$L^p$ metric geometry of big and nef cohomology classes

TL;DR

This paper introduces a new metric on the space of big and nef cohomology classes on compact Kähler manifolds, establishing a complete geodesic metric space structure for finite energy classes.

Contribution

It defines a novel $L^p$ metric on the space of big and nef classes, extending the geometric understanding of cohomology classes in Kähler geometry.

Findings

The space $ig( ext{finite energy classes}ig)$ becomes a complete geodesic metric space under the $d_p$ metric.

The $d_p$ metric generalizes previous metrics in Kähler geometry.

Provides tools for studying the geometry of cohomology classes in complex geometry.

Abstract

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$, and $\theta$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,\theta)$, making it a complete geodesic metric space.