Cohomology of hypersurfaces of weighted projective space and the intersection form on $H^2$

TL;DR

This paper computes the intersection form on the second cohomology of hypersurfaces in weighted projective spaces, aiding the classification of Fano manifolds obtained via smoothing singular Fanos.

Contribution

It provides explicit formulas for the intersection form and cohomology groups of hypersurfaces in weighted projective spaces, advancing understanding of their topological properties.

Findings

Computed the intersection form on H^2 for hypersurfaces in weighted projective spaces.

Described the integer cohomology groups H^k for k<n.

Derived explicit formulas for the pullback map i^*.

Abstract

Given a hypersurface $i \colon X \hookrightarrow \widetilde{P}^n$ in a weighted projective space, we compute the intersection form on the second cohomology $H^2(X, \mathbb{Z})^{\otimes n-1} \to \mathbb{Z}$ for the purpose of identifying Fano manifolds obtained from smoothing singular Fanos. In the process, we describe the integer cohomology groups $H^k(X, \mathbb{Z})$ for $k<n$ and give an explicit formula for the pullback map $i^*$.