text: 6 A. AKHMEDOV AND W. ZHANG are only ﬁnitely many such surfaces up to diﬀeomorphism. In particular, the number of possible fundamental groups is ﬁnite. For detailed discussion on these surfaces, especially regarding the possible (algebraic) fundamental groups, see the beautiful survey paper [13]. One of − the outstanding problems asks for the greatest positive number a(= 9 − b ) such that c2 ≤ a implies the fundamental group to be ﬁnite, and the smallest 1 positive number b such that c2 ≥ b implies the fundamental group to be 1 inﬁnite. Currently, we know and . b ≥ 7 a ≤ 3 There is a remarkable corollary of Yau’s solution of Calabi conjecture [47]. Theorem 2.4. Let S be an algebraic surface of general type. Then K2 = S if and only if the universal covering of is the complex ball. 9χ (S) S h Especially, when p = 0, c2 = 9 =⇒ |π | = ∞. Recently, Prasad and Ye- g 1 1 ung (and Cartwright-Steger) give a complete classiﬁcation of these so called fake projective planes. It is known that there are exactly 100 up to biholo- morphism [38, 14]. Let us now take a look at the possible list of symplectic -manifolds with 4 b+ = 1. We discuss it using the notion of symplectic Kodaira dimension. For κs = −∞, as we mentioned, there are nothing more than rational and ruled surfaces. For κs = 0, although there are no classiﬁcation at this moment, in addition to hyperelliptic surfaces, we know that there are certain (non-complex) T2 bundles over T2 with b = 2. All of them are symplectic [22]. They could 1 have diﬀerent fundamental groups from hyperelliptic surfaces since Nil4, Nil3 × E1, Sol3 × E1 and E4 are all the possible geometric types and on the other hand all hyperelliptic surfaces have geometric type E4. Let us digress on the deﬁnition of a geometry structure (in the sense of Thurston). A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group on with compact sta- G X bilizers. A model geometry is called maximal if is maximal among groups G acting smoothly and transitively on with compact stabilizers. Sometimes X this condition is included in the deﬁnition of a model geometry. A geometric structure on a manifold is a diﬀeomorphism from to for some M M X/Γ model geometry , where is a discrete subgroup of acting freely on . X Γ G X If a given manifold admits a geometric structure, then it admits one whose model is maximal. In other word, diﬀerent geometries are distinguished by their fundamental groups. It is remarkable that T.-J. Li [32] proved that all symplectic manifolds with κs = 0, SCY surfaces, have the same rational homology types as K3 surfaces, Enrique surfaces or T2 bundles over T2. Moreover, the paper of S. Friedl and S. Vidussi discussed the restrictions on the fundamental groups of SCY -manifolds [20]. 4

text: A.

text: AKHMEDOV

text: AND

text: W.

text: ZHANG

text: are

text: only

text: ﬁnitely

text: many

text: such

text: surfaces

text: up

text: to

text: diﬀeomorphism.

text: In

text: particular,

text: the

text: number

text: of

text: possible

text: fundamental

text: groups

text: is

text: ﬁnite.

text: For

text: detailed

text: discussion

text: on

text: these

text: surfaces,

text: especially

text: regarding

text: the

text: possible

text: (algebraic)

text: fundamental

text: groups,

text: see

text: the

text: beautiful

text: survey

text: paper

text: One

text: of

text: the

text: outstanding

text: problems

text: asks

text: for

text: the

text: greatest

text: positive

text: number

text: a(=

text: b

text: such

text: that

text: a

text: implies

text: the

text: fundamental

text: group

text: to

text: be

text: ﬁnite,

text: and

text: the

text: smallest

text: positive

text: number

text: b

text: such

text: that

text: b

text: implies

text: the

text: fundamental

text: group

text: to

text: be

text: inﬁnite.

text: Currently,

text: we

text: know

text: and

text: b

text: a

text: There

text: is

text: a

text: remarkable

text: corollary

text: of

text: Yau’s

text: solution

text: of

text: Calabi

text: conjecture

text: Theorem

text: Let

text: S

text: be

text: an

text: algebraic

text: surface

text: of

text: general

text: type.

text: Then

text: S

text: if

text: and

text: only

text: if

text: the

text: universal

text: covering

text: of

text: is

text: the

text: complex

text: ball.

text: 9χ

text: (S)

text: S

text: h

text: Especially,

text: when

text: p

text: |π

text: Recently,

text: Prasad

text: and

text: Ye-

text: g

text: ung

text: (and

text: Cartwright-Steger)

text: give

text: a

text: complete

text: classiﬁcation

text: of

text: these

text: so

text: called

text: fake

text: projective

text: planes.

text: It

text: is

text: known

text: that

text: there

text: are

text: exactly

text: up

text: to

text: biholo-

text: morphism

text: Let

text: us

text: now

text: take

text: a

text: look

text: at

text: the

text: possible

text: list

text: of

text: symplectic

text: -manifolds

text: with

text: b+

text: We

text: discuss

text: it

text: using

text: the

text: notion

text: of

text: symplectic

text: Kodaira

text: dimension.

text: For

text: κs

text: as

text: we

text: mentioned,

text: there

text: are

text: nothing

text: more

text: than

text: rational

text: and

text: ruled

text: surfaces.

text: For

text: κs

text: although

text: there

text: are

text: no

text: classiﬁcation

text: at

text: this

text: moment,

text: in

text: addition

text: to

text: hyperelliptic

text: surfaces,

text: we

text: know

text: that

text: there

text: are

text: certain

text: (non-complex)

text: bundles

text: over

text: with

text: b

text: All

text: of

text: them

text: are

text: symplectic

text: They

text: could

text: have

text: diﬀerent

text: fundamental

text: groups

text: from

text: hyperelliptic

text: surfaces

text: since

text: and

text: are

text: all

text: the

text: possible

text: geometric

text: types

text: and

text: on

text: the

text: other

text: hand

text: all

text: hyperelliptic

text: surfaces

text: have

text: geometric

text: type

text: Let

text: us

text: digress

text: on

text: the

text: deﬁnition

text: of

text: a

text: geometry

text: structure

text: (in

text: the

text: sense

text: of

text: Thurston).

text: A

text: model

text: geometry

text: is

text: a

text: simply

text: connected

text: smooth

text: manifold

text: X

text: together

text: with

text: a

text: transitive

text: action

text: of

text: a

text: Lie

text: group

text: on

text: with

text: compact

text: sta-

text: G

text: X

text: bilizers.

text: A

text: model

text: geometry

text: is

text: called

text: maximal

text: if

text: is

text: maximal

text: among

text: groups

text: G

text: acting

text: smoothly

text: and

text: transitively

text: on

text: with

text: compact

text: stabilizers.

text: Sometimes

text: X

text: this

text: condition

text: is

text: included

text: in

text: the

text: deﬁnition

text: of

text: a

text: model

text: geometry.

text: A

text: geometric

text: structure

text: on

text: a

text: manifold

text: is

text: a

text: diﬀeomorphism

text: from

text: to

text: for

text: some

text: M

text: M

text: X/Γ

text: model

text: geometry

text: where

text: is

text: a

text: discrete

text: subgroup

text: of

text: acting

text: freely

text: on

text: X

text: Γ

text: G

text: X

text: If

text: a

text: given

text: manifold

text: admits

text: a

text: geometric

text: structure,

text: then

text: it

text: admits

text: one

text: whose

text: model

text: is

text: maximal.

text: In

text: other

text: word,

text: diﬀerent

text: geometries

text: are

text: distinguished

text: by

text: their

text: fundamental

text: groups.

text: It

text: is

text: remarkable

text: that

text: T.-J.

text: Li

text: proved

text: that

text: all

text: symplectic

text: manifolds

text: with

text: κs

text: SCY

text: surfaces,

text: have

text: the

text: same

text: rational

text: homology

text: types

text: as

text: surfaces,

text: Enrique

text: surfaces

text: or

text: bundles

text: over

text: Moreover,

text: the

text: paper

text: of

text: S.

text: Friedl

text: and

text: S.

text: Vidussi

text: discussed

text: the

text: restrictions

text: on

text: the

text: fundamental

text: groups

text: of

text: SCY

text: -manifolds


text: Least Energy Approximation for Processes with Stationary Increments ∗ † Zakhar Kabluchko Mikhail Lifshits May 27, 2020 Abstract A function f = f is called least energy approximation to a function (cid:90) T B on the interval (cid:2)[0, T] with penalty Q i(cid:3)f it solves the variational problem T (cid:48) − (cid:38) f (t)2 + Q(f(t) B(t)) dt min . 0 For quadratic penalty the least energy approximation can be found explicitly. If B is a random process with stationary increments, then on large intervals f also is close to a process of the same class and T the relation between the corresponding spectral measures can be found. → ∞ We show that in a long run (when T ) the expectation of energy of optimal approximation per unit of time converges to some limit which we compute explicitly. For Gaussian and L´evy processes we complete this result with almost sure and L1 convergence. As an example, the asymptotic expression of approximation energy is found for fractional Brownian motion. 2010 AMS Mathematics Subject Classiﬁcation: Primary: 60G10; Secondary: 60G15, 49J40, 41A00. Key words and phrases: least energy approximation, Gaussian process, L´evy process, fractional Brownian motion, process with stationary incre- ments, taut string, variational calculus. ∗ Mu¨nster University, Orl´eans-Ring 10, 48149 Mu¨nster, Germany, email zakhar.kabluchko@uni-muenster.de † St.Petersburg State University, Russia, Stary Peterhof, Bibliotechnaya pl.,2, email mikhail@lifshits.org and MAI, Linko¨ping University. 1 ##LTLine##

text: Least

label: title

text: Energy

text: Approximation

text: for

text: Processes

text: with

text: Stationary

text: Increments

label: author

label: paragraph

text: Zakhar

text: Kabluchko

text: Mikhail

text: Lifshits

text: May

label: date

text: Abstract

text: A

label: abstract

text: function

text: f

text: f

text: is

text: called

text: least

text: energy

text: approximation

text: to

text: a

text: function

text: T

text: B

text: on

text: the

text: interval

text: T]

text: with

text: penalty

text: Q

text: it

text: solves

text: the

text: variational

text: problem

text: T

text: f

text: Q(f(t)

text: B(t))

text: dt

text: min

text: For

text: quadratic

text: penalty

text: the

text: least

text: energy

text: approximation

text: can

text: be

text: found

text: explicitly.

text: If

text: B

text: is

text: a

text: random

text: process

text: with

text: stationary

text: increments,

text: then

text: on

text: large

text: intervals

text: f

text: also

text: is

text: close

text: to

text: a

text: process

text: of

text: the

text: same

text: class

text: and

text: T

text: the

text: relation

text: between

text: the

text: corresponding

text: spectral

text: measures

text: can

text: be

text: found.

text: We

text: show

text: that

text: in

text: a

text: long

text: run

text: (when

text: T

text: the

text: expectation

text: of

text: energy

text: of

text: optimal

text: approximation

text: per

text: unit

text: of

text: time

text: converges

text: to

text: some

text: limit

text: which

text: we

text: compute

text: explicitly.

text: For

text: Gaussian

text: and

text: L´evy

text: processes

text: we

text: complete

text: this

text: result

text: with

text: almost

text: sure

text: and

text: convergence.

text: As

text: an

text: example,

text: the

text: asymptotic

text: expression

text: of

text: approximation

text: energy

text: is

text: found

text: for

text: fractional

text: Brownian

text: motion.

