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Fano manifolds of Calabi-Yau Hodge type. (English) Zbl 1314.14085
Given a smooth projective variety \(X\), the authors define it to be Calabi-Yau Hodge types if it satisfies the following three conditions.
1.
The middle dimensional Hodge structure is numerically similar to that of a Calabi-Yau threefold, that is \(h^{n+2,n-1}(X)=1\), and \(h^{n+p+1,n-p}(X) = 0\) for \(p \geq 2\).
2.
For any generator \(\omega \in H^{n+2,n-1}(X)\), the contraction map \(H^1(X, TX) \rightarrow H^{n-1}(X, \Omega^{n+1}_X)\) is an isomorphism.
3.
The Hodge numbers \(h^{k, 0}(X) = 0\) for \(1 \leq k \leq 2n\).
The author study some basic properties of varieties of this type, give examples among complete intersections and hypersurfaces in homogeneous spaces, and study the derived categories of some of the examples.
Reviewer: Zhiyu Tian (Bonn)

MSC:
14J45 Fano varieties
14J40 \(n\)-folds (\(n>4\))
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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