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I have a question about Chern classes. The first one is general. If $X$ is a projective scheme and $F$ a coherent sheaf over $X$. We define the Hilbert polynomial $P_F(n):= \chi(X,F(m))$, where $F(m)$ is the twisted sheaf. If I denote with $ch(F(m))$ the Chen character and with $td(X)$ the Todd class, we have the equality $$ \chi(X,F(m))= \int_X ch(F(m)) \, \dot \, td(X) \,\,\,\, ?$$

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This is the statement of the Grothendieck-Riemann-Roch Theorem applied to the structure morphism $f:X \to \operatorname{Spec} k$. This is a generalization of the Hirzebruch-Riemann-Roch Theorem which gives the same statement but for holomorphic vector bundles on a compact complex manifold.

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