

text: r

text: R

text: gives

text: the

text: jump

text: conditions

text: between

text: ψ

text: and

text: ψ

text: In

text: in

text: out

text: a

text: small

text: enough

text: neighborhood

text: of

text: r

text: R

text: we

text: can

text: keep

text: only

text: second

text: derivatives

text: in

text: r

text: and

text: the

text: Stern-

text: Gerlach

text: term,

text: while

text: neglecting

text: all

text: the

text: other

text: terms

text: with

text: respect

text: to

text: them,

text: to

text: get:

text: ∂2

text: gB)ψ

text: r

text: Integrating

text: ∂2

text: in

text: a

text: small

text: neighborhood

text: around

text: r

text: R,

text: we

text: get

text: the

text: jump

text: condition

text: of

text: ψ

text: between

text: r

text: the

text: two

text: domains,

text: r

text: R

text: and

text: r

text: R

text: −g

text: Φ

text: (∂rψout

text: ∂rψin)

text: r=R

text: 2πRψ

text: In

text: order

text: to

text: ﬁnd

text: the

text: wavefunction

text: in

text: the

text: two

text: domains,

text: we

text: will

text: use

text: the

text: separation

text: of

text: variables:

text: ψ

text: f

text: (r)

text: eiθm

text: (m

text: Z),

text: and

text: ψ

text: C

text: rmeimθ

text: D

text: r

text: meimθ

text: in

text: −Φ

text: −Φ

text: ψout

text: 2π

text: eimθ

text: (m

text: From

text: the

text: restrictions

text: of

text: continuity,

text: normalization

text: and

text: matching

text: of

text: the

text: jump

text: condition

text: we

text: must

text: take

text: D

text: F

text: and

text: m

text: The

text: jump

text: condition

text: gives

text: us

text: C

text: E

text: which

text: also

text: give

text: us

text: continuity.

text: Note

text: that

text: the

text: existence

text: of

text: the

text: Aharonov

text: and

text: Casher

text: zero

text: mode

text: seems

text: to

text: be

text: very

text: sensitive

text: to

text: the

text: jump

text: condition,

text: determined

text: by

text: the

text: value

text: of

text: g.

text: The

text: remaining

text: wavefunction

text: is

text: normalizable

text: if,

text: and

text: only

text: if,

text: ψ

text: is

text: normalizable.

text: Therefore

text: we

text: require:

text: out

text: ˆ

text: ˆ

text: −Φ

text: ψ

text: r(m

text: R

text: R

text: or

text: m

text: Φ

text: This

text: gives

text: us

text: the

text: Φ

text: diﬀerent

text: solutions

text: predicted

text: by

text: Aharonov

text: and

text: 2π

text: 2π

text: Casher.

text: Bounded

text: states

text: with

text: correction

text: to

text: the

text: gyromagnetic

text: factor

text: In

text: this

text: section

text: we

text: will

text: ﬁnd

text: the

text: wavefunctions

text: and

text: binding

text: energies

text: for

text: the

text: case

text: of

text: g

text: (the

text: case

text: of

text: g

text: does

text: not

text: yield

text: a

text: Using

text: the

text: same

text: separation

text: of

text: variables

text: as

text: before:

text: ψ

text: f

text: (r)

text: eiθm,

text: we

text: get:

text: −1

text: r

text: R

text: r∂

text: f

text: (r)

text: Ef

text: (r)

text: r

text: r

text: r

text: in

text: −1

text: r

text: R

text: r∂

text: (m

text: f

text: (r)

text: Ef

text: (r)

