Geometry of Deformations via Incidence Varieties

TL;DR

This paper unifies the geometric understanding of classical deformation complexes by constructing incidence varieties that encode algebraic structures and their cohomologies, revealing new insights into algebra rigidity and moduli spaces.

Contribution

It introduces GL-equivariant incidence varieties that recover algebra structures and relate fibers to cohomology spaces, providing a geometric framework for deformation theory.

Findings

Incidence varieties encode algebra structures and their cohomologies.

Fibers of the incidence map are isomorphic to cohomology spaces.

Generic points in moduli spaces are rigid, forming open orbits.

Abstract

We provide a unified geometric realization of the classical deformation complexes. We construct GL-equivariant bilinear incidence varieties whose diagonal slices recover the varieties of associative, commutative, Leibniz, and Lie algebra structures on a finite-dimensional vector space. We prove that the fiber of the incidence map at a given algebra law is canonically isomorphic to the space of 2-cocycles in the corresponding cohomology theory (Hochschild, Harrison, Leibniz, or Chevalley--Eilenberg). Furthermore, we introduce invariant bilinear forms to define open strata of separable and semisimple algebras, and demonstrate that these strata consist of open GL-orbits, establishing the rigidity of generic points in the coarse moduli spaces for all four geometries.