text: AMS

text: Mathematics

text: Subject

text: Classiﬁcation:

text: Primary:

text: Secondary:

text: Key

text: words

text: and

text: phrases:

text: least

text: energy

text: approximation,

text: Gaussian

text: process,

text: L´evy

text: process,

text: fractional

text: Brownian

text: motion,

text: process

text: with

text: stationary

text: incre-

text: ments,

text: taut

text: string,

text: variational

text: calculus.

text: Mu¨nster

text: University,

text: Orl´eans-Ring

text: Mu¨nster,

text: Germany,

text: email

text: zakhar.kabluchko@uni-muenster.de

text: St.Petersburg

text: State

text: University,

text: Russia,

text: Stary

text: Peterhof,

text: Bibliotechnaya

text: email

text: mikhail@lifshits.org

text: and

text: MAI,

text: Linko¨ping

text: University.

text: ##LTLine##


text: (cid:88) By using the identity ∞ (2γ ) = 1 (37) j j=1 we may rewrite the least energy as (cid:88) (cid:88) ∞ ∞ E − 2T 2 T 2 1 (f , W ) = S2 + γj w2 . T T T 2 + 1 T 2 + 1 j j=1 γj j=1 γj E Recall that w2 = γ . We ﬁ(cid:0)nd the avera(cid:1)ge(cid:88)least energy j j ∞ E E − E 1 (f , W ) = 2 [S2] + 1 . T T 1 + 1 (cid:88) (cid:90) j=1 T2γj Since ∞ ∞ 1 ∼ T dx T → ∞ = , as T , (38) 1 + 1 π 1 + x2 2 j=1 T2γj 0 and, as we will see, E ∼ 1 → ∞ S2 , as T , (39) T we arrive at (31). It remains to analyze the behavior of S. By using the deﬁnition of S (cid:32) (cid:112) (cid:33) in (34) and formulae ((cid:88)36)(cid:112)we get an equation ∞ S 2γ + w − S = 2γ T 2 j j w . j T 2 + 1 j j=1 γj Solving it in S we get A(T ) S = D(T ) with (cid:112) (cid:88) ∞ w 2γ  j j  A(T ) = , (cid:88)T 2γ + 1 j=1 j  ∞ 2γ − j D(T ) = 1. 1 + 1 j=1 T2γj 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: By

text: using

text: the

text: identity

text: j

text: we

text: may

text: rewrite

text: the

text: least

text: energy

text: as

text: E

text: T

text: (f

text: W

text: γj

text: T

text: T

text: T

text: T

text: j

text: γj

text: γj

text: E

text: Recall

text: that

text: γ

text: We

text: the

text: energy

text: j

text: j

text: E

text: E

text: E

text: (f

text: W

text: T

text: T

text: Since

text: T

text: dx

text: T

text: as

text: T

text: π

text: and,

text: as

text: we

text: will

text: see,

text: E

text: as

text: T

text: T

text: we

text: arrive

text: at

text: It

text: remains

text: to

text: analyze

text: the

text: behavior

text: of

text: S.

text: By

text: using

text: the

text: deﬁnition

text: of

text: S

text: in

text: and

text: formulae

text: get

text: an

text: equation

text: S

text: 2γ

text: w

text: S

text: 2γ

text: T

text: j

text: j

text: w

text: j

text: T

text: j

text: γj

text: Solving

text: it

text: in

text: S

text: we

text: get

text: A(T

text: S

text: D(T

text: with

text: w

text: 2γ

text: j

text: j

text: A(T

text: 2γ

text: j

text: 2γ

text: j

text: D(T

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: (cid:90) Notice that A(T )(cid:88)is a centered normal random variable with variance ∞ ∞ E |A(T )|2 = 2γj2 ∼ 2 dx T −3 ∼ 1 (T 2γ + 1)2 π (1 + x2)2 T 3 j 0 j=1 and for the non-random denomina(cid:88)tor D(T ) by using (37) and (38) we have ∞ − 2 1 ∼ − 1 D(T ) = . T 2 1 + 1 T j=1 T2γj Now (39) is conﬁrmed and we are done. 5 Addendum: Proof of Proposition 2.1 (cid:90) Proof of Proposition 2.1: We ﬁrst notice that E ≤ E T − ∞ inf (f) (0) = Q( B(t)) dt < . ∈W T T f 12[0,T] 0 · − · Indeed, if B( ) is bounded, the(cid:101)n Q( B( )) is bounded. If (1) holds, then we have (cid:101) ≤ | | ∈ R Q(x) A ( x p+1 + 1), x , (cid:90) ∈(cid:90) with A := 2A + Q(0). Hence, if B L [0, T ], then (cid:101) p+1 T − ≤ T | | ∞ Q( B(t)) dt A ( B(t) p+1 + 1) dt < . 0 0 Second, we(cid:26)show that we (cid:90)may restrict the minimization in(cid:27)(2) on a subclass W ∈ W T (cid:48) ≤ | | ≤ := f 1[0, T ] : f (t)2dt C, max f(t) M (40) C,M 2 ≤ ≤ 0 0 t T with suﬃciently large parameters C and M. To justify this statement, it is suﬃcient to show that for each r > 0 there exist C, M such that { E ≤ } ⊂ W f : (f) r . T C,M E (cid:90)≤ Let f be such that (f) r. Then T T (cid:48) ≤ E ≤ f (t)2dt (f) r T 0 ≤ ≤ | − | ≤ √ and we may let C := r. Moreover, for any 0 s, t T we have f(s) f(t) | | ≥ √ rT by using H¨older inequality. Therefore, if f(s) M holds for some ∈ | | ≥ − ∈ s [0, T ], then f(t) M rT for all t [0, T ]. We see that 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Notice

text: that

text: A(T

text: a

text: centered

text: normal

text: random

text: variable

text: with

text: variance

text: E

text: |A(T

text: dx

text: T

text: −3

text: (T

text: 2γ

text: π

text: T

text: j

text: and

text: for

text: the

text: non-random

text: D(T

text: by

text: using

text: and

text: we

text: have

text: D(T

text: T

text: T

text: Now

text: is

text: conﬁrmed

text: and

text: we

text: are

text: done.

label: section

text: Addendum:

text: Proof

text: of

text: Proposition

text: Proof

text: of

text: Proposition

text: We

text: ﬁrst

text: notice

text: that

text: E

text: E

text: T

text: inf

text: (f)

text: Q(

text: B(t))

text: dt

text: ∈W

text: T

text: T

text: f

text: Indeed,

text: if

text: B(

text: is

text: bounded,

text: Q(

text: B(

text: is

text: bounded.

text: If

text: holds,

text: then

text: we

text: have

text: R

text: Q(x)

text: A

text: x

text: x

text: with

text: A

text: Hence,

text: if

text: B

text: L

text: T

text: then

text: T

text: T

text: Q(

text: B(t))

text: dt

text: A

text: B(t)

text: dt

text: Second,

text: that

text: we

text: restrict

text: the

text: minimization

text: on

text: a

text: subclass

text: W

text: W

text: T

text: f

text: T

text: f

text: C,

text: max

text: f(t)

text: M

text: C,M

text: t

text: T

text: with

text: suﬃciently

text: large

text: parameters

text: C

text: and

text: M.

text: To

text: justify

text: this

text: statement,

text: it

text: is

text: suﬃcient

text: to

text: show

text: that

text: for

text: each

text: r

text: there

text: exist

text: C,

text: M

text: such

text: that

text: E

text: W

text: f

text: (f)

text: r

text: T

text: C,M

text: E

text: Let

text: f

text: be

text: such

text: that

text: (f)

text: r.

text: Then

text: T

text: T

text: E

text: f

text: (f)

text: r

text: T

text: and

text: we

text: may

text: let

text: C

text: r.

text: Moreover,

text: for

text: any

text: s,

text: t

text: T

text: we

text: have

text: f(s)

text: f(t)

text: rT

text: by

text: using

text: H¨older

text: inequality.

text: Therefore,

text: if

text: f(s)

text: M

text: holds

text: for

text: some

text: s

text: T

text: then

text: f(t)

text: M

text: rT

text: for

text: all

text: t

text: T

text: We

text: see

text: that

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Fig. 5.— Histograms of the apparent spectral indices (left panel) and de-reddened spectral indices (right panel) of all YSOs (hatched columns) and H ﬂow sources (ﬁlled black columns) identiﬁed in this paper. 2 The dashed lines show the criteria of YSO classiﬁcation scheme suggested by Lada (1987). − Fig. 6.— Infrared nebulae associated with outﬂow sources. The backgrounds are K band images. North s is up and east to the left. The outﬂow sources are located in the center of panels. 23 ##LTFigure## ##LTFigure## ##LTFigure##

text: Fig.

text: Histograms

label: caption

text: of

text: the

text: apparent

text: spectral

text: indices

text: (left

text: panel)

text: and

text: de-reddened

text: spectral

text: indices

text: (right

text: panel)

text: of

text: all

text: YSOs

text: (hatched

text: columns)

text: and

text: H

text: ﬂow

text: sources

text: (ﬁlled

text: black

text: columns)

text: identiﬁed

text: in

text: this

text: paper.

text: The

text: dashed

text: lines

text: show

text: the

text: criteria

text: of

text: YSO

text: classiﬁcation

text: scheme

text: suggested

text: by

text: Lada

text: Fig.

text: Infrared

text: nebulae

text: associated

text: with

text: outﬂow

text: sources.

text: The

text: backgrounds

text: are

text: K

text: band

text: images.

text: North

text: s

text: is

text: up

text: and

text: east

text: to

text: the

text: left.

text: The

text: outﬂow

text: sources

text: are

text: located

text: in

text: the

text: center

text: of

text: panels.

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##

text: ##LTFigure##


text: Fig. 7.— A disk-jet system candidate that consists of MHO 3251b1 and its driving source ID#17. The left panel shows the H narrow-band image while the right panel shows the K broadband image. The white 2 s plus in the right panel marks the ﬂux-weighted mean IRAC and MIPS 24 µm position for ID#17. The blue cross in the right panel marks the mean IRAC position for ID#17 while the red cross labels the MIPS 24 µm position for ID#17. The solid line in the left panel shows the orientation of a collimated jet. 24 ##LTFigure##

text: Fig.

text: A

text: disk-jet

text: system

text: candidate

text: that

text: consists

text: of

text: MHO

text: and

text: its

text: driving

text: source

text: The

text: left

text: panel

text: shows

text: the

text: H

text: narrow-band

text: image

text: while

text: the

text: right

text: panel

text: shows

text: the

text: K

text: broadband

text: image.

text: The

text: white

text: s

text: plus

text: in

text: the

text: right

text: panel

text: marks

text: the

text: ﬂux-weighted

text: mean

text: IRAC

text: and

text: MIPS

text: µm

text: position

text: for

text: The

text: blue

text: cross

text: in

text: the

text: right

text: panel

text: marks

text: the

text: mean

text: IRAC

text: position

text: for

text: while

text: the

text: red

text: cross

text: labels

text: the

text: MIPS

text: µm

text: position

text: for

text: The

text: solid

text: line

text: in

text: the

text: left

text: panel

text: shows

text: the

text: orientation

text: of

text: a

text: collimated

text: jet.