text: r

text: r

text: r

text: out

text: out

text: m

text: the

text: second

text: term

text: of

text: ψin

text: is

text: ln(r),

text: and

text: is

text: still

text: not

text: continuous

text: at

text: A

text: similar

text: correction

text: should

text: be

text: noted

text: for

text: the

text: solutions

text: of

text: ψout,

text: where

text: the

text: solution

text: containing

text: a

text: logarithmic

text: function

text: is

text: rejected

text: for

text: not

text: being

text: normalizable.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Looking at E < 0, we express f (r) as a linear combination of the modiﬁed Bessel functions of the ﬁrst and second kind (I K ): (cid:16) (cid:17) (cid:16) (cid:17) n, n √ √ (cid:16) (cid:17) (cid:16) (cid:17) − − f (r) = C I Er + C K Er in 1 m √ 2 m √ − − f(r) = C I − Er + C K − Er out 3 (m (Φ/2π)) 4 (m (Φ/2π)) For normalization and continuity at 0, we must cho(cid:0)ose C2 =(cid:1) C3 = 0. From continuity at r = R we get: (cid:0) √ (cid:1) − C K − √ ER 1 = (m (Φ/2π)−) C I ER 4 m The binding energyE depends on the following parameters: n, Φ, g . Naturally, we are interested in the function E(g), for a value of g slightly above 2. It is technically simpler to examine g (m, Φ, E), which we get explicitly from jump condition (4): − − | 4πR g(E) = (∂rψout ∂rψin) r=R Φ /ψ(R) (cid:32) (cid:0) (cid:1) (cid:0) (cid:1) (cid:33) or: (cid:0)√ (cid:1) (cid:0)√ (cid:1) − | − | ∂rIm √ Er r=R − ∂rK(m−(Φ/2π)) √ Er r=R 4πR g(E) = − − I ER K − ER Φ m (m (Φ/2π)) For m = 0, Expanding to ﬁrst order in E we get: − R2E g(E) = 2 − + O(E 2Φπ ) 2( Φ 1) (cid:16) (cid:17) 2π or: − · − −(g 2) ( Φ 1) − E(g) = 2π + O (g 2)2Φπ R2 As an example, we can plot the electron’s energy as function of the ﬂux (noted as phi in the graph), for the gyromagnetic ratio of an isolated electron given by QED and no orbital angular (cid:126) · − momentum. We plot it for R = 1/α2 (or R = = 3.626 10 9meter) 2. Φ is shown in units of Φ (or φ = h = 4.135 · 10−15Wb ) and E2misecoαf2 order of magnitude of m c2α5 and shown in 2π unit e e eV . 2See section 3.1 for the reason we chose this radius 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Looking

text: at

text: E

text: we

text: express

text: f

text: (r)

text: as

text: a

text: linear

text: combination

text: of

text: the

text: modiﬁed

text: Bessel

text: functions

text: of

text: the

text: ﬁrst

text: and

text: second

text: kind

text: (I

text: K

text: n,

text: n

text: f

text: (r)

text: C

text: I

text: Er

text: C

text: K

text: Er

text: in

text: m

text: m

text: f(r)

text: C

text: I

text: Er

text: C

text: K

text: Er

text: out

text: (m

text: (m

text: For

text: normalization

text: and

text: continuity

text: at

text: we

text: must

text: From

text: continuity

text: at

text: r

text: R

text: we

text: get:

text: C

text: K

text: ER

text: (m

text: C

text: I

text: ER

text: m

text: The

text: binding

text: energyE

text: depends

text: on

text: the

text: following

text: parameters:

text: n,

text: Φ,

text: g

text: Naturally,

text: we

text: are

text: interested

text: in

text: the

text: function

text: E(g),

text: for

text: a

text: value

text: of

text: g

text: slightly

text: above

text: It

text: is

text: technically

text: simpler

text: to

text: examine

text: g

text: (m,

text: Φ,

text: E),

text: which

text: we

text: get

text: explicitly

text: from

text: jump

text: condition

text: g(E)

text: (∂rψout

text: ∂rψin)

text: r=R

text: Φ

text: /ψ(R)

text: or:

text: ∂rIm

text: Er

text: r=R

text: Er

text: r=R

text: g(E)

text: I

text: ER

text: K

text: ER

text: Φ

text: m

text: (m

text: For

text: m

text: Expanding

text: to

text: ﬁrst

text: order

text: in

text: E

text: we

text: get:

text: g(E)

text: O(E

text: 2Φπ

text: Φ

text: 2π

text: or:

text: −(g

text: Φ

text: E(g)

text: 2π

text: O

text: (g

text: As

text: an

text: example,

text: we

text: can

text: plot

text: the

text: electron’s

text: energy

text: as

text: function

text: of

text: the

text: ﬂux

text: (noted

text: as

text: phi

text: in

text: the

text: graph),

text: for

text: the

text: gyromagnetic

text: ratio

text: of

text: an

text: isolated

text: electron

text: given

text: by

text: QED

text: and

text: no

text: orbital

text: angular

text: momentum.

