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Geometric Rigidity in Moduli Stacks of Algebras

Atabey Kaygun

Abstract

We study quadratic moduli schemes $X$ of algebra laws on a fixed vector space $W$ under the transport-of-structure action of $GL(W)$ on $Hom(W^{\otimes 2},W)$. We construct an intrinsic three-term deformation complex on $X$ whose fibers encode transverse first-order classes and primary obstructions, and whose cohomology agrees on the operadic loci with the standard low-degree deformation cohomology (à la Gerstenhaber and Nijenhuis--Richardson). We then define a canonical quadratic map $κ^{inc}_{2,μ}\colon H^2_{inc}(μ)\to H^3_{inc}(μ)$ that controls second-order lifts modulo isotriviality. If $μ$ is smooth point in a reduced component and $(κ^{inc}_{2,μ})^{-1}(0)=\{0\}$, then the $G$-orbit of $μ$ is Zariski open in that component. This provides a coordinate-free explanation of Richardson-type geometric rigidity even when the second deformation cohomology does not vanish.

Geometric Rigidity in Moduli Stacks of Algebras

Abstract

We study quadratic moduli schemes of algebra laws on a fixed vector space under the transport-of-structure action of on . We construct an intrinsic three-term deformation complex on whose fibers encode transverse first-order classes and primary obstructions, and whose cohomology agrees on the operadic loci with the standard low-degree deformation cohomology (à la Gerstenhaber and Nijenhuis--Richardson). We then define a canonical quadratic map that controls second-order lifts modulo isotriviality. If is smooth point in a reduced component and , then the -orbit of is Zariski open in that component. This provides a coordinate-free explanation of Richardson-type geometric rigidity even when the second deformation cohomology does not vanish.
Paper Structure (25 sections, 25 theorems, 59 equations)

This paper contains 25 sections, 25 theorems, 59 equations.

Table of Contents

  1. The Fundamental Deformation Complex
  2. The incidence scheme
  3. The fundamental exact sequence
  4. The incidence deformation complex
  5. Cohomological vs. Geometric Rigidity
  6. Rank loci
  7. Two rank stratifications
  8. The constant-rank locus
  9. Rigidity implications
  10. Quadratic Obstructions and Anisotropy
  11. The quadratic obstruction
  12. Bilinear obstruction pairing
  13. Anisotropy and open orbits
  14. The anisotropic locus
  15. A cohomologically non-rigid anisotropic point
  16. ...and 10 more sections

Key Result

Lemma 1.1

The incidence locus $\mathsf{Inc}(Q) \subset A_W \times A_W$ is defined as the scheme-theoretic zero locus $Z(s_\Theta)$ of the section $s_\Theta$stacks-project. Its $\Bbbk$-points are precisely the pairs $(\mu,\nu)$ such that $q(\mu,\nu)=0$ for all $q\in Q$.

Theorems & Definitions (63)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Lemma 1.5
  • proof
  • Definition 1.6
  • ...and 53 more