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):

1. the arithmetic genus of $X$
2. the constant coefficient of the Hilbert polynomial of $X$
3. $\chi(X, \mathscr{O}_X)$
4. 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 dimension $k$ 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. 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.)