Multiplications of Maximal Rank in the Cohomology of P^1\times P ^1

TL;DR

This paper proves that multiplication by a general bi-homogeneous form induces a maximal rank linear map on certain cohomology groups of a smooth quadric in projective space, addressing a specific case of a broader open problem.

Contribution

It establishes the maximal rank property for a specific multiplication map in the cohomology of a smooth quadric, a novel result in this area.

Findings

The multiplication map has maximal rank in the studied case.

Provides an interpretation using bi-homogeneous linear differential operators.

Addresses a particular case of a broader open problem.

Abstract

We show that the linear map defined by multiplication with a general bi-homogeneous form between two bi-graduated pieces of the first cohomology of a nonsingular quadric in the projective space is of maximal rank. This is the first non trivial case of a more general open problem on natural linear maps between vector spaces of tensors defined in terms of multiplications and contractions. An interpretation in terms of bi-homogeneous linear differential operators with polynomial coefficients is also given.