text: We

text: plot

text: it

text: for

text: R

text: (or

text: R

text: Φ

text: is

text: shown

text: in

text: units

text: of

text: Φ

text: (or

text: φ

text: h

text: and

text: order

text: of

text: magnitude

text: of

text: m

text: and

text: shown

text: in

text: 2π

text: unit

text: e

text: e

text: eV

text: section

text: for

text: the

text: reason

text: we

text: chose

text: this

text: radius

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Simple Scalings for Various Regimes of Electron Acceleration in Surface Plasma Waves C. Riconda,1 M. Raynaud,2 T. Vialis,1 and M. Grech3 1LULI, Sorbonne Universit´e, Universit´e Pierre et Marie Curie, Ecole Polytechnique, CNRS UMR 7605, CEA, Paris, 75005 France 2Laboratoire des Solides Irradi´es, CNRS UMR 7642, CEA-DSM-IRAMIS, Ecole Polytechnique, Universit´e Paris-Saclay, 91128 Palaiseau, France 3LULI, CNRS UMR 7605, Universit´e Pierre et Marie Curie, Ecole Polytechnique, CEA, 91128 Palaiseau, France (Dated: May 27, 2020) Abstract Diﬀerent electron acceleration regimes in the evanescent ﬁeld of a surface plasma wave are studied by considering the interaction of a test electron with the high-frequency electromagnetic ﬁeld of a surface wave. The non-relativistic and relativistic limits are investigated. Simple scalings are found demonstrating the possibility to achieve an eﬃcient conversion of the surface wave ﬁeld energy into electron kinetic energy. This mechanism of electron acceleration can provide a high-frequency pulsed source of relativistic electrons with a well deﬁned energy. In the relativistic limit, the most energetic electrons are obtained in the so-called electromagnetic regime for surface waves. In this regime the particles are accelerated to velocities larger than the wave phase velocity, mainly in the direction parallel to the plasma-vacuum interface. PACS numbers: 52.65.Cc,73.20.Mf,52.35.Mw 1

text: Simple

text: Scalings

text: for

text: Various

text: Regimes

text: of

text: Electron

text: Acceleration

text: in

text: Surface

text: Plasma

text: Waves

text: C.

text: M.

text: T.

text: and

text: M.

text: Sorbonne

text: Universit´e,

text: Universit´e

text: Pierre

text: et

text: Marie

text: Curie,

text: Ecole

text: Polytechnique,

text: CNRS

text: UMR

text: CEA,

text: Paris,

text: France

text: des

text: Solides

text: Irradi´es,

text: CNRS

text: UMR

text: CEA-DSM-IRAMIS,

text: Ecole

text: Polytechnique,

text: Universit´e

text: Paris-Saclay,

text: Palaiseau,

text: France

text: CNRS

text: UMR

text: Universit´e

text: Pierre

text: et

text: Marie

text: Curie,

text: Ecole

text: Polytechnique,

text: CEA,

text: Palaiseau,

text: France

text: (Dated:

text: May

text: Abstract

text: Diﬀerent

text: electron

text: acceleration

text: regimes

text: in

text: the

text: evanescent

text: ﬁeld

text: of

text: a

text: surface

text: plasma

text: wave

text: are

text: studied

text: by

text: considering

text: the

text: interaction

text: of

text: a

text: test

text: electron

text: with

text: the

text: high-frequency

text: electromagnetic

text: ﬁeld

text: of

text: a

text: surface

text: wave.

text: The

text: non-relativistic

text: and

text: relativistic

text: limits

text: are

text: investigated.

