$L^2$-cohomology for affine spaces and an application to monads

Abstract

Given a locally free sheaf $\mathcal{ℱ}$ on a projective space ${\mathbb{P}}_n$, a natural question is to compute the $L^2$-cohomology of its restriction to an affine subspace ${ℂ}^n={\mathbb{P}}_n\setminus {\mathbb{P}}_{n-1}$ equipped with a flat Kähler form ${\omega }_{{ℂ}^n}^{}$. This problem arises naturally when studying the geometry of moduli of monads. We identify the known construction of a natural hermitian structure on the moduli space of the associated vector bundles with the Weil–Petersson approach to a Kähler metric based on harmonic theory for Kodaira–Spencer tensors. We show that for a holomorphic family of monads on ${\mathbb{P}}_2$ or on a modification ${\tilde{\mathbb{P}}}_2$ of ${\mathbb{P}}_2$, there is a Kodaira–Spencer map providing the base with a Kähler structure.