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Koszul duality between Betti and cohomology numbers in the Calabi-Yau case. (English) Zbl 1442.14129
Let $$X$$ be a smooth projective Calabi-Yau variety over a field $$k$$ and let $$\mathcal{O}(1)$$ be a very ample line bundle on $$X$$. Let $$A=\bigoplus_{m\geq 0}H^0(X,\mathcal{O}(m))$$ be the homogeneous coordinate ring. It follows from the results of R.-O. Buchweitz [“Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings”, Preprint, https://tspace.library.utoronto.ca/handle/1807/16682], and D. Orlov [Prog. Math. 270, 503–531 (2009; Zbl 1200.18007)] that the triangulated category $$D^b(X)=D^b(\text{Coh}(X))$$ of the coherent sheaves on $$X$$ is equivalent as a triangulated category to the stable category of maximal Cohen-Macaulay modules on the affine cone $$\tilde{X}=\text{Spec}(A)$$. This equivalence is called Orlov’s equivalence.
The main result of this paper shows that if $$\mathcal{O}(1)$$ is additionally Koszul, satisfying $$H^i(X,\mathcal{O}(j))=0$$ for $$i,j>0$$, then there are formulas for Betti numbers $$\beta_{ij}(M)$$ of a maximal Cohen-Macaulay module $$M$$, that are similar to that of the cohomology numbers $$h^{ij}(F)$$, where $$F\in D^b(X)$$ and $$M$$ corresponds to $$F$$ under Orlov’s equivalence. More precisely, in this case, $\beta_{ij}(M)=\dim\,\text{Hom}(R_{-j}, F[d-1+j-i]),$ for $$i,j\in\mathbb{Z}$$, where $$\dim(X)=d$$. The term $$R_{-j}$$ in the above expression comes from the box-product resolution of the structure sheaf of the diagonal $$\Delta_X\subset X\times X$$.
##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 13D02 Syzygies, resolutions, complexes and commutative rings 14F06 Sheaves in algebraic geometry
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##### References:
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