Forms and Complex Manifolds

TL;DR

This paper explores the intersection form on the second cohomology of compact Kähler manifolds, focusing on higher dimensions where less is known, and discusses implications for birational geometry and open problems.

Contribution

It investigates fundamental properties of the intersection form in dimensions four and higher, extending understanding beyond the well-studied lower dimensions.

Findings

Analysis of the intersection form in higher dimensions

Applications to birational geometry

Presentation of open problems in the field

Abstract

We study the intersection form $F_X$ on the second cohomology group $H^2(X, \mathbb{Z})$ of a compact K\"ahler manifold $X$ of dimension $n$. Although the structure of $F_X$ is relatively well understood in dimensions two and three, much less is known for $n \geq 4$. We investigate the fundamental properties of $F_X$ in higher dimensions and discuss several applications to birational geometry. Finally, we present a number of open problems concerning the relationship between birational invariants and topological invariants of K\"ahler manifolds.