Editing Atiyah–Singer index theorem (section)

===Hirzebruch–Riemann–Roch theorem===
Take ''X'' to be a [[complex manifold]] of (complex) dimension ''n'' with a holomorphic vector bundle ''V''. We let the vector bundles ''E'' and ''F'' be the sums of the bundles of differential forms with coefficients in ''V'' of type (0, ''i'') with ''i'' even or odd, and we let the differential operator ''D'' be the sum

:<math>\overline\partial + \overline\partial^*</math>

restricted to ''E''. 

This derivation of the Hirzebruch–Riemann–Roch theorem is more natural if we use the index theorem for elliptic complexes rather than elliptic operators. We can take the complex to be

:<math>0 \rightarrow V \rightarrow V \otimes \Lambda^{0,1}T^*(X) \rightarrow V \otimes \Lambda^{0,2}T^*(X) \rightarrow \dotsm</math>

with the differential given by <math>\overline\partial</math>. Then the ''i'''th cohomology group is just the coherent cohomology group H<sup>''i''</sup>(''X'', ''V''), so the analytical index of this complex is the [[holomorphic Euler characteristic]] of ''V'':

:<math>\operatorname{index}(D) = \sum_p (-1)^p \dim H^p(X, V) = \chi(X, V)</math>

Since we are dealing with complex bundles, the computation of the topological index is simpler. Using Chern roots and doing similar computations as in the previous example, the Euler class is given by <math display="inline">e(TX) = \prod_{i}^{n}x_i(TX)</math> and

:<math>\begin{align}
  \operatorname{ch}\left(\sum_{j}^{n} (-1)^j V \otimes \Lambda^{j}\overline{T^*X}\right) &= \operatorname{ch}(V)\prod_{j}^{n}\left(1 - e^{x_j}\right)(TX) \\
  \operatorname{Td}(TX \otimes \mathbb{C}) = \operatorname{Td}(TX)\operatorname{Td}\left(\overline{TX}\right) &= \prod_i^n\frac{x_i}{1 - e^{-x_i}} \prod_j^n\frac{-x_j}{1 - e^{x_j}}(TX)
\end{align}</math>

Applying the index theorem, we obtain the [[Hirzebruch-Riemann-Roch theorem]]:
:<math>\chi(X, V)=\int _X \operatorname{ch}(V)\operatorname{Td}(TX)</math>

In fact we get a generalization of it to all complex manifolds: Hirzebruch's proof only worked for '''projective''' complex manifolds ''X''.