[Paper Review] Deformations of vector bundles on coisotropic subvarieties via the Atiyah class

Jeremy Pecharich|arXiv (Cornell University)|Oct 18, 2010

Homotopy and Cohomology in Algebraic Topology17 references3 citations

TL;DR

This paper establishes a criterion for first- and second-order noncommutative deformations of vector bundles on coisotropic subvarieties of algebraic Poisson varieties using the Atiyah class. It shows that such deformations are governed by a curved differential graded Lie algebra, with the Atiyah class of $E \otimes L^\vee$ vanishing in $H^1(Y, N(E))$ as a necessary and sufficient condition for first-order deformations when $H^2(Y, \mathcal{O}_Y(E)) = 0$, generalizing classical deformation theory to the Poisson setting.

ABSTRACT

Using the Atiyah class we give a criterion for a vector bundle on a coisotropic subvariety, $Y$, of an algebraic Poisson variety $X$ to admit a first and second order noncommutative deformation. We also show noncommutative deformations of a vector bundle are governed by a curved dg Lie algebra which reduces to the classical relative Hochschild complex when the Poisson structure on $X$ is trivial.

Motivation & Objective

• To extend deformation theory of vector bundles to coisotropic subvarieties in algebraic Poisson varieties.
• To characterize when a vector bundle on a coisotropic subvariety admits a first- and second-order noncommutative deformation.
• To show that noncommutative deformations are governed by a curved dg Lie algebra that reduces to the relative Hochschild complex when the Poisson structure is trivial.
• To provide a cohomological criterion involving the Atiyah class for the existence of such deformations.

Proposed method

• Uses the Atiyah class of $E \otimes L^\vee$ to define a morphism $at_N(E \otimes L^\vee)$ in $H^1(Y, N(E))$.
• Applies spectral sequences to analyze obstructions in $H^0(Y, \wedge^2 N(E))$, $H^1(Y, N(E))$, and $H^2(Y, \mathcal{O}_Y(E))$.
• Introduces the normal complex $\mathcal{N}_E^\bullet$ with a differential $d_{N_E}$ defined via the curvature of a first-order deformation operator $\gamma$.
• Relies on the assumption $\beta_1^X \equiv 0$ to simplify the Lie algebra structure on the conormal bundle $N^\vee$, enabling the definition of curvature $c(\gamma)$.
• Uses local equations (A.1)–(A.8) to describe compatibility of module actions and transition functions in second-order deformations.
• Applies Lemma A.1 and A.2 to ensure existence of differential operators satisfying cocycle conditions, assuming $H^2(Y, \mathcal{O}_Y(E)) = 0$.

Experimental results

Research questions

• RQ1 When does a vector bundle $E$ on a coisotropic subvariety $Y \subset X$ admit a first-order noncommutative deformation?
• RQ2 What is the role