text: ##LTFigure##


text: southwest of ID#28, MHO 3277 and MHO 3278. It seems that MHO 3253, ID#28, MHO 3254b-c, and MHO 3277-3278 align well with each other. Thus here we propose another possibility that MHO 3253, MHO 3254b-c, and MHO 3277-3278 are driven by ID#28. Moreover, based on the location of MHO 3254a, we propose that MHO 3254a belongs to the outﬂow driven by ID#23 (see the description about MHO 3255). • MHO 3255: Teixeira et al. (2012) ﬁrstly detected this object, but only the features corresponding to ∼ (cid:48) MHO 3255a and b in Fig. A10. We identify a new feature, MHO 3255c, which is located 4 to the northeast of MHO 3255b. Teixeira et al. (2012) suggested a protostar “P5” that is associated with our identiﬁed YSO, ID#23, as the possible driving source of MHO 3255. Based on the good alignment among MHO 3255, MHO 3254a, and ID#23, we suggest that ID#23 drives a bipolar outﬂow that consists of MHO 3255 and MHO 3254a. • MHO 3259: Fig. A11 shows the region of MHO 3259 that exhibits a bow-like structure. ID#1 is the nearest YSO, lying in the southwest of MHO 3259. We propose ID#1 as the possible driving source of MHO 3259. • MHO 3260: Fig. A12 shows the region of MHO 3260 which consists of four features. MHO 3260c and d are two knots. MHO 3260a and b show approximate point symmetric structures, which indicates that the driving source should be located between them. Bontemps et al. (2010) detected a protostar which is labeled as ID#2 in Fig. A12. Note that ID#2 is invisible at 3.6-24µm, but visible in Herschel PACS and SPIRE images, which indicates that ID#2 may stay at a very early evolutionary stage. We suggest ID#2 as the driving source of MHO 3260. • MHO 3261: Fig. A13 shows the region of MHO 3261 which consists of two features. The YSO ID#4 is located at the northeast of MHO 3261b and associated with an infrared nebula (see Fig. 6). The infrared nebula exhibits the NE-SW elongated structure. We suggest ID#4 as the driving source of MHO 3261. • MHO 3262: This object is an elongated knot in Fig. A14. ID#5 is the nearest YSO to it and we suggest ID#5 as the possible driving source of MHO 3262. • MHO 3263: Fig. A15 shows the region of MHO 3263 which consists of three features. The YSO ID#6 and MHO 3263 are roughly located on a straight line. Thus we suggest ID#6 as the possible driving source of MHO 3263. • MHO 3264: Fig. A16 shows the region of MHO 3264 which consists of ﬁve features. The YSO ID#7 is roughly aligned with the features of MHO 3264. ID#7 is also associated with an infrared nebula (see Fig. 6) which seems to trace the cavity wall of the west lobe of outﬂow. Thus we suggest ID#7 as the possible driving source of MHO 3264. • MHO 3265: Fig. A17 shows the region of MHO 3265 which consists of two features. The YSO ID#8 is located between MHO 3265a and b. Note that we suggest MHO 3265a belongs to another outﬂow rather than the outﬂow that consists of MHO 2214 based on its morphology although MHO 3265a is located between MHO 2214r3 and MHO 2214r4 (see the description about MHO 2214). Considering the good alignment of ID#8 and MHO 3265, we propose ID#8 as the possible driving source of MHO 3265. • MHO 3266: MHO 3266 shows a complex structure in Fig. A18. The YSO ID#10 is located in the southwest of MHO 3266 and seems to be located on the extension line of the major axis of MHO 3266. Thus we suggest ID#10 as the possible driving source of MHO 3266. • MHO 3267: Fig. A19 shows the region of MHO 3267 which consists of three features. The YSO ID#11 is associated with IRAS 18270-0153 (Connelley et al. 2007; Connelley & Greene 2010) and an infrared 30

text: southwest

label: list

text: of

text: MHO

text: and

text: MHO

text: It

text: seems

text: that

text: MHO

text: MHO

text: and

text: MHO

text: align

text: well

text: with

text: each

text: other.

text: Thus

text: here

text: we

text: propose

text: another

text: possibility

text: that

text: MHO

text: MHO

text: and

text: MHO

text: are

text: driven

text: by

text: Moreover,

text: based

text: on

text: the

text: location

text: of

text: MHO

text: we

text: propose

text: that

text: MHO

text: belongs

text: to

text: the

text: outﬂow

text: driven

text: by

text: (see

text: the

text: description

text: about

text: MHO

text: MHO

text: Teixeira

text: et

text: al.

text: ﬁrstly

text: detected

text: this

text: object,

text: but

text: only

text: the

text: features

text: corresponding

text: to

text: MHO

text: and

text: b

text: in

text: Fig.

text: We

text: identify

text: a

text: new

text: feature,

text: MHO

text: which

text: is

text: located

text: to

text: the

text: northeast

text: of

text: MHO

text: Teixeira

text: et

text: al.

text: suggested

text: a

text: protostar

text: “P5”

text: that

text: is

text: associated

text: with

text: our

text: identiﬁed

text: YSO,

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: Based

text: on

text: the

text: good

text: alignment

text: among

text: MHO

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text: and

text: we

text: suggest

text: that

text: drives

text: a

text: bipolar

text: outﬂow

text: that

text: consists

text: of

text: MHO

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text: MHO

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text: shows

text: the

text: region

text: of

text: MHO

text: that

text: exhibits

text: a

text: bow-like

text: structure.

text: is

text: the

text: nearest

text: YSO,

text: lying

text: in

text: the

text: southwest

text: of

text: MHO

text: We

text: propose

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: Fig.

text: shows

text: the

text: region

text: of

text: MHO

text: which

text: consists

text: of

text: four

text: features.

text: MHO

text: and

text: d

text: are

text: two

text: knots.

text: MHO

text: and

text: b

text: show

text: approximate

text: point

text: symmetric

text: structures,

text: which

text: indicates

text: that

text: the

text: driving

text: source

text: should

text: be

text: located

text: between

text: them.

text: Bontemps

text: et

text: al.

text: detected

text: a

text: protostar

text: which

text: is

text: labeled

text: as

text: in

text: Fig.

text: Note

text: that

text: is

text: invisible

text: at

text: but

text: visible

text: in

text: Herschel

text: PACS

text: and

text: SPIRE

text: images,

text: which

text: indicates

text: that

text: may

text: stay

text: at

text: a

text: very

text: early

text: evolutionary

text: stage.

text: We

text: suggest

text: as

text: the

text: driving

text: source

text: of

text: MHO

text: MHO

text: Fig.

text: shows

text: the

text: region

text: of

text: MHO

text: which

text: consists

text: of

text: two

text: features.

text: The

text: YSO

text: is

text: located

text: at

text: the

text: northeast

text: of

text: MHO

text: and

text: associated

text: with

text: an

text: infrared

text: nebula

text: (see

text: Fig.

text: The

text: infrared

text: nebula

text: exhibits

text: the

text: NE-SW

text: elongated

text: structure.

text: We

text: suggest

text: as

text: the

text: driving

text: source

text: of

text: MHO

text: MHO

text: This

text: object

text: is

text: an

text: elongated

text: knot

text: in

text: Fig.

text: is

text: the

text: nearest

text: YSO

text: to

text: it

text: and

text: we

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: Fig.

text: shows

text: the

text: region

text: of

text: MHO

text: which

text: consists

text: of

text: three

text: features.

text: The

text: YSO

text: and

text: MHO

text: are

text: roughly

text: located

text: on

text: a

text: straight

text: line.

text: Thus

text: we

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: Fig.

text: shows

text: the

text: region

text: of

text: MHO

text: which

text: consists

text: of

text: ﬁve

text: features.

text: The

text: YSO

text: is

text: roughly

text: aligned

text: with

text: the

text: features

text: of

text: MHO

text: is

text: also

text: associated

text: with

text: an

text: infrared

text: nebula

text: (see

text: Fig.

text: which

text: seems

text: to

text: trace

text: the

text: cavity

text: wall

text: of

text: the

text: west

text: lobe

text: of

text: outﬂow.

text: Thus

text: we

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: Fig.

text: shows

text: the

text: region

text: of

text: MHO

text: which

text: consists

text: of

text: two

text: features.

text: The

text: YSO

text: is

text: located

text: between

text: MHO

text: and

text: b.

text: Note

text: that

text: we

text: suggest

text: MHO

text: belongs

text: to

text: another

text: outﬂow

text: rather

text: than

text: the

text: outﬂow

text: that

text: consists

text: of

text: MHO

text: based

text: on

text: its

text: morphology

text: although

text: MHO

text: is

text: located

text: between

text: MHO

text: and

text: MHO

text: (see

text: the

text: description

text: about

text: MHO

text: Considering

text: the

text: good

text: alignment

text: of

text: and

text: MHO

text: we

text: propose

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: MHO

text: shows

text: a

text: complex

text: structure

text: in

text: Fig.

text: The

text: YSO

text: is

text: located

text: in

text: the

text: southwest

text: of

text: MHO

text: and

text: seems

text: to

text: be

text: located

text: on

text: the

text: extension

text: line

text: of

text: the

text: major

text: axis

text: of

text: MHO

text: Thus

text: we

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: Fig.

text: shows

text: the

text: region

text: of

text: MHO

text: which

text: consists

text: of

text: three

text: features.

text: The

text: YSO

text: is

text: associated

text: with

text: IRAS

text: (Connelley

text: et

text: al.

text: Connelley

text: Greene

text: and

text: an

text: infrared


text: • MHO 3281: Fig. A31 shows the region of MHO 3281 that consists of three features. The YSO ID#31 is located between MHO 3281a and b. Note that ID#31 is also associated with an infrared nebula (see Fig. 6). It seems that MHO 3281 and ID#31 constitute a bipolar outﬂow system. • MHO 3282: This object seems to be a bright knot with a faint ring-like structure in Fig. A32. ID#32 is the nearest YSO and we suggest it as the possible driving source of MHO 3282. • MHO 3283: Fig. A33 shows the region of MHO 3283 that consists of two features. MHO 3283a exhibits the multiple bow shock morphology, which indicates that its possible driving source should be located to its east. The YSO ID#33 in the southeast of MHO 3283 is located roughly on the line connecting MHO 3283a and MHO 3283b. Thus we suggest ID#33 as the possible driving source of MHO 3283. • MHO 3284 and MHO 3285: MHO 3284 shows complex morphology in Fig. A34 while MHO 3285 consists of some knots in Fig. A35. No YSOs have been identiﬁed in the ambient of MHO 3284 and MHO 3285. Therefore, we can not infer any driving source candidates for MHO 3284 and MHO 3285. • MHO 3286: This object is a chain of two bright knots in Fig. A36, which indicates that its possible driving source may be located to its southeast or northwest. There is a YSO, ID#34, to the southeast of MHO 3286. ID#34 and the knots of MHO 3286 seems to align with each other. Thus we suggest ID#34 as the possible driving source of MHO 3286. • MHO 3287: Figure A37 shows the region of MHO 3287 that consists of three features. MHO 3287a is a collimated jet in the vicinity of the YSO ID#35 that is associated with a protostar identiﬁed by Bontemps et al. (2010) using the Herschel data. There is good alignment between ID#35 and MHO 3287. Thus we suggest ID#35 as the driving source of MHO 3287. Note that the circular bright blot in the corresponding position of ID#35 is due to the instrument eﬀect, “persistence”, which occurs due to charges being trapped in high ﬂux or saturated regions in one exposure that are then slowly released in subsequent exposures. Since exposures are dithered, the released charges appear as new “artiﬁcial sources” in the subsequent images. • MHO 3288: This object is a patch in Fig. A38. The nearest YSO to MHO 3288 is ID#36, which is also associated with an infrared nebula (see Fig. 6). We suggest ID#36 as the possible driving source of MHO 3288. • MHO 3289: This object shows a bow-like structure in Fig. A39, which indicates that the driving source of MHO 3289 may be located to the southeast of it. ID#37 is located to the southeast of MHO 3289 and associated with a SE-NW bipolar infrared nebula (see Fig. 6). We suggest ID#37 as the possible driving source of MHO 3289. • MHO 3290: This object consists of three features that align well with each other in Fig. A40. The YSO ID#38 is located between MHO 3290b and MHO 3290c and associated with an infrared nebula that extends towards southeast from ID#38 in Fig. 6. It seems that ID#38 and MHO 3290 constitute a bipolar outﬂow system. We suggest ID#38 as the driving source of MHO 3290. • MHO 3291: This object is a knot in Fig. A41. ID#39 is the nearest YSO to MHO 3291. We suggest ID#39 as the possible driving source of MHO 3291. • MHO 3292: Figure A42 shows the region of MHO 3292. The YSO ID#40 is located roughly on the line connecting the features of MHO 3292. ID#40 is also associated with an infrared nebula that extends towards the northeast from ID#40 in Fig. 6. Thus we suggest ID#40 as the possible driving source of MHO 3292. 32

text: MHO

text: Fig.