text: Simple

text: scalings

text: are

text: found

text: demonstrating

text: the

text: possibility

text: to

text: achieve

text: an

text: eﬃcient

text: conversion

text: of

text: the

text: surface

text: wave

text: ﬁeld

text: energy

text: into

text: electron

text: kinetic

text: energy.

text: This

text: mechanism

text: of

text: electron

text: acceleration

text: can

text: provide

text: a

text: high-frequency

text: pulsed

text: source

text: of

text: relativistic

text: electrons

text: with

text: a

text: well

text: deﬁned

text: energy.

text: In

text: the

text: relativistic

text: limit,

text: the

text: most

text: energetic

text: electrons

text: are

text: obtained

text: in

text: the

text: so-called

text: electromagnetic

text: regime

text: for

text: surface

text: waves.

text: In

text: this

text: regime

text: the

text: particles

text: are

text: accelerated

text: to

text: velocities

text: larger

text: than

text: the

text: wave

text: phase

text: velocity,

text: mainly

text: in

text: the

text: direction

text: parallel

text: to

text: the

text: plasma-vacuum

text: interface.

text: PACS

text: numbers:


text: INTRODUCTION Electron acceleration by laser-plasma interaction has been studied extensively within the context of laser absorption by a plasma. It has also been studied for the development of techniques aimed at producing hot electrons in order to obtain improved energetic electron sources and to enhance secondary processes including X-ray, γ-ray, positron production and ion acceleration [1]. In under-dense plasmas, various methods of wakeﬁeld acceleration [2–4] ∼ have been proposed and electrons up to 4.2 GeV have been experimentally observed in the optimal conﬁguration [5]. However, the electron acceleration mechanism invoked involves high laser intensity, well into the relativistic regime, and short pulse duration τ = 40 fs L ∼ resulting in a small current (total current of 10pC). It is thus of interest to search for alternative conﬁgurations wherein the current can be enhanced. In dense plasmas, an energetic electron population may be created by resonant ab- sorption [6]. In this case, the electrons are accelerated by the resonant plasma waves excited at densities around the critical density by the interaction of a “long” laser pulse with a gentle-gradient dense plasma. With the development of lasers of high intensity − ∼ (Iλ2 > 1016W cm 2µm2), short pulse duration (< 100fs), and high contrast ( 1012), new 0 electron heating mechanisms have also been proposed [7–11]. They involve very sharp den- sity gradients and over-dense plasmas which do not necessarily require a resonant plasma response. Among these mechanisms are vacuum heating [7], and the so-called ponderomotive × (cid:126) (cid:126) or J B heating [8]. Numerical simulations [12] have shown that, in many cases, vacuum heating can be more eﬃcient than resonant absorption. The advantage of these mechanisms is that they involve dense plasmas such that they bear the potential to generate very high currents. In order to improve the electron acceleration to relativistic values by a laser interacting with an over-dense plasma, particular targets designs can be used [13, 14]. Also surface electron plasma waves resonantly excited by the laser on a structured target can be used to reach that goal [15, 16]. The electrons interact in this case with the surface plasma wave ﬁeld that is highly localized at the vacuum-plasma interface and oscillates with the laser frequency. The local ﬁeld amplitude is higher than that of the incoming laser such that the typical electron quiver motion is much faster than the electron thermal velocity. Experimental evidence for the feasibility to accelerate electrons and ions by this kind of 2

text: INTRODUCTION

text: Electron

text: acceleration

text: by

text: laser-plasma

text: interaction

text: has

text: been

text: studied

text: extensively

text: within

text: the

text: context

text: of

text: laser

text: absorption

text: by

text: a

text: plasma.

