Tangent bundle of punctual Hilbert scheme and distinguishing products of varieties

2026/04/16 by Supravat Sarkar
#math.AG

paper · pdf

Abstract

We describe the indecomposable components of the tangent bundle of the punctual Hilbert scheme of a smooth projective surface. As an application, we prove a recent conjecture about classification of products of punctual Hilbert schemes of smooth projective surfaces. We also determine when two products of symmetric powers of a smooth variety can be isomorphic. As a key step in our proof, we give a new characterization of abelian varieties, which states that in dimension ≥ 2, a smooth complex projective variety whose tangent bundle is trivial upto a line bundle twist is an abelian variety.

Citations

• Algebraic Families on an Algebraic Surface
• Vector Bundles Over an Elliptic Curve

Related

• Automorphisms of punctual Hilbert schemes and symmetric powers of varieties
• Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces
• The Degree of the Tangent and Secant Variety to a Projective Surface
• Symmetric products and puncturing Campana-special varieties
• The tangent space of the punctual Hilbert scheme