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

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 Kähler 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 zerofiber, 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 Kähler 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 Kähler 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 Kähler manifold $X$ is said to be big if it contains the sum of a Kähler 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 Kähler 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–Mustaţă.