text: shows

text: the

text: region

text: of

text: MHO

text: that

text: consists

text: of

text: three

text: features.

text: The

text: YSO

text: is

text: located

text: between

text: MHO

text: and

text: b.

text: Note

text: that

text: is

text: also

text: associated

text: with

text: an

text: infrared

text: nebula

text: (see

text: Fig.

text: It

text: seems

text: that

text: MHO

text: and

text: constitute

text: a

text: bipolar

text: outﬂow

text: system.

text: MHO

text: This

text: object

text: seems

text: to

text: be

text: a

text: bright

text: knot

text: with

text: a

text: faint

text: ring-like

text: structure

text: in

text: Fig.

text: is

text: the

text: nearest

text: YSO

text: and

text: we

text: suggest

text: it

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: Fig.

text: shows

text: the

text: region

text: of

text: MHO

text: that

text: consists

text: of

text: two

text: features.

text: MHO

text: exhibits

text: the

text: multiple

text: bow

text: shock

text: morphology,

text: which

text: indicates

text: that

text: its

text: possible

text: driving

text: source

text: should

text: be

text: located

text: to

text: its

text: east.

text: The

text: YSO

text: in

text: the

text: southeast

text: of

text: MHO

text: is

text: located

text: roughly

text: on

text: the

text: line

text: connecting

text: MHO

text: and

text: MHO

text: Thus

text: we

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: and

text: MHO

text: MHO

text: shows

text: complex

text: morphology

text: in

text: Fig.

text: while

text: MHO

text: consists

text: of

text: some

text: knots

text: in

text: Fig.

text: No

text: YSOs

text: have

text: been

text: identiﬁed

text: in

text: the

text: ambient

text: of

text: MHO

text: and

text: MHO

text: Therefore,

text: we

text: can

text: not

text: infer

text: any

text: driving

text: source

text: candidates

text: for

text: MHO

text: and

text: MHO

text: MHO

text: This

text: object

text: is

text: a

text: chain

text: of

text: two

text: bright

text: knots

text: in

text: Fig.

text: which

text: indicates

text: that

text: its

text: possible

text: driving

text: source

text: may

text: be

text: located

text: to

text: its

text: southeast

text: or

text: northwest.

text: There

text: is

text: a

text: YSO,

text: to

text: the

text: southeast

text: of

text: MHO

text: and

text: the

text: knots

text: of

text: MHO

text: seems

text: to

text: align

text: with

text: each

text: other.

text: Thus

text: we

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: Figure

text: shows

text: the

text: region

text: of

text: MHO

text: that

text: consists

text: of

text: three

text: features.

text: MHO

text: is

text: a

text: collimated

text: jet

text: in

text: the

text: vicinity

text: of

text: the

text: YSO

text: that

text: is

text: associated

text: with

text: a

text: protostar

text: identiﬁed

text: by

text: Bontemps

text: et

text: al.

text: using

text: the

text: Herschel

text: data.

text: There

text: is

text: good

text: alignment

text: between

text: and

text: MHO

text: Thus

text: we

text: suggest

text: as

text: the

text: driving

text: source

text: of

text: MHO

text: Note

text: that

text: the

text: circular

text: bright

text: blot

text: in

text: the

text: corresponding

text: position

text: of

text: is

text: due

text: to

text: the

text: instrument

text: eﬀect,

text: “persistence”,

text: which

text: occurs

text: due

text: to

text: charges

text: being

text: trapped

text: in

text: high

text: ﬂux

text: or

text: saturated

text: regions

text: in

text: one

text: exposure

text: that

text: are

text: then

text: slowly

text: released

text: in

text: subsequent

text: exposures.

text: Since

text: exposures

text: are

text: dithered,

text: the

text: released

text: charges

text: appear

text: as

text: new

text: “artiﬁcial

text: sources”

text: in

text: the

text: subsequent

text: images.

text: MHO

text: This

text: object

text: is

text: a

text: patch

text: in

text: Fig.

text: The

text: nearest

text: YSO

text: to

text: MHO

text: is

text: which

text: is

text: also

text: associated

text: with

text: an

text: infrared

text: nebula

text: (see

text: Fig.

text: We

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: This

text: object

text: shows

text: a

text: bow-like

text: structure

text: in

text: Fig.

text: which

text: indicates

text: that

text: the

text: driving

text: source

text: of

text: MHO

text: may

text: be

text: located

text: to

text: the

text: southeast

text: of

text: it.

text: is

text: located

text: to

text: the

text: southeast

text: of

text: MHO

text: and

text: associated

text: with

text: a

text: SE-NW

text: bipolar

text: infrared

text: nebula

text: (see

text: Fig.

text: We

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: This

text: object

text: consists

text: of

text: three

text: features

text: that

text: align

text: well

text: with

text: each

text: other

text: in

text: Fig.

text: The

text: YSO

text: is

text: located

text: between

text: MHO

text: and

text: MHO

text: and

text: associated

text: with

text: an

text: infrared

text: nebula

text: that

text: extends

text: towards

text: southeast

text: from

text: in

text: Fig.

text: It

text: seems

text: that

text: and

text: MHO

text: constitute

text: a

text: bipolar

text: outﬂow

text: system.

text: We

text: suggest

text: as

text: the

text: driving

text: source

text: of

text: MHO

text: MHO

text: This

text: object

text: is

text: a

text: knot

text: in

text: Fig.

text: is

text: the

text: nearest

text: YSO

text: to

text: MHO

text: We

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO

text: MHO

text: Figure

text: shows

text: the

text: region

text: of

text: MHO

text: The

text: YSO

text: is

text: located

text: roughly

text: on

text: the

text: line

text: connecting

text: the

text: features

text: of

text: MHO

text: is

text: also

text: associated

text: with

text: an

text: infrared

text: nebula

text: that

text: extends

text: towards

text: the

text: northeast

text: from

text: in

text: Fig.

text: Thus

text: we

text: suggest

text: as

text: the

text: possible

text: driving

text: source

text: of

text: MHO


text: Fig. A6.— Same as Fig. A2, but for the region of MHO 3250. Note that the red circles mark the holes inside the polygons. 40 ##LTFigure##

text: Fig.

text: Same

text: as

text: Fig.

text: but

text: for

text: the

text: region

text: of

text: MHO

text: Note

text: that

text: the

text: red

text: circles

text: mark

text: the

text: holes

text: inside

text: the

text: polygons.

text: ##LTFigure##


text: Fig. A19.— Same as Fig. A2, but for the region of MHO 3267. 53 ##LTFigure##

text: Fig.

text: Same

text: as

text: Fig.

text: but

text: for

text: the

text: region

text: of

text: MHO

text: ##LTFigure##


text: Fig. A22.— Same as Fig. A2, but for the region of MHO 3270. Note that the red circle marks the hole inside the polygon. 56 ##LTFigure##

text: Fig.

text: Same

text: as

text: Fig.

text: but

text: for

text: the

text: region

text: of

text: MHO

text: Note

text: that

text: the

text: red

text: circle

text: marks

text: the

text: hole

text: inside

text: the

text: polygon.

text: ##LTFigure##


text: Fig. A26.— Same as Fig. A1, but for the region of MHO 3274. 60 ##LTFigure##

text: Fig.

text: Same

text: as

text: Fig.

text: but

text: for

text: the

text: region

text: of

text: MHO

text: ##LTFigure##


text: Fig. A29.— Same as Fig. A1, but for the region of MHO 3279. 63 ##LTFigure##

text: Fig.

text: Same

text: as

text: Fig.

text: but

text: for

text: the

text: region

text: of

text: MHO

text: ##LTFigure##


text: ralucric ehT .nogylop eht edisni eloh eht skram elcric der eht taht etoN .7823 OHM fo noiger eht rof tub ,2A .giF sa emaS —.73A .giF .tceﬀe tnemurtsni ”ecnetsisrep“ eht ot eud si 53#DI fo noitisop gnidnopserroc eht ni tolb thgirb 71 ##LTFigure##

text: ralucric

text: ehT

text: .nogylop

text: eht

text: edisni

text: eloh

text: eht

text: skram

text: elcric

text: der

text: eht

text: taht

text: etoN

text: OHM

text: fo

text: noiger

text: eht

text: rof

text: tub

text: .giF

text: sa

text: emaS

text: .giF

text: .tceﬀe

text: tnemurtsni

text: ”ecnetsisrep“

text: eht

text: ot

text: eud

text: si

text: fo

text: noitisop

text: gnidnopserroc

text: eht

text: ni

text: tolb

text: thgirb

text: ##LTFigure##


text: between the ﬁeld and the atom in an optical cavity. However, this increase of time will inevitably lead also to the increase in the probability of photon leakage out of the cavity. Our task is to ﬁnd the optimal time for ﬁnding the atom in an optical cavity, depending on the intensity of photons leak. The array consists in four wires, two idle ones x , y and two wires x , y which 2 2 1 1 are acted upon by the optical components. The idle wires cannot be disregarded, as the photons inside of them contribute to the overall energy of the system. There are two linear gates: beam splitters BS1 and BS2, whose action in Fock space looks as (cid:107) (cid:105) | (cid:105) √ 1 | (cid:105) | (cid:105) −→ n m = (a+)n(a+)m 0 0 a1 a1 n!m! 1 2 a1 a2 (11) −→ √ 1 √1 √1 − | (cid:105) | (cid:105) [ (a+ + a+)]n[ (a+ a+)]m 0 0 n!m! 2 1 2 2 1 2 a1 a2 and two optical cavities, which perform the nonlinear shifts: NS1, NS2. The nonlinear shift in the ideal case acts on Fock states of photons as | (cid:105) → | (cid:105) | (cid:105) → | (cid:105) | (cid:105) → −| (cid:105) 0 0 , 1 1 , 2 2 (12) This is realized by Jaynes-Cummins Hamiltonian (1). The gates work sequen- tially, from the left to the right, as shown in the Figure 2. We assume that the beam splitters work instantly, whereas the NS gates work simultaneously, and their work lasts for the time ∆t. Nonlinear shifts are realized by optical cavities with resonant frequency ω , which is close to the frequency of atom transition from the excited to c the ground state ω with a small detuning ∆. The atom, initially prepared in the a ground state, ﬂies through the cavity along the path, and during the ﬂight interacts with ﬁeld inside the cavity via (7). We interrupt this interaction by the deﬂation of photons from the cavity in the moment t = τ that is chosen in advance; the choice of τ determines the accuracy of the gate array. At this stage we meet the unavoidable decoherence factor, which comes from the residual entanglement between atom and ﬁeld, mentioned above. To choose τ we consider three arithmetic progressions A : t0, t0, . . . , t0, . . . , 0 1 s B : t1, t1, . . . , t1, . . . , (13) 0 1 n C : t2, t2, . . . , t2 , . . . , 0 1 m where A, B and C consist of the sequential time moments, when the unitary | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) −| (cid:105) evolution of JC transforms photonic states as 0 to 0 , 1 to 1 and 2 to 2 re- spectively. If we choose the time τ, when these progressions approximately coincide, we will obtain the required duration. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: between

text: the

text: ﬁeld

text: and

text: the

text: atom

text: in

text: an

text: optical

text: cavity.

