TITLE: A geometric characterization for arithmetic genus
QUESTION [37 upvotes]: Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others):

the arithmetic genus of $X$
the constant coefficient of the Hilbert polynomial of $X$
$\chi(X, \mathscr{O}_X)$
the "Todd genus" $\int_X \operatorname{td}(T_X)$, where $T_X$ is the tangent bundle of $X$ and $\operatorname{td}$ denotes the Todd class.


Is there a geometric characterization for any of these numbers?

If I understand correctly, characteristic classes (and in particular, Todd classes) can be defined entirely from the topology of $X$, or at least its structure as a smooth manifold. [Edit: This is not true--see the answer of "anonymous." If I understand correctly, the Todd class of a complex vector bundle is a smooth invariant. However, different complex structures on the same real manifold $X$ can give rise to non-isomorphic complex vector bundle structures on $T_X$; in fact, a complex vector bundle structure on $T_X$ is, by definition, an almost complex structure on a real manifold $X$.]  Thus, in some sense, item 4 provides a "geometric characterization" for the arithmetic genus of $X$ (and the other items on the list).  However, I personally find this description so far abstracted from actual geometric properties of $X$ as to be hardly geometric at all. (If anyone disagrees with me and can articulate a geometric intuition for the Todd genus, that would be a reasonable answer.) 
By comparison, I do consider the following characterizations of various properties "geometric":

The self-intersection number of the diagonal embedding of $X$ into $X \times X$. (the Euler characteristic)
The number of points in which a general linear space of complementary dimension meets $X \subset \mathbb P^n$. (the degree of $X \hookrightarrow \mathbb P^n$)
The genus of the curve $X \cap L$, where $L$ is a general linear space of dimension one greater than $\operatorname{codim} X$. (I don't know of a standard name for this, but in a particular sense, it is one of the coefficients of the Hilbert polynomial of $X$.)
The maximum number of copies of $S^1$ that can be removed from $X$ without disconnecting it. (the genus of $X$ if $X$ is a smooth curve, i.e., Riemann surface)

Note that either the first or the last point gives a geometric characterization for (information equivalent to) the genus of a curve. Without one of these, I would not consider the third bullet a "geometric characterization" of anything.  In a way, this provides part of my motivation for asking this question. Let $L_k$ be a general linear space of codimension $k$ in $\mathbb P^n$. Unless I am mistaken, knowing the Hilbert polynomial for $X$ is equivalent to knowing the arithmetic genus of $X \cap L_k$, for every $k \leq n$ such that this intersection is nonempty, via the formula
$$ \chi(\mathscr O_X(n)) = \sum_{k \ge 0} \chi(\mathscr O_{X \cap L_k}) \binom{n+k-1}{k}\;\text.$$
Thus, a geometric characterization for arithmetic genus would automatically give a geometric characterization for the Hilbert polynomial. (Again, in some sense, this is already provided by the Hirzebruch-Riemann-Roch Theorem; but I find this formula so abstracted as to be hardly geometric at all.)

REPLY [8 votes]: It is not true that quantities like the Todd genus (or the arithmetic genus) depend only on the underlying smooth manifold. There are examples of diffeomorphic Kaehler manifolds with different Todd genera.
Check the following papers for further information:
D. Kotschick, Topologically invariant Chern numbers of projective varieties, Adv. Math. 2012, doi 10.1016/j.aim.2011.10.020
D. Kotschick and S. Schreieder, The Hodge ring of Kaehler manifolds, Compos. Math. 2013, doi 10.1112/S0010437X12000759