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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 \(\Delta_X\subset X\times X\).
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
13D02 Syzygies, resolutions, complexes and commutative rings
14F06 Sheaves in algebraic geometry
Full Text: DOI
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