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On Hodge numbers of complete intersections and Landau-Ginzburg models. (English) Zbl 1343.14038
Homological Mirror Symmetry predicts that for any nonsingular Fano variety \(X\), there exists a Landau-Ginzburg model \(Y\to \mathbb{A}^1\) denoted by LG\((X)\), and the equivalences of categories \[ \mathrm{Fuk}(X)\cong D^b_{\mathrm{sing}}(Y),\quad D^b(X)\cong \mathrm {FS}(Y), \] where these categories are respectively the Fukaya category, derived category of singularities, derived category of of coherent sheaves, and Fukaya-Seidel category. As a consequence one obtains coincidence of non-commutative Hodge structures. Define \(k_{\mathrm{LG}(X)}\) to be the difference of the number of irreducible components of all reducible fibers of \(\mathrm{LG}(X)\) and the number of reducible fibers of \(\mathrm{LG}(X)\). It was conjectured by Gross-Katzarkov-Ruddat that if \(\dim X\geq 3\) then \(h^{1,\dim X-1}(X)=k_{\mathrm{LG}(X)}\). The conjecture is proven in some cases such as when \(X\) is a general type hypersurface under the assumption that the central fibers of the Landau-Ginzburg models are semistable with normal crossing singularities.
The main result of the paper under review is as follows: Suppose that \(X\) is complete intersection and \(\mathrm{LG}(X)\) is a Calabi-Yau compactification of Givental’s Landau-Ginzburg model for \(X\), then, \(h^{1,\dim X-1}(X)=k_{\mathrm{LG}(X)}\) if \(\dim X\geq 3\), and \(h^{1,1}(X)=k_{\mathrm{LG}(X)}+1\) if \(\dim X=2\). This proves the conjecture above in this case in which the normal crossing condition is not used. The Calabi-Yau compactifications of Givental’s Landau-Ginzburg models are constructed by first finding suitable singular compactifications whose total spaces are Calabi-Yau, and then by taking the crepant resolution of their singularities.

14J33 Mirror symmetry (algebro-geometric aspects)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M10 Complete intersections
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J45 Fano varieties
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