Let a_X : X → AlbX be the Albanese map of a smooth complex projective variety. Roughly speaking, in this note we prove that for all i >= 0 and alpha in Pic^0 X, the cohomology ranks h^i(AlbX, {a_x}_* omega_X otimes P_{alpha}) are derived invariants. This proves conjectures of Popa and Lombardi-Popa - including the derived invariance of the Hodge numbers h^{0, j} - in the case of varieties of maximal Albanese dimension and a weaker version of them for arbitrary varieties. Finally, we provide an application to the derived invariance of certain irregular fibrations.
Derived invariants arising from the Albanese map / F. Caucci, G. Pareschi. - In: ALGEBRAIC GEOMETRY. - ISSN 2214-2584. - 6:6(2019), pp. 730-746. [10.14231/AG-2019-031]
Derived invariants arising from the Albanese map
F. Caucci;
2019
Abstract
Let a_X : X → AlbX be the Albanese map of a smooth complex projective variety. Roughly speaking, in this note we prove that for all i >= 0 and alpha in Pic^0 X, the cohomology ranks h^i(AlbX, {a_x}_* omega_X otimes P_{alpha}) are derived invariants. This proves conjectures of Popa and Lombardi-Popa - including the derived invariance of the Hodge numbers h^{0, j} - in the case of varieties of maximal Albanese dimension and a weaker version of them for arbitrary varieties. Finally, we provide an application to the derived invariance of certain irregular fibrations.
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