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I wonder what holomorphic Euler characteristic $\chi(\mathcal{O}_X)$ of a variety represents. For example, I have seen someone fix $\chi(\mathcal{O}_C)=n$ for a complex curve $C$. What does this mean geometrically? Remember that $\chi(\mathcal{O}_X)$ is defined as the alternating sum $$ \chi(\mathcal{O}_X)=\sum_{i=0}^{\dim X}(-1)^i\dim \mathrm{H}^i(X,\mathcal{O}_X). $$

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If you have any sheaf $\mathcal F$ (if it is locally free, it is essentially the same as talking about vector bundles), then a very important invariant would be the dimension of independent global sections, $\dim \Gamma (X,\mathcal F)$. Now that is a difficult number to calculate since taking global sections of an exact sequence of sheaves is not in general an exact sequence of the global section spaces, for example: $$0\rightarrow \mathcal F\rightarrow \mathcal G\rightarrow \mathcal H\rightarrow 0,\;\; \text{but} \\ 0\rightarrow \Gamma(X,\mathcal F)\rightarrow \Gamma(X,\mathcal G)\rightarrow \Gamma(X,\mathcal H).$$ This is the beginning of sheaf cohomology (which can be built via derived functors and usually computed via Čech cohomology): the inexactness of the global section sequence is measured by new groups $H^i(X,\mathcal F)$, where $H^0(X,\mathcal F):=\Gamma (X,\mathcal F)$: $$0\rightarrow H^0(\mathcal F)\rightarrow H^0(\mathcal G)\rightarrow H^0(\mathcal H)\rightarrow H^1(\mathcal F)\rightarrow H^1(\mathcal G)\rightarrow H^1(\mathcal H)\rightarrow H^2(\mathcal F)\rightarrow \cdots\rightarrow 0$$ Thus although you may not be able to compute $\dim H^0(X,\mathcal F)=:h^0(X,\mathcal F)$, there are many techniques and important theorems to compute its Euler-Poincare characteristic $$\chi(\mathcal F):=\sum_{i=0}^{\dim X}(-1)^i\cdot \dim H^i(X,\mathcal F),$$ because $\chi$ is additive on exact sequences, i.e. $\chi (\mathcal G)=\chi (\mathcal F)+\chi (\mathcal H)$. For example one can prove Bézout's theorem using two appropriate exact sequences and this previous equations, since the $H^i(\mathbb P^2,\mathcal O_{\mathbb P^2}(d))$ can be computed explicitly. The major theorem for computing $\chi$ in general is the Hirzebruch-Riemann-Roch theorem with its many applications relating the degrees of the variety and of its divisors.

So you can think of $\chi(\mathcal F)$ as the number of independent sections of the sheaf $\mathcal F$ modulo cohomological correction terms. The whole subject of vanishing theorems studies results which give the conditions when certain $H^i(X,\mathcal F)=0$ for $i\geq 1$, so that your $\chi$ is as close as possible to your desired dimension (e.g. that is why the defining alternating sum is finite, ending up at least at $\dim X$). Now, $\mathcal O_X$ is the structure sheaf of the variety, i.e. the collection of regular function rings for every open subset of $X$ with its usual restriction to open subsets. Thus, $\chi (\mathcal O_X)$ measures, up to cohomological corrections, the number of independent global regular/holomorphic functions on your variety $X$, which is a very important invariant. For example the number of independent differential forms of top order, $p_g(X):=\dim H^0(X,\omega_X)$, is called the geometric genus of $X$ ($\omega_X$ is a line bundle in this case). It is a remarkable theorem that for complex projective algebraic curves (equivalently, compact Riemann surfaces or closed orientable real surfaces) this geometric genus coincides with the topological genus $g=p_g$, the number of "doughnut holes" the real surface of the complex curve has. There is another number called the arithmetic genus $p_a$, defined via Hilbert's polynomials, which also coincides for the case of curves $g=p_g=p_a$. Such results and concepts appear in the Riemann-Roch theorem which gives the value of $\chi (\mathcal L)$ for a line bundle on a curve $X$ in terms of such genera $g$ and the degree of the associated divisor $\deg [\mathcal L]$. Besides, it makes use of a duality theorem by Serre so that one can express $H^{n-k}(X,\mathcal L)$ in terms of $H^k(X,\omega_X\otimes\mathcal L^\vee)$, and in this case of curves put $\chi (\mathcal L)$ just in terms of dimensions of global sections! $$\chi (\mathcal L)=\dim H^0(X,\mathcal L)-\dim H^0(X,\omega_X\otimes\mathcal L^\vee)=\deg [\mathcal L]+1-g.$$ If instead of the one-dimensional sheaves (line bundles) $\mathcal L$, we write the theorem in terms of the divisors associated to the zero locus of a generic of its sections, $D=[\mathcal L]$, the result is: $$\chi (D)=h^0(D)-h^0(K-D)=\deg D+1-g.$$ Since $H^0(X,\cdot)=\Gamma (X,\cdot)$ measures the number of independent sections, this Riemann-Roch theorem tells you the following: the number of independent rational/meromorphic functions with zeros and poles at, and bounded by, the divisor $D$, minus the corresponding number but for the divisor $K-D$ (where $K$ is a canonical divisor, i.e. the generic zero locus of a differential 1-form on the curve) is just the degree of the divisor $D$, plus one, minus the (geometric = topological = arithmetic) genus of the complex algebraic curve. So in this case, if you fix $\chi_C (\mathcal L)=n$ for a curve $C$, you are considering sets of points of zeros and poles, i.e. divisors $D=[\mathcal L]$ given by rational sections of $\mathcal L$, with total fixed degree $\deg D=(g_C-1)+n$. Notice that in algebraic topology, your complex curve $C$ seen as a closed oriented real surface has for any triangulation of $V$ no. of vertices, $E$ no. of edges and $F$ no. of faces a corresponding topological characteristics: $$\chi_{\text{top.}}(C)=V-E+F=-2(g-1)=-\deg K,$$ which is the degree of a generic canonical divisor on $C$; this $g-1$ also appears in many other applications like Gauß-Bonnet theorem. These are remarkable results with very important consequences.

Its generalization to higher dimensional varieties is the cornerstone Hirzebruch-Riemann-Roch theorem which computes $\chi_X (\mathcal F)$ in terms of intersection numbers of characteristic classes of $\mathcal F$ on $X$. Its generalization to compact manifolds in general (even nonalgebraic or with boundary) is the remarkably famous Atiyah-Singer index theorem, from which Hirzebruch-Riemann-Roch can be obtained for the Dolbeault complex of holomorphic differential operators. I have briefly explained a bit about this theorem in this other answer, where the focus is on computing the Euler characteristic of a sequence of elliptic pseudodifferential operators, i.e. the number of independent solutions to homogeneous partial differential equations up to cohomological corrections, on compact manifolds in terms of their purely topological data.

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This really is one great answer! –  Nils Matthes Nov 22 '12 at 11:17
    
Thank you for the detailed answer! –  M. K. Nov 22 '12 at 19:39
    
Thanks a lot to both of you! I love math and explaining it... (hard to stop sometimes). –  Javier Álvarez Nov 22 '12 at 20:02
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