Forms and Complex Manifolds

TL;DR

This work analyzes the intersection form $F_X$ on $H^2(X,\mathbb{Z})$ for compact K"ahler manifolds of dimension $n$, extending understanding to higher dimensions ($n\ge 4$) and connecting topology to birational geometry. It develops a Hessian-based framework $\mathcal{H}_{F_X}$ and proves a finiteness result: there exist nonzero classes $e_1,\dots,e_q$ with $q\le b+1$ such that any exceptional divisor of a divisorial contraction to a point is represented by one of these classes, linking topological data to Minimal Model Program steps. The paper also analyzes how $F_X$ behaves under blow-ups, provides rank bounds for the Hessian in birational modifications, and surveys open problems on volume, Chern numbers, and the MMP that illuminate the interplay between topology and birational geometry in higher dimensions. Together these results establish new topological constraints on birational transformations and set a program for extending volume and Chern-number bounds to higher-dimensional K"ahler manifolds.

Abstract

We study the intersection form $F_X$ on the second cohomology group $H^2(X, \mathbb{Z})$ of a compact Kähler 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 Kähler manifolds.

Theorems & Definitions (35)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Definition 3.1
• Example 3.2
• Lemma 3.3
• proof