text: However,

text: this

text: increase

text: of

text: time

text: will

text: inevitably

text: lead

text: also

text: to

text: the

text: increase

text: in

text: the

text: probability

text: of

text: photon

text: leakage

text: out

text: of

text: the

text: cavity.

text: Our

text: task

text: is

text: to

text: ﬁnd

text: the

text: optimal

text: time

text: for

text: ﬁnding

text: the

text: atom

text: in

text: an

text: optical

text: cavity,

text: depending

text: on

text: the

text: intensity

text: of

text: photons

text: leak.

text: The

text: array

text: consists

text: in

text: four

text: wires,

text: two

text: idle

text: ones

text: x

text: y

text: and

text: two

text: wires

text: x

text: y

text: which

text: are

text: acted

text: upon

text: by

text: the

text: optical

text: components.

text: The

text: idle

text: wires

text: cannot

text: be

text: disregarded,

text: as

text: the

text: photons

text: inside

text: of

text: them

text: contribute

text: to

text: the

text: overall

text: energy

text: of

text: the

text: system.

text: There

text: are

text: two

text: linear

text: gates:

text: beam

text: splitters

text: and

text: whose

text: action

text: in

text: Fock

text: space

text: looks

text: as

text: n

text: m

text: (a+)n(a+)m

text: n!m!

text: √1

text: √1

text: (a+

text: a+)]n[

text: (a+

text: a+)]m

text: n!m!

text: and

text: two

text: optical

text: cavities,

text: which

text: perform

text: the

text: nonlinear

text: shifts:

text: The

text: nonlinear

text: shift

text: in

text: the

text: ideal

text: case

text: acts

text: on

text: Fock

text: states

text: of

text: photons

text: as

text: This

text: is

text: realized

text: by

text: Jaynes-Cummins

text: Hamiltonian

text: The

text: gates

text: work

text: sequen-

text: tially,

text: from

text: the

text: left

text: to

text: the

text: right,

text: as

text: shown

text: in

text: the

text: Figure

text: We

text: assume

text: that

text: the

text: beam

text: splitters

text: work

text: instantly,

text: whereas

text: the

text: NS

text: gates

text: work

text: simultaneously,

text: and

text: their

text: work

text: lasts

text: for

text: the

text: time

text: ∆t.

text: Nonlinear

text: shifts

text: are

text: realized

text: by

text: optical

text: cavities

text: with

text: resonant

text: frequency

text: ω

text: which

text: is

text: close

text: to

text: the

text: frequency

text: of

text: atom

text: transition

text: from

text: the

text: excited

text: to

text: c

text: the

text: ground

text: state

text: ω

text: with

text: a

text: small

text: detuning

text: The

text: atom,

text: initially

text: prepared

text: in

text: the

text: a

text: ground

text: state,

text: ﬂies

text: through

text: the

text: cavity

text: along

text: the

text: path,

text: and

text: during

text: the

text: ﬂight

text: interacts

text: with

text: ﬁeld

text: inside

text: the

text: cavity

text: via

text: We

text: interrupt

text: this

text: interaction

text: by

text: the

text: deﬂation

text: of

text: photons

text: from

text: the

text: cavity

text: in

text: the

text: moment

text: t

text: τ

text: that

text: is

text: chosen

text: in

text: advance;

text: the

text: choice

text: of

text: τ

text: determines

text: the

text: accuracy

text: of

text: the

text: gate

text: array.

text: At

text: this

text: stage

text: we

text: meet

text: the

text: unavoidable

text: decoherence

text: factor,

text: which

text: comes

text: from

text: the

text: residual

text: entanglement

text: between

text: atom

text: and

text: ﬁeld,

text: mentioned

text: above.

text: To

text: choose

text: τ

text: we

text: consider

text: three

text: arithmetic

text: progressions

text: A

text: s

text: B

text: n

text: C

text: m

text: where

text: A,

text: B

text: and

text: C

text: consist

text: of

text: the

text: sequential

text: time

text: moments,

text: when

text: the

text: unitary

text: evolution

text: of

text: JC

text: transforms

text: photonic

text: states

text: as

text: to

text: to

text: and

text: to

text: re-

text: spectively.

text: If

text: we

text: choose

text: the

text: time

text: τ,

text: when

text: these

text: progressions

text: approximately

text: coincide,

text: we

text: will

text: obtain

text: the

text: required

text: duration.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MOTIFS ET ADJOINTS 17 Démonstration. — En utilisant le fait que préserve le produit tensoriel T ∈ M et la dualité, on a des isomorphismes pour : P M ⊗ (cid:39) N ⊗ (S(N T M), P ) (N T M, T P ) (cid:39) N ∗ ⊗ (N, (T M) T P ) (cid:39) N ∗ ⊗ (N, T (M P )) (cid:39) M ∗ ⊗ (SN, M P ) (cid:39) M ⊗ (SN M, P ). La conclusion en résulte par Yoneda. C Proposition 3.7. — a) Soit ( ) ∈ un 2-syst inductif de catries (c’est- → α α (cid:83)A C ire un pseudo-foncteur ), de -colimite . Pour tout , soit A Cat 2 α S C → α une classe de morphismes de , et supposons que tout morphisme α β α ⊂ C de envoie dans . Soit . Alors on a une A S S S = Im(S ) F l( ) α β α α ivalence canonique de catries − C (cid:39) − − C S 1 2 l−i→m S 1 . α α C b) Avec les notations prdentes, soient une autre catrie et, pour tout , α C → C 0 C un foncteur. Soit une classe de morphismes de . On f : S α 0 α 0 suppose que les envoient dans et sont compatibles aux foncteurs f S S α 0 α − C → − C de transition, d’o un foncteur induit f : S 1 S 1 . On suppose de 0 0 plus que: (i) Pour tout , admet un adjoint uche qui envoie dans . α f S S α α 0 → − C → − C (ii) Pour tout morphisme α β, le foncteur S 1 S 1 est α α β β une ivalence de catries. Alors admet un adjoint uche. f Démonstration. — a) Les deux membres de l’ivalence ont chacun la pro- pri 2-universelle de l’autre. b) rlte immatement de a). Dans le dernier ncous nous cantonnons au cadre des thies motiviques pour ter de perdre le lecteur dans une hypergralitutile. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MOTIFS

text: ET

text: ADJOINTS

text: Démonstration.

text: En

text: utilisant

text: le

text: fait

text: que

text: préserve

text: le

text: produit

text: tensoriel

text: T

text: M

text: et

text: la

text: dualité,

text: on

text: a

text: des

text: isomorphismes

text: pour

text: P

text: M

text: N

text: (S(N

text: T

text: M),

text: P

text: (N

text: T

text: M,

text: T

text: P

text: N

text: (N,

text: (T

text: M)

text: T

text: P

text: N

text: (N,

text: T

text: (M

text: P

text: M

text: (SN,

text: M

text: P

text: M

text: (SN

text: M,

text: P

text: La

text: conclusion

text: en

text: résulte

text: par

text: Yoneda.

text: C

text: Proposition

text: a)

text: Soit

text: un

text: inductif

text: de

text: catries

text: (c’est-

text: α

text: α

text: C

text: ire

text: un

text: pseudo-foncteur

text: de

text: -colimite

text: Pour

text: tout

text: soit

text: A

text: Cat

text: α

text: S

text: C

text: α

text: une

text: classe

text: de

text: morphismes

text: de

text: et

text: supposons

text: que

text: tout

text: morphisme

text: α

text: β

text: α

text: C

text: de

text: envoie

text: dans

text: Soit

text: Alors

text: on

text: a

text: une

text: A

text: S

text: S

text: S

text: Im(S

text: F

text: l(

text: α

text: β

text: α

text: α

text: ivalence

text: canonique

text: de

text: catries

text: C

text: C

text: S

text: l−i→m

text: S

text: α

text: α

text: C

text: b)

text: Avec

text: les

text: notations

text: prdentes,

text: soient

text: une

text: autre

text: catrie

text: et,

text: pour

text: tout

text: α

text: C

text: C

text: C

text: un

text: foncteur.

text: Soit

text: une

text: classe

text: de

text: morphismes

text: de

text: On

text: f

text: S

text: α

text: α

text: suppose

text: que

text: les

text: envoient

text: dans

text: et

text: sont

text: compatibles

text: aux

text: foncteurs

text: f

text: S

text: S

text: α

text: α

text: C

text: C

text: de

text: transition,

text: d’o

text: un

text: foncteur

text: induit

text: f

text: S

text: S

text: On

text: suppose

text: de

text: plus

text: que:

text: (i)

text: Pour

text: tout

text: admet

text: un

text: adjoint

text: uche

text: qui

text: envoie

text: dans

text: α

text: f

text: S

text: S

text: α

text: α

text: C

text: C

text: (ii)

text: Pour

text: tout

text: morphisme

text: α

text: β,

text: le

text: foncteur

text: S

text: S

text: est

text: α

text: α

text: β

text: β

text: une

text: ivalence

text: de

text: catries.

text: Alors

text: admet

text: un

text: adjoint

text: uche.

text: f

text: Démonstration.

text: a)

text: Les

text: deux

text: membres

text: de

text: l’ivalence

text: ont

text: chacun

text: la

text: pro-

text: pri

text: de

text: l’autre.

text: b)

text: rlte

text: immatement

text: de

text: a).

text: Dans

text: le

text: dernier

text: ncous

text: nous

text: cantonnons

text: au

text: cadre

text: des

text: thies

text: motiviques

text: pour

text: ter

text: de

text: perdre

text: le

text: lecteur

text: dans

text: une

text: hypergralitutile.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 26 BRUNO KAHN est bijectif, où d = dim Y . La proposition 1.7.2 de [26] nous ramène au cas où est régulière. Soit une clôture séparable de , la K/k K K k s s fermeture séparable de dans , et . k K G = Gal(K /K) G = Gal(k /k) s K s k s On a un diagramme commutatif (cid:120) (cid:120) Z × −−∼−→ Z ×    ∼d ((X Y ) , A) ∼d ((X Y ) , A)GK K Ks Z × −−∼−→ Z × ∼d (X Y, A) ∼d ((X Y ) , A)Gk ks où les ﬂèches horizontales sont des isomorphismes. De plus, il est bien connu que la ﬂèche Z × → Z × ∼d ((X Y ) , A) ∼d ((X Y ) , A) ks Ks ∼ est un isomorphisme, puisque est moins ﬁne que l’équivalence algé- → brique [29, p. 448]. Comme K/k est régulière, G G est surjectif, K k donc la ﬂèche verticale de droite du diagramme est bijective, ainsi que celle de gauche. ∼ À partir de maintenant, (équivalence numérique) et ; = num A = Q on l’omet des notations. → Théorème 5.6. — Pour toute extension primaire f : k K, le fonc- ∗ → teur admet un adjoint à gauche et un f : Mot (k) Mot (K) f num num (cid:93) adjoint à droite f∗; ces adjoints sont canoniquement isomorphes. Le même énoncé vaut en remplaçant Mot par Moteﬀ (motifs eﬀec- num num tifs). Démonstration. — Résulte du théorème de semi-simplicité de Jannsen [17] et des propositions 5.3 et 5.5. ∈ × Proposition 5.7. — a) Pour tout (M, N) Mot (k) Mot (K), num num on a la formule de projection ⊗ ∗ (cid:39) ⊗ f (N f M) f N M. (cid:93) (cid:93) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BRUNO

text: KAHN

text: est

text: bijectif,

text: où

text: d

text: dim

text: Y

text: La

text: proposition

text: de

text: nous

text: ramène

text: au

text: cas

text: où

text: est

text: régulière.

