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From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves replaced by vector bundles. Unfortunately I don't know enough algebraic geometry to carryout the translation (for example I'm not sure what the pushforward of quasi-coherent sheafs corresponds to).

Does GHRR just amount to the Atiyah-Singer families index theorem for the fiberwise Dolbeault-Dirac operator?

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A quasicoherent sheaf does not correspond to a vector bundle, in general: for one thing, quasicoherent sheaves can be infinite-dimensional, and for another, the dimension of a fibre can change from point to point. The pushforward is not easy to describe in terms of bundles, but in terms of sections, given a continuous map $f : X \to Y$, the pushforward sheaf $f_* \mathscr{F}$ is the sheaf on $Y$ such that the sections of $f_* \mathscr{F}$ over an open set $V$ are the same as the sections of $\mathscr{F}$ over $f^{-1} V$. – Zhen Lin Mar 16 '13 at 19:12
I don't really know the general algebraic geometry version, but for me Hirzebruch-Riemann-Roch has always been the Atiyah-Singer index theorem for the (twisted) Dolbeault complex. See for example pages 151-153 of Berline-Getzler-Vergne's Heat Kernels and Dirac Operators. – Henry T. Horton Mar 16 '13 at 19:14
@HenryT.Horton Thanks for your comment. I'm familiar with HRR being a special case of the index theorem for the Clifford bundle of anti-holomorphic forms. So my question is really if HRR generalizes to GHRR in the same way that the Dirac picture of HRR generalizes to a special case of the families index theorem that is equivalent to GHRR. – Eric O. Korman Mar 16 '13 at 19:53

Let $E\rightarrow X$ be a holomorphic vector bundle over a compact complex manifold. $H^i(X, \mathcal{O}(E))$ be the $i-$th sheaf cohomology, where $\mathcal{O}(E)$ is the sheaf of holomorphic sections of $E$. Then $$\sum(-1)^i\dim H^i(X, E)=\int_X td(X)ch(E)$$, where $td(X)$ is the total Todd class of M, and $ch(E)$ is the Chern character of $E.$

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