Hochschild cohomology of projective hypersurfaces

TL;DR

This paper computes the Hochschild cohomology of projective hypersurfaces, revealing its relation to smoothness, deformations, and providing explicit calculations for quartic cases.

Contribution

It offers a detailed computation of Hochschild cohomology for projective hypersurfaces, linking it to smoothness criteria and deformation structures, with explicit examples for quartic hypersurfaces.

Findings

A projective hypersurface is smooth iff the HKR decomposition holds for its Hochschild cohomology.

The first Hodge component encodes various types of first-order deformations.

Explicit dimensions of Hochschild cohomology groups are computed for quartic hypersurfaces.

Abstract

We compute Hochschild cohomology of projective hypersurfaces starting from the Gerstenhaber-Schack complex of the (restricted) structure sheaf. We are particularly interested in the second cohomology group and its relation with deformations. We show that a projective hypersurface is smooth if and only if the classical HKR decomposition holds for this group. In general, the first Hodge component describing scheme deformations has an interesting inner structure corresponding to the various ways in which first order deformations can be realized: deforming local multiplications, deforming restriction maps, or deforming both. We make our computations precise in the case of quartic hypersurfaces, and compute explicit dimensions in many examples.