text: Soit

text: une

text: clôture

text: séparable

text: de

text: la

text: K/k

text: K

text: K

text: k

text: s

text: s

text: fermeture

text: séparable

text: de

text: dans

text: et

text: k

text: K

text: G

text: Gal(K

text: /K)

text: G

text: Gal(k

text: /k)

text: s

text: K

text: s

text: k

text: s

text: On

text: a

text: un

text: diagramme

text: commutatif

text: Z

text: Z

text: ∼d

text: ((X

text: Y

text: A)

text: ∼d

text: ((X

text: Y

text: A)GK

text: K

text: Ks

text: Z

text: Z

text: ∼d

text: (X

text: Y,

text: A)

text: ∼d

text: ((X

text: Y

text: A)Gk

text: ks

text: où

text: les

text: ﬂèches

text: horizontales

text: sont

text: des

text: isomorphismes.

text: De

text: plus,

text: il

text: est

text: bien

text: connu

text: que

text: la

text: ﬂèche

text: Z

text: Z

text: ∼d

text: ((X

text: Y

text: A)

text: ∼d

text: ((X

text: Y

text: A)

text: ks

text: Ks

text: est

text: un

text: isomorphisme,

text: puisque

text: est

text: moins

text: ﬁne

text: que

text: l’équivalence

text: algé-

text: brique

text: p.

text: Comme

text: K/k

text: est

text: régulière,

text: G

text: G

text: est

text: surjectif,

text: K

text: k

text: donc

text: la

text: ﬂèche

text: verticale

text: de

text: droite

text: du

text: diagramme

text: est

text: bijective,

text: ainsi

text: que

text: celle

text: de

text: gauche.

text: À

text: partir

text: de

text: maintenant,

text: (équivalence

text: numérique)

text: et

text: num

text: A

text: Q

text: on

text: l’omet

text: des

text: notations.

text: Théorème

text: Pour

text: toute

text: extension

text: primaire

text: f

text: k

text: K,

text: le

text: fonc-

text: teur

text: admet

text: un

text: adjoint

text: à

text: gauche

text: et

text: un

text: f

text: Mot

text: (k)

text: Mot

text: (K)

text: f

text: num

text: num

text: adjoint

text: à

text: droite

text: f∗;

text: ces

text: adjoints

text: sont

text: canoniquement

text: isomorphes.

text: Le

text: même

text: énoncé

text: vaut

text: en

text: remplaçant

text: Mot

text: par

text: Moteﬀ

text: (motifs

text: eﬀec-

text: num

text: num

text: tifs).

text: Démonstration.

text: Résulte

text: du

text: théorème

text: de

text: semi-simplicité

text: de

text: Jannsen

text: et

text: des

text: propositions

text: et

text: Proposition

text: a)

text: Pour

text: tout

text: (M,

text: N)

text: Mot

text: (k)

text: Mot

text: (K),

text: num

text: num

text: on

text: a

text: la

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text: f

text: (N

text: f

text: M)

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text: MOTIFS ET ADJOINTS 27 ∈ ∈ b) Pour tout et pour tout , on a un isomorphisme M Mot (K) n Z num (cid:39) f (M(n)) (f M)(n). (cid:93) (cid:93) Démonstration. — a) Cela résulte de la proposition 3.6. b) C’est le cas L⊗ L particulier N = n où est le motif de Lefschetz. Pour énoncer le résultat, nous avons besoin de nous restreindre à une sous-théorie pleine convenable de (conjecturalement égale à Mot num ): Mot num Pour une des théories motiviques homologiques de , notons comme hom ∗ dans [2, rem. 1] Mot (k) la sous-catégorie pleine de Mot (k) formé num num des objets dont un relevé dans a tous ses projecteurs de Kün- Mot (k) hom ∗ neth algébriques. D’après [2, prop. 6], Mot (k) ne dépend pas du num choix de la théorie homologique de . (La preuve de cette proposition, in- complète en caractéristique > 0 dans [2], est complétée dans [1, app. B].) Les projecteurs de Künneth déﬁnissent une -graduation sur le foncteur Z ∗ identique de . Il est clair que l’extension des scalaires respecte Mot ∗ num et la graduation ci-dessus. Mot num → Proposition 5.8. — Soit f : k K une extension primaire. Alors ∗ ⊂ ∗ , et respecte les -graduations. f Mot (K) Mot (k) f Z (cid:93) num num (cid:93) ∗ ∗ (cid:39) (Comme est pleinement ﬁdèle, et l’inclusion est une f f f Id (cid:93) égalité.) ∈ Démonstration. — Il est équivalent de montrer que, si M Mot (K) num est pur de poids , alors est pur de poids . On peut supposer w f M w (cid:93) simple. Si , il n’y a rien à démontrer. Sinon, d’après la M f M = 0 (cid:93) (cid:39) ∗ (cid:39) preuve de la proposition 5.3, avec simple, et . Soit M f S S f M S ∈ ∗ ∈(cid:93) ˜ un objet s’envoyant sur . Alors ˜ S Mot (k) S f S Mot (K) hom hom s’envoie sur , et M ∗ ∗ (cid:39) ∗ ⊗ ˜ ˜ H (f S) H (S) F (K). F(k) (cid:54) Ceci montre que Hi(S˜) = 0 pour i = w, donc, par déﬁnition, que est pur de poids . S = f M w (cid:93) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MOTIFS

text: ET

text: ADJOINTS

text: b)

text: Pour

text: tout

text: et

text: pour

text: tout

text: on

text: a

text: un

text: isomorphisme

text: M

text: Mot

text: (K)

text: n

text: Z

text: num

text: f

text: (M(n))

text: (f

text: M)(n).

text: Démonstration.

text: a)

text: Cela

text: résulte

text: de

text: la

text: proposition

text: b)

text: C’est

text: le

text: cas

text: L⊗

text: L

text: particulier

text: N

text: n

text: où

text: est

text: le

text: motif

text: de

text: Lefschetz.

text: Pour

text: énoncer

text: le

text: résultat,

text: nous

text: avons

text: besoin

text: de

text: nous

text: restreindre

text: à

text: une

text: sous-théorie

text: pleine

text: convenable

text: de

text: (conjecturalement

text: égale

text: à

text: Mot

text: num

text: Mot

text: num

text: Pour

text: une

text: des

text: théories

text: motiviques

text: homologiques

text: de

text: notons

text: comme

text: hom

text: dans

text: rem.

text: Mot

text: (k)

text: la

text: sous-catégorie

text: pleine

text: de

text: Mot

text: (k)

text: formé

text: num

text: num

text: des

text: objets

text: dont

text: un

text: relevé

text: dans

text: a

text: tous

text: ses

text: projecteurs

text: de

text: Kün-

text: Mot

text: (k)

text: hom

text: neth

text: algébriques.

text: D’après

text: prop.

text: Mot

text: (k)

text: ne

text: dépend

text: pas

text: du

text: num

text: choix

text: de

text: la

text: théorie

text: homologique

text: de

text: (La

text: preuve

text: de

text: cette

text: proposition,

text: in-

text: complète

text: en

text: caractéristique

text: dans

text: est

text: complétée

text: dans

text: app.

text: B].)

text: Les

text: projecteurs

text: de

text: Künneth

text: déﬁnissent

text: une

text: -graduation

text: sur

text: le

text: foncteur

text: Z

text: identique

text: de

text: Il

text: est

text: clair

text: que

text: l’extension

text: des

text: scalaires

text: respecte

text: Mot

text: num

text: et

text: la

text: graduation

text: ci-dessus.

text: Mot

text: num

text: Proposition

text: Soit

text: f

text: k

text: K

text: une

text: extension

text: primaire.

text: Alors

text: et

text: respecte

text: les

text: -graduations.

text: f

text: Mot

text: (K)

text: Mot

text: (k)

text: f

text: Z

text: num

text: num

text: (Comme

text: est

text: pleinement

text: ﬁdèle,

text: et

text: l’inclusion

text: est

text: une

text: f

text: f

text: f

text: Id

text: égalité.)

text: Démonstration.

text: Il

text: est

text: équivalent

text: de

text: montrer

text: que,

text: si

text: M

text: Mot

text: (K)

text: num

text: est

text: pur

text: de

text: poids

text: alors

text: est

text: pur

text: de

text: poids

text: On

text: peut

text: supposer

text: w

text: f

text: M

text: w

text: simple.

text: Si

text: il

text: n’y

text: a

text: rien

text: à

text: démontrer.

text: Sinon,

text: d’après

text: la

text: M

text: f

text: M

text: preuve

text: de

text: la

text: proposition

text: avec

text: simple,

text: et

text: Soit

text: M

text: f

text: S

text: S

text: f

text: M

text: S

text: un

text: objet

text: s’envoyant

text: sur

text: Alors

text: S

text: Mot

text: (k)

text: S

text: f

text: S

text: Mot

text: (K)

text: hom

text: hom

text: s’envoie

text: sur

text: et

text: M

text: H

text: (f

text: S)

text: H

text: (S)

text: F

text: (K).

