Deformations of Kähler manifolds to normal bundles and restricted volumes of big classes

Journal article, 2024

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 Kahler manifold and Y is a submanifold. The deformation space X is fibered over 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 Kahler forms on modifications of 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 Kahler 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) co- homology class on a compact Kahler manifold X is said to be big if it contains the sum of a Kahler 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 Kahler 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-Mustata.