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Geometry of Deformations via Incidence Varieties

Atabey Kaygun

TL;DR

This work provides a unified geometric framework for deformation theory across associative, commutative, Leibniz, and Lie algebras by embedding their law spaces as diagonal slices of GL(V)-equivariant incidence varieties. The central construction, the incidence variety $\mathsf{As}(V)$ (and its Leibniz/Lie analogues), yields fibers isomorphic to the appropriate deformation cohomology spaces $Z^2$, so tangent spaces to the law varieties coincide with classical cocycles $Z^2$ (e.g., Hochschild, Harrison, Leibniz, or Chevalley–Eilenberg). A principal trace form discriminant identifies the open separable (semisimple) strata, where Hochschild and Harrison cohomology vanish, producing rigid, open GL-orbits in the moduli; similarly, the Killing form characterizes the semisimple Lie strata within Leibniz and Lie varieties, yielding Whitehead-type rigidity results. The framework recovers refined cohomology descriptions in the commutative case via Harrison theory and suggests a geometric path to study singular strata through intersection cohomology, linking classical deformation theory with algebro-geometric moduli via GL-orbit structure and block decompositions (Wedderburn/CE). Collectively, the results illuminate how geometric incidence structures encode deformation complexes and moduli behavior across multiple algebraic geometries, offering precise tangential and rigidity statements and opening avenues for further geometric invariants on singular strata.

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.

Geometry of Deformations via Incidence Varieties

TL;DR

This work provides a unified geometric framework for deformation theory across associative, commutative, Leibniz, and Lie algebras by embedding their law spaces as diagonal slices of GL(V)-equivariant incidence varieties. The central construction, the incidence variety (and its Leibniz/Lie analogues), yields fibers isomorphic to the appropriate deformation cohomology spaces , so tangent spaces to the law varieties coincide with classical cocycles (e.g., Hochschild, Harrison, Leibniz, or Chevalley–Eilenberg). A principal trace form discriminant identifies the open separable (semisimple) strata, where Hochschild and Harrison cohomology vanish, producing rigid, open GL-orbits in the moduli; similarly, the Killing form characterizes the semisimple Lie strata within Leibniz and Lie varieties, yielding Whitehead-type rigidity results. The framework recovers refined cohomology descriptions in the commutative case via Harrison theory and suggests a geometric path to study singular strata through intersection cohomology, linking classical deformation theory with algebro-geometric moduli via GL-orbit structure and block decompositions (Wedderburn/CE). Collectively, the results illuminate how geometric incidence structures encode deformation complexes and moduli behavior across multiple algebraic geometries, offering precise tangential and rigidity statements and opening avenues for further geometric invariants on singular strata.

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.

Paper Structure

This paper contains 13 sections, 20 theorems, 111 equations.

Table of Contents

  1. The Associative Case
  2. The quadratic associativity locus
  3. Bilinearization and its symmetries
  4. The incidence as a sheaf and its cohomological fiber
  5. The principal trace form and semisimplicity
  6. Tangent spaces, rigidity, and moduli
  7. The commutative case, Harrison cohomology, and rigidity
  8. The Leibniz Case
  9. The quadratic Leibniz locus
  10. Bilinearization and cohomological interpretation
  11. Canonical cocycles and the Killing form
  12. Rigidity on the semisimple Lie stratum
  13. The Lie quotient and rigidity

Key Result

Proposition 1.1

Let $V^{3,1}:=\mathop{\mathrm{Hom}}\nolimits(V^{\otimes 3},V)$ and define $\Phi\colon V^{2,1}\to V^{3,1}$ by The associativity locus is a closed subscheme cut out by $n^4$ quadratic polynomials

Theorems & Definitions (38)

  • Proposition 1.1
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Theorem 1.5: Aguiar1998
  • Theorem 1.6
  • proof
  • ...and 28 more