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Derived invariants arising from the Albanese map. (English) Zbl 1441.14060

Summary: Let \(a_X: X\to \to \operatorname{Alb} X\) be the Albanese map of a smooth complex projective variety. Roughly speaking, in this note we prove that for all \(i\geqslant 0\) and \(\alpha\in\operatorname{Pic}^0 X\), the cohomology ranks \(h^i(\operatorname{Alb} X, a_{X_\ast}\omega_X\otimes P_\alpha)\) are derived invariants. This proves conjectures of M. Popa [Clay Math. Proc. 18, 567–575 (2013; Zbl 1317.14038)] and L. Lombardi and M. Popa [Lond. Math. Soc. Lect. Note Ser. 417, 291–306 (2014; Zbl 1326.14013)] – 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.

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14D06 Fibrations, degenerations in algebraic geometry
14E05 Rational and birational maps
14F17 Vanishing theorems in algebraic geometry
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