Deformations of K\"ahler manifolds to normal bundles and restricted volumes of big classes

TL;DR

This paper explores how deformations of Kähler manifolds to their normal bundles influence the volume of big classes, establishing a link between restricted volumes and volume derivatives, with implications for complex geometry.

Contribution

It introduces a method to deform Kähler manifolds towards their normal bundles and relates restricted volumes of big classes to volume derivatives, extending previous results.

Findings

Kähler forms can be deformed to concentrate mass in the normal bundle.

Restricted volume along a hypersurface equals the volume derivative in the class direction.

The volume of big classes relates to derivatives along submanifolds.

Abstract

The deformation of a variety $X$ to the normal cone of a subvariety $Y$ is a classical construction in algebraic geometry. In this paper we study the case when $(X,\omega)$ is a compact K\"ahler manifold and $Y$ is a submanifold. The deformation space $\mathcal{X}$ is fibered over $\mathbb{P}^1$ and all the fibers $X_{\tau}$ are isomorphic to $X$, except the zero-fiber, which has the projective completion of the normal bundle $N_{Y|X}$ as one of its components. The first main result of this paper is that one can find K\"ahler forms on modifications of $\mathcal{X}$ which restricts to $\omega$ on $X_1$ and which makes the volume of the normal bundle in the zero-fiber come arbitrarily close to the volume of $X$. Phrased differently, we find K\"ahler deformations of $(X,\omega)$ such that almost all of the mass ends up in the normal bundle. The proof relies on a general result on the volume of big cohomology classes, which is the other main result of the paper. A $(1,1)$ cohomology class on a compact K\"ahler manifold $X$ is said to be big if it contains the sum of a K\"ahler form and a closed positive current. A quantative measure of bigness is provided by the volume function, and there is also a related notion of restricted volume along a submanifold. We prove that if $Y$ is a smooth hypersurface which intersects the K\"ahler locus of a big class $\alpha$ then up to a dimensional constant, the restricted volume of $\alpha$ along $Y$ is equal to the derivative of the volume at $\alpha$ in the direction of the cohomology class of $Y$. This generalizes the corresponding result on the volume of line bundles due to Boucksom-Favre-Jonsson and independently Lazarsfeld-Musta\c{t}\u{a}.