text: F(k)

text: Ceci

text: montre

text: que

text: Hi(S˜)

text: pour

text: i

text: w,

text: donc,

text: par

text: déﬁnition,

text: que

text: est

text: pur

text: de

text: poids

text: S

text: f

text: M

text: w

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: 30 BRUNO KAHN d) Résulte soit de la démonstration de c), soit de la compatibilité des adjoints à la composition. ∈ × e) Soit (M, N) Moto (K) Moto (k). D’après le corollaire 5.11, num num on a un isomorphisme dans Moteﬀ (k) num ⊗ ∗ (cid:39) ⊗ feﬀ(i M f eﬀi N) (feﬀi M) i N. (cid:93) K k (cid:93) K k On obtient la formule voulue en appliquant à cet isomorphisme, en π k utilisant que commute au produit tensoriel et en utilisant c). π k Conjecturalement, ivalences homologique et numque coident; les ad- joints uche et oite du foncteur d’extension des scalaires devraient donc exister pour les motifs homologiques. Voici tout ce que je sais dire: (cid:54) Proposition 5.13. — Soient p = 0 ou un nombre premier, l = p un nombre premier, et la thie motivique de la dnition 2.3. Alors, pour F →l toute extension de type ﬁni dans , le foncteur d’“extension f : k K corps ∗ → p des scalaires” admet un adjoint uche et un adjoint f : F (k) F (K) f l l (cid:93) oite f∗. Ces adjoints sont compatibles aux poids et respectent la sous-thie ⇒ F ss de la remarque 2.4; le morphisme canonique f∗ f , restreint ss(K), l (cid:93) l est un isomorphisme. Démonstration. — On se ram au cas o est de type ﬁni sur son sous- k corps premier, puis en utilisant le th 4.7 au cas o est rli. Soit K/k → H = Ker(G G ): on voit tout de suite que f et f∗ sont donnar K k (cid:93) f V = V , f∗V = V H. (cid:93) H La compatibilitx poids est dente, ainsi que le fait que ces foncteurs envoient F ss(K) dans F ss(k) (une sous-reprntation et une reprntation l l quotient d’une reprntation semi-simple sont semi-simples). Malheureusement, la conjecture de Tate ne semble pas suﬃsante pour en dire l’existence d’adjoints pour les motifs homologiques: outre le probl (cid:48) de descendre les coeﬃcients de k Q , seposeunprobld essentiellesurjectivituand estuncorpsfini, onpeutlerudregrilIetauthdeHonda : l eneffet, lesvaleurspropresdeF robeniusopntsurVH sont des nombres de Weil, comme on le voit par un argument de spalisation. Si est un corps k de nombres, on peut peut-e utiliser la conjecture de Fontaine-Mazur (en plus de la conjecture de Tate!) pour conclure. Pour d’autres corps , je k ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BRUNO

text: KAHN

text: d)

text: Résulte

text: soit

text: de

text: la

text: démonstration

text: de

text: c),

text: soit

text: de

text: la

text: compatibilité

text: des

text: adjoints

text: à

text: la

text: composition.

text: e)

text: Soit

text: (M,

text: N)

text: Moto

text: (K)

text: Moto

text: (k).

text: D’après

text: le

text: corollaire

text: num

text: num

text: on

text: a

text: un

text: isomorphisme

text: dans

text: Moteﬀ

text: (k)

text: num

text: feﬀ(i

text: M

text: f

text: eﬀi

text: N)

text: (feﬀi

text: M)

text: i

text: N.

text: K

text: k

text: K

text: k

text: On

text: obtient

text: la

text: formule

text: voulue

text: en

text: appliquant

text: à

text: cet

text: isomorphisme,

text: en

text: π

text: k

text: utilisant

text: que

text: commute

text: au

text: produit

text: tensoriel

text: et

text: en

text: utilisant

text: c).

text: π

text: k

text: Conjecturalement,

text: ivalences

text: homologique

text: et

text: numque

text: coident;

text: les

text: ad-

text: joints

text: uche

text: et

text: oite

text: du

text: foncteur

text: d’extension

text: des

text: scalaires

text: devraient

text: donc

text: exister

text: pour

text: les

text: motifs

text: homologiques.

text: Voici

text: tout

text: ce

text: que

text: je

text: sais

text: dire:

text: Proposition

text: Soient

text: p

text: ou

text: un

text: nombre

text: premier,

text: l

text: p

text: un

text: nombre

text: premier,

text: et

text: la

text: thie

text: motivique

text: de

text: la

text: dnition

text: Alors,

text: pour

text: F

text: →l

text: toute

text: extension

text: de

text: type

text: ﬁni

text: dans

text: le

text: foncteur

text: d’“extension

text: f

text: k

text: K

text: corps

text: p

text: des

text: scalaires”

text: admet

text: un

text: adjoint

text: uche

text: et

text: un

text: adjoint

text: f

text: F

text: (k)

text: F

text: (K)

text: f

text: l

text: l

text: oite

text: f∗.

text: Ces

text: adjoints

text: sont

text: compatibles

text: aux

text: poids

text: et

text: respectent

text: la

text: sous-thie

text: F

text: ss

text: de

text: la

text: remarque

text: le

text: morphisme

text: canonique

text: f∗

text: f

text: restreint

text: ss(K),

text: l

text: l

text: est

text: un

text: isomorphisme.

text: Démonstration.

text: On

text: se

text: ram

text: au

text: cas

text: o

text: est

text: de

text: type

text: ﬁni

text: sur

text: son

text: sous-

text: k

text: corps

text: premier,

text: puis

text: en

text: utilisant

text: le

text: th

text: au

text: cas

text: o

text: est

text: rli.

text: Soit

text: K/k

text: H

text: Ker(G

text: G

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text: voit

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text: de

text: suite

text: que

text: f

text: et

text: f∗

text: sont

text: donnar

text: K

text: k

text: f

text: V

text: V

text: f∗V

text: V

text: H.

text: H

text: La

text: compatibilitx

text: poids

text: est

text: dente,

text: ainsi

text: que

text: le

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text: ces

text: foncteurs

text: envoient

text: F

text: ss(K)

text: dans

text: F

text: ss(k)

text: (une

text: sous-reprntation

text: et

text: une

text: reprntation

text: l

text: l

text: quotient

text: d’une

text: reprntation

text: semi-simple

text: sont

text: semi-simples).

text: Malheureusement,

text: la

text: conjecture

text: de

text: Tate

text: ne

text: semble

text: pas

text: suﬃsante

text: pour

text: en

text: dire

text: l’existence

text: d’adjoints

text: pour

text: les

text: motifs

text: homologiques:

text: outre

text: le

text: probl

text: de

text: descendre

text: les

text: coeﬃcients

text: de

text: k

text: Q

text: seposeunprobld

text: essentiellesurjectivituand

text: estuncorpsfini,

text: onpeutlerudregrilIetauthdeHonda

text: l

text: eneffet,

text: lesvaleurspropresdeF

text: robeniusopntsurVH

text: sont

text: des

text: nombres

text: de

text: Weil,

text: comme

text: on

text: le

text: voit

text: par

text: un

text: argument

text: de

text: spalisation.

text: Si

text: est

text: un

text: corps

text: k

text: de

text: nombres,

text: on

text: peut

text: peut-e

text: utiliser

text: la

text: conjecture

text: de

text: Fontaine-Mazur

text: (en

text: plus

text: de

text: la

text: conjecture

text: de

text: Tate!)

text: pour

text: conclure.

text: Pour

text: d’autres

text: corps

text: je

text: k

text: ##LTLine##

text: ##LTLine##

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text: 4 BRUNO KAHN Voir th. 5.6, prop. 5.7 et prop. 5.8. En se restreignant aux motifs eﬀectifs de poids , on retrouve la -trace et la -image des vars 1 K/k K/k abennes (oge pr: ex. 5.9. M − Théorème 2. — Soit p = 1 ou un nombre premier, et soit = S 1 Sm, − b Chowo( , Z[1/p]) ou DMo la thie des varis lisses birationnelles [25], gm des motifs de Chow birationnels eﬃcients Z[1/p] [26] ou des motifs trian- guliques birationnels [27]. Alors, pour toute extension de type ﬁni srable de corps d’exposant caractstique , le foncteur K/k p M → M (k) (K) admet un adjoint uche. Voir th. 6.1, 6.6 et 6.9. Une application de ces rltats a d donnans [26, prop. 5.1.4]. Pour le corollaire suivant, on dit qu’une -vari propre est universellement - k X R ⊗ triviale (resp. universellement -triviale) si est rit point CH A X(E)/R 0 ⊗ −→∼ pour toute extension , o est la -ivalence (resp. si E/k R R CH (X ) A 0 E pour toute extension , o est un anneau commutatif). Soit A E/k A p = 1 ou un nombre premier. X (cid:57)(cid:57)(cid:75) Corollaire 1. — Soit π : Y une application rationnelle domi- nante de -varis propres, lisses et connexes, dont la restriction ouvert k de dnition est propre. Supposons que la ﬁbre grique de soit uni- X π ⊗ versellement -triviale (resp. universellement -triviale, o est R CH A A 0 X −→∼ une -alge). Alors induit un isomorphisme Z[1/p] π (L)/R Y (L)/R X ⊗ −→∼ ⊗ (resp. ) pour toute extension de corps CH ( ) A CH (Y ) A L/k 0 0 d’exposant caractstique . p Voir cor. 6.3 et 6.8. En particulier, si et sont universellement - X Y R X triviales, il en est de m de : c’est une variante d’un cbre th de Graber, Harris et Starr [11, cor. 1.3], mais ici le corps de base est quelconque. Tandis que le prnt contexte catrique n’est pas trop diﬃcile iter pour traiter le cas de , en utilisant le lemme de dacement pour les -cycles CH 0 0 [41], cela semble moins dent dans le cas des classes de R-ivalence. Le th qui suit construit un “motif de Tate-Šafarevič”, introduit dans [20] de mani diﬀnte (voir remarque 9.4). Notons Choweﬀ (resp. Moteﬀ) H la thie des motifs de Chow eﬀectifs (resp. celle des motifs eﬀectifs modulo l’ivalence homologique relative e cohomologie de Weil ). H

text: BRUNO

text: KAHN

text: Voir

text: th.

text: prop.

text: et

text: prop.

text: En

text: se

text: restreignant

text: aux

text: motifs

text: eﬀectifs

text: de

text: poids

text: on

text: retrouve

text: la

text: -trace

text: et

text: la

text: -image

text: des

text: vars

text: K/k

text: K/k

text: abennes

text: (oge

text: pr:

text: ex.

text: M

text: Théorème

text: Soit

text: p

text: ou

text: un

text: nombre

text: premier,

text: et

text: soit

text: S

text: Sm,

text: b

text: Chowo(

text: ou

text: DMo

text: la

text: thie

text: des

text: varis

text: lisses

text: birationnelles

text: gm

text: des

text: motifs

text: de

text: Chow

text: birationnels

text: eﬃcients

text: ou

text: des

text: motifs

text: trian-

text: guliques

text: birationnels

text: Alors,

text: pour

text: toute

text: extension

text: de

text: type

text: ﬁni

text: srable

text: de

text: corps

text: d’exposant

text: caractstique

text: le

text: foncteur

text: K/k

text: p

text: M

text: M

text: (k)

text: (K)

text: admet

text: un

text: adjoint

text: uche.

text: Voir

text: th.

text: et

text: Une

text: application

text: de

text: ces

text: rltats

text: a

text: d

text: donnans

text: prop.

text: Pour

text: le

text: corollaire

text: suivant,

text: on

text: dit

text: qu’une

text: -vari

text: propre

text: est

text: universellement

text: k

text: X

text: R

text: triviale

text: (resp.

text: universellement

text: -triviale)

text: si

text: est

text: rit

text: point

text: CH

text: A

text: X(E)/R

text: pour

text: toute

text: extension

text: o

text: est

text: la

text: -ivalence

text: (resp.

text: si

text: E/k

text: R

text: R

text: CH

text: (X

text: A

text: E

text: pour

text: toute

text: extension

text: o

text: est

text: un

text: anneau

text: commutatif).

text: Soit

text: A

text: E/k

text: A

text: p

text: ou

text: un

text: nombre

text: premier.

text: X

text: Corollaire

text: Soit

text: π

text: Y

text: une

text: application

text: rationnelle

text: domi-

text: nante

text: de

text: -varis

text: propres,

text: lisses

text: et

text: connexes,

text: dont

text: la

text: restriction

text: ouvert

text: k

text: de

text: dnition

text: est

text: propre.

text: Supposons

text: que

text: la

text: ﬁbre

text: grique

text: de

text: soit

text: uni-

text: X

text: π

text: versellement

text: -triviale

text: (resp.

text: universellement

text: -triviale,

text: o

text: est

text: R

text: CH

text: A

text: A

text: X

text: une

text: -alge).

