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Derived invariance of the number of holomorphic 1 -forms and vector fields

Mihnea Popa, Christian Schnell (2011)

Annales scientifiques de l'École Normale Supérieure

We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.

Duality for the de Rham cohomology of an abelian scheme

Robert F. Coleman (1998)

Annales de l'institut Fourier

In this paper the equality is established of three different pairings between the first de Rham cohomology group of an abelian scheme over a base flat over ℤ and that of its dual. These pairings have appeared and been used either explicitly or implicitly in the literature.In the last section we deduce a generalization to arbitrary characteristic of Serre’s formula for the Poincaré pairing on the first de Rham cohomology group of a curve over a field of characteristic zero.

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