2021/03/05 by David Witt Nyström, Nyström, David Witt · 4 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2103.03660
openalex publication_date 2021/03/05 · openalex created_date 2021/03/15 · openalex updated_date 2026/07/28
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,ω) is a compact Kähler manifold and Y is a submanifold. The deformation space X is fibered over ℙ1 and all the fibers Xτ are isomorphic to X, except the zero-fiber, which has the projective completion of the normal bundle NY|X as one of its components. The first main result of this paper is that one can find Kähler forms on modifications of X which restricts to ω on X1 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,ω) 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 α then up to a dimensional constant, the restricted volume of α along Y is equal to the derivative of the volume at α 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.