text: Alors

text: induit

text: un

text: isomorphisme

text: π

text: (L)/R

text: Y

text: (L)/R

text: X

text: (resp.

text: pour

text: toute

text: extension

text: de

text: corps

text: CH

text: A

text: CH

text: (Y

text: A

text: L/k

text: d’exposant

text: caractstique

text: p

text: Voir

text: cor.

text: et

text: En

text: particulier,

text: si

text: et

text: sont

text: universellement

text: X

text: Y

text: R

text: X

text: triviales,

text: il

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text: de

text: m

text: de

text: c’est

text: une

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text: d’un

text: cbre

text: th

text: de

text: Graber,

text: Harris

text: et

text: Starr

text: cor.

text: mais

text: ici

text: le

text: corps

text: de

text: base

text: est

text: quelconque.

text: Tandis

text: que

text: le

text: prnt

text: contexte

text: catrique

text: n’est

text: pas

text: trop

text: diﬃcile

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text: le

text: lemme

text: de

text: dacement

text: pour

text: les

text: -cycles

text: CH

text: cela

text: semble

text: moins

text: dent

text: dans

text: le

text: cas

text: des

text: classes

text: de

text: R-ivalence.

text: Le

text: th

text: qui

text: suit

text: construit

text: un

text: “motif

text: de

text: Tate-Šafarevič”,

text: introduit

text: dans

text: de

text: mani

text: diﬀnte

text: (voir

text: remarque

text: Notons

text: Choweﬀ

text: (resp.

text: Moteﬀ)

text: H

text: la

text: thie

text: des

text: motifs

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text: Chow

text: eﬀectifs

text: (resp.

text: celle

text: des

text: motifs

text: eﬀectifs

text: modulo

text: l’ivalence

text: homologique

text: relative

text: e

text: cohomologie

text: de

text: Weil

text: H


text: MOTIFS ET ADJOINTS 31 ne sais pas. Peut-on remplacer ces rustines un peu fortes par, disons, la conjecture de Tate gralis 6. Catégories birationnelles → Théorème 6.1. — Soit f : k K une extension de type ﬁni srable. Soit l’ensemble des morphismes birationnels. Alors le foncteur S b ∗ − → − f : S 1 Sm(k) S 1 Sm(K) b b ∈ − X admet un adjoint à gauche f . Pour X S 1 Sm(K), on a f X = X (cid:93) b (cid:93) pour tout -mod lisse de . k X Nous allons donner deux démonstrations. La première utilise la réso- lution des singularités de Hironaka, et n’est donc valable qu’en caracté- ristique zéro; la seconde est valable en toute caractéristique. Première démonstration. — D’apr[th. 6.6.3]birat, on a une bijection ca- nonique − (cid:39) (6.1) S 1 Sm(k)(V, W ) W (k(V ))/R b pour tout couple de -varis lisses, avec propre, o le membre (V, W ) k W de droite est l’ensemble des classes de -ivalence. Supposons maintenant R k de caractstique 0. D’après [24, prop. 8.5], on peut remplacer les ∈ variétés lisses Sm par les variétés projectives lisses Smproj. Soit X − S 1 Smproj(K). On veut montrer que le foncteur b (cid:55)→ Y Hom(X, Y ) K − de S 1 Smproj(k) vers la catégorie des ensembles est coreprésentable. b D’après (6.1), la valeur de ce foncteur sur est Y Y (K(X))/R K X où désigne la R-équivalence. Soit une -variété projective lisse telle R k X (cid:39) que (résolution des singularités de Hironaka). On a alors k( ) K(X) (cid:39) X Y (K(X))/R Y (k( ))/R K

text: MOTIFS

text: ET

text: ADJOINTS

text: ne

text: sais

text: pas.

text: Peut-on

text: remplacer

text: ces

text: rustines

text: un

text: peu

text: fortes

text: par,

text: disons,

text: la

text: conjecture

text: de

text: Tate

text: gralis

text: Catégories

text: birationnelles

text: Théorème

text: Soit

text: f

text: k

text: K

text: une

text: extension

text: de

text: type

text: ﬁni

text: srable.

text: Soit

text: l’ensemble

text: des

text: morphismes

text: birationnels.

text: Alors

text: le

text: foncteur

text: S

text: b

text: f

text: S

text: Sm(k)

text: S

text: Sm(K)

text: b

text: b

text: X

text: admet

text: un

text: adjoint

text: à

text: gauche

text: f

text: Pour

text: X

text: S

text: Sm(K),

text: on

text: a

text: f

text: X

text: X

text: b

text: pour

text: tout

text: -mod

text: lisse

text: de

text: k

text: X

text: Nous

text: allons

text: donner

text: deux

text: démonstrations.

text: La

text: première

text: utilise

text: la

text: réso-

text: lution

text: des

text: singularités

text: de

text: Hironaka,

text: et

text: n’est

text: donc

text: valable

text: qu’en

text: caracté-

text: ristique

text: zéro;

text: la

text: seconde

text: est

text: valable

text: en

text: toute

text: caractéristique.

text: Première

text: démonstration.

text: D’apr[th.

text: on

text: a

text: une

text: bijection

text: ca-

text: nonique

text: S

text: Sm(k)(V,

text: W

text: W

text: (k(V

text: ))/R

text: b

text: pour

text: tout

text: couple

text: de

text: -varis

text: lisses,

text: avec

text: propre,

text: o

text: le

text: membre

text: (V,

text: W

text: k

text: W

text: de

text: droite

text: est

text: l’ensemble

text: des

text: classes

text: de

text: -ivalence.

text: Supposons

text: maintenant

text: R

text: k

text: de

text: caractstique

text: D’après

text: prop.

text: on

text: peut

text: remplacer

text: les

text: variétés

text: lisses

text: Sm

text: par

text: les

text: variétés

text: projectives

text: lisses

text: Smproj.

text: Soit

text: X

text: S

text: Smproj(K).

text: On

text: veut

text: montrer

text: que

text: le

text: foncteur

text: b

text: Y

text: Hom(X,

text: Y

text: K

text: de

text: S

text: Smproj(k)

text: vers

text: la

text: catégorie

text: des

text: ensembles

text: est

text: coreprésentable.

text: b

text: D’après

text: la

text: valeur

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text: ce

text: foncteur

text: sur

text: est

text: Y

text: Y

text: (K(X))/R

text: K

text: X

text: où

text: désigne

text: la

text: R-équivalence.

text: Soit

text: une

text: -variété

text: projective

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text: k

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text: que

text: (résolution

text: des

text: singularités

text: de

text: Hironaka).

text: On

text: a

text: alors

text: k(

text: K(X)

text: X

text: Y

text: (K(X))/R

text: Y

text: (k(

text: ))/R

text: K


text: 32 BRUNO KAHN X et notre foncteur est coreprésenté par . ⇐⇒ Seconde démonstration. — Rappelons que “srable de type ﬁni” “admet un mod lisse” [25, lemma 4.8.1]. Pour tout mod lisse U de K/k, soit la catégorie des -schémas lisses séparés de type ﬁni. On a Sm(U) U − −→∼ 2 l−i→m Sm(U) Sm(K) U → (limite prise sur les immersions ouvertes). Si est le p : U Spec k U morphisme structural, on a un couple de foncteurs adjoints: ∗ (cid:28) p : Sm(k) Sm(U) : (p ) U U (cid:93) donnés par l’extension et la restriction des scalaires ( est adjoint (p ) ∗ (cid:48) → U (cid:93) uche de ). De m, si est une immersion ouverte, on a un p j : U U U couple de foncteurs adjoints ∗ (cid:28) (cid:48) j : Sm(U) Sm(U ) : j . (cid:93) Tous ces foncteurs respectent les morphismes birationnels. D’apra pro- position 3.7, pour conclure il suﬃt de dntrer que les foncteurs − ∗ − → − (cid:48) (6.2) S 1j : S 1 Sm(U) S 1 Sm(U ) b b b induits par les immersions ouvertes sont des ivalences de catries. j ∗ − − ∗ Mais c’est dent: l’adjonction (j , j ) induit une adjonction (S 1j , S 1j ) (cid:93) b (cid:93) b et il suﬃt de montrer que son unit sa conitnt des isomorphismes naturels. ⇒ ∗ ∗ ⇒ Or l’unitd est d un isomorphisme naturel, et la conit (cid:48) j j j Sm(U ) (cid:93) ∈ (cid:93) est une immersion ouverte quand on l’lue sur tout . Id X Sm(U) Sm(U) La derni assertion du th est dente par construction. Pour le corollaire suivant, soient une extension srable de type ﬁni K/k et deux -varis propres et lisses munies de -mods propres et X , X K k 1X 2 X (cid:57)(cid:57)(cid:75) lisses et . Soit une -application rationnelle: rap- α : X X K 1 2 1 2 − pelons que α dnit un morphisme dans S 1 Sm(K), donc une application → b α∗(L) : X (L)/R X (L)/R pour toute extension srable de type ﬁni 1 2 L/K ([7, prop. 10] en caractstique z, [25, 6.3 et cor. 6.6.6] en gral). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BRUNO

text: KAHN

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text: et

text: notre

text: foncteur

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text: démonstration.

text: Rappelons

text: que

text: “srable

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text: type

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text: “admet

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text: Si

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text: ces

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text: D’apra

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text: j

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text: MOTIFS ET ADJOINTS 37 Soit une -alge commutative, o ou un nombre premier. A Z[1/p] p = 1 Pour tout corps k d’exposant caractstique p, on a des foncteurs [27, cor. 3.4.2 et th. 4.2.2] − → → S 1 Sm(k) Chowo(k, A) DMo (k, A) b gm (cid:54) qui dnissent des morphismes de thies motiviques (sur si , corps p = 1 p sur si ) corps p = 1 0 − → − → − (6.4) S 1 Sm Chowo( , A) DMo ( , A). b gm Proposition 6.11. — Les morphismes de changement de base associ (6.4) via les ths 6.1, 6.6 et 6.9 sont des isomorphismes pour toute exten- sion srable . K/k Démonstration. — Étant donns dnstrations des trois ths cit il suﬃt de le vﬁer pour les adjoints relatifs mod lisse U de K/k (en remplat Choweﬀ − par Cor ). Dans le cas de S 1 Sm et de Cor , l’adjoint uche envoie f b f un -sch-lisse sur lui-m; c’est encore le cas pour le motif de dans U X X DMeﬀ (U). gm 7. Non existence d’adjoints Après avoir montré l’existence d’adjoints dans des situations variées, il peut être instructif de donner des exemples de non existence. En voici de deux sortes, avec des catégories de motifs purs: On a: → Lemme 7.1. — Si f : k K est une telle extension, l’adjoint à gauche ∗ → ∈ de n’est pas déﬁni en . f f : Mot (k) Mot (K) 1 Mot (K) (cid:93) num num num Démonstration. — Pour simpliﬁer, supposons que soit la clôture sé- K parable de (on pourrait généraliser l’argument). Il s’agit de voir que le k foncteur (cid:51) (cid:55)→ ∗ Mot (k) M Mot (K)(1, f M) num num ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MOTIFS

text: ET

text: ADJOINTS

text: Soit

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text: -alge

text: commutative,

text: o

text: ou

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text: premier.

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text: p

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text: Sm

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text: b

text: gm

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text: ths

text: et

text: sont

text: des

text: isomorphismes

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text: exten-

text: sion

text: srable

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text: Démonstration.

text: Étant

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