text: It

text: has

text: also

text: been

text: studied

text: for

text: the

text: development

text: of

text: techniques

text: aimed

text: at

text: producing

text: hot

text: electrons

text: in

text: order

text: to

text: obtain

text: improved

text: energetic

text: electron

text: sources

text: and

text: to

text: enhance

text: secondary

text: processes

text: including

text: X-ray,

text: γ-ray,

text: positron

text: production

text: and

text: ion

text: acceleration

text: In

text: under-dense

text: plasmas,

text: various

text: methods

text: of

text: wakeﬁeld

text: acceleration

text: have

text: been

text: proposed

text: and

text: electrons

text: up

text: to

text: GeV

text: have

text: been

text: experimentally

text: observed

text: in

text: the

text: optimal

text: conﬁguration

text: However,

text: the

text: electron

text: acceleration

text: mechanism

text: invoked

text: involves

text: high

text: laser

text: intensity,

text: well

text: into

text: the

text: relativistic

text: regime,

text: and

text: short

text: pulse

text: duration

text: τ

text: fs

text: L

text: resulting

text: in

text: a

text: small

text: current

text: (total

text: current

text: of

text: It

text: is

text: thus

text: of

text: interest

text: to

text: search

text: for

text: alternative

text: conﬁgurations

text: wherein

text: the

text: current

text: can

text: be

text: enhanced.

text: In

text: dense

text: plasmas,

text: an

text: energetic

text: electron

text: population

text: may

text: be

text: created

text: by

text: resonant

text: ab-

text: sorption

text: In

text: this

text: case,

text: the

text: electrons

text: are

text: accelerated

text: by

text: the

text: resonant

text: plasma

text: waves

text: excited

text: at

text: densities

text: around

text: the

text: critical

text: density

text: by

text: the

text: interaction

text: of

text: a

text: “long”

text: laser

text: pulse

text: with

text: a

text: gentle-gradient

text: dense

text: plasma.

text: With

text: the

text: development

text: of

text: lasers

text: of

text: high

text: intensity

text: cm

text: short

text: pulse

text: duration

text: and

text: high

text: contrast

text: new

text: electron

text: heating

text: mechanisms

text: have

text: also

text: been

text: proposed

text: They

text: involve

text: very

text: sharp

text: den-

text: sity

text: gradients

text: and

text: over-dense

text: plasmas

text: which

text: do

text: not

text: necessarily

text: require

text: a

text: resonant

text: plasma

text: response.

text: Among

text: these

text: mechanisms

text: are

text: vacuum

text: heating

text: and

text: the

text: so-called

text: ponderomotive

text: or

text: J

text: B

text: heating

text: Numerical

text: simulations

text: have

text: shown

text: that,

text: in

text: many

text: cases,

text: vacuum

text: heating

text: can

text: be

text: more

text: eﬃcient

text: than

text: resonant

text: absorption.

text: The

text: advantage

text: of

text: these

text: mechanisms

text: is

text: that

text: they

text: involve

text: dense

text: plasmas

text: such

text: that

text: they

text: bear

text: the

text: potential

text: to

text: generate

text: very

text: high

text: currents.

text: In

text: order

text: to

text: improve

text: the

text: electron

text: acceleration

text: to

text: relativistic

text: values

text: by

text: a

text: laser

text: interacting

text: with

text: an

text: over-dense

text: plasma,

text: particular

text: targets

text: designs

text: can

text: be

text: used

text: Also

text: surface

text: electron

text: plasma

text: waves

text: resonantly

text: excited

text: by

text: the

text: laser

text: on

text: a

text: structured

text: target

text: can

text: be

text: used

text: to

text: reach

text: that

text: goal

text: The

text: electrons

text: interact

text: in

text: this

text: case

text: with

text: the

text: surface

text: plasma

text: wave

text: ﬁeld

text: that

text: is

text: highly

text: localized

text: at

text: the

text: vacuum-plasma

text: interface

text: and

text: oscillates

text: with

text: the

text: laser

text: frequency.

text: The

text: local

text: ﬁeld

text: amplitude

text: is

text: higher

text: than

text: that

text: of

text: the

text: incoming

text: laser

text: such

text: that

text: the

text: typical

text: electron

text: quiver

text: motion

text: is

text: much

text: faster

text: than

text: the

text: electron

text: thermal

text: velocity.

text: Experimental

text: evidence

text: for

text: the

text: feasibility

text: to

text: accelerate

text: electrons

text: and

text: ions

text: by

text: this

text: kind

text: of