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 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.)



A geometric characterization for arithmetic genus
First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either 


a) The number $\chi (X, \mathcal O\_X)$: the Hirzebruch arithmetic number in which you are interested .  

b) The number $p\_a(X)=(-1)^n(\chi (X, \mathcal O\_X)-1)$, the Severi arithmetic genus, which has historical precedence but of course was defined non-cohomologically.  

 For example, for projective space we have $\chi (\mathbb P^n, \mathcal O\_{\mathbb P^n})=1$ but $p\_a (\mathbb P^n)=0$. 


Hirzebruch introduced his definition mainly because it has the powerful multiplicativity property $$\chi (X\times Y,\mathcal O\_{X\times Y})= \chi (X,\mathcal O\_{X})\cdot \chi ( Y,\mathcal O\_{ Y})$$ which certainly is a step toward the geometric interpretation you are seeking. 


Another step in the right direction is that for a finite covering $X\to Y$ of degree $d$ we have the pleasant relation $\chi (X,\mathcal O\_{X})=d\cdot \chi ( Y,\mathcal O\_{ Y})$.  

But the most important geometric property is that $\chi (X,\mathcal O\_{X})$ is a birational invariant, because each number $dim\_\mathbb C H^i(X,\mathcal O\_{X})$ is already a birational invariant. 


Arithmetic genus is reasonably easy to compute: for a hypersurface $H\subset \mathbb P^n$ of degree $d$ you have $p\_a(H)=\binom {d-1}{n}$, which for $n=2$ gives the well-known elementary formula $p\_a(C)=\frac {(d-1)(d-2)}{2}$ for the plane curve $C$.  

[This formula (and others) can be found in Hartshorne, Chapter I, Exercise 7.2, page 54]  

For a surface you have Max Noether's formula $\chi (S, \mathcal O\_S)=\frac {c\_1^2(S)+c\_2(S)}{12}$, where $c\_2(S)$ (=second Chern class of $S$) is also the purely *topological* Euler-Poincaré characteristic of $S$, equal to the alternating sum of the Betti numbers of the underlying toplogical space.$S\_{top}$. 


Finally, Fulton has given an axiomatic characterization of the arithmetic genus in algebraic geometry over an arbitrary algebraically closed field [here](http://www.jstor.org/discover/10.2307/2374145?uid=3738016&uid=2129&uid=2&uid=70&uid=4&sid=55977165963).  

 In a sense it may be considered an explanation of the geometric significance of the arithmetic genus: if you want it to satisfy certain geometric properties, the definition is forced upon you.


**Edit** (added by Charles with Georges's permission):
Fulton's axiomatic characterization may be described as follows: There is a unique assignment of a number $\mathcal{A}(X)$ to every [smooth irreducible projective variety over a fixed algebraically closed field] (hereafter simply "variety"), such that the following three axioms are satisfied:


1. $\mathcal{A}$ respects isomorphism classes.
2. If $X$ is a point, then $\mathcal{A}(X) = 1$.
3. Let $X$, $Y$, and $Z$ be (smooth) varieties of the same dimension. 
Suppose that $X$, $Y$, and $Z$ can be embedded as codimension-one subvarieties 
of a common (smooth) variety $W$, such that 


	* $X$ and $Y+Z$ are linearly equivalent as divisors in $W$, and
	* $Y$ and $Z$ intersect transversely in a disjoint union of (smooth) varieties 
	$V\_1, \dotsc, V\_{\ell}$.Then $$\mathcal{A}(X) = \mathcal{A}(Y) + \mathcal{A}(Z) - \sum\_i \mathcal{A}(V\_i).$$


This assignment takes $X$ to its "Hirzebruch arithmetic number" $\mathcal{A}(X) = \chi(X, \mathcal{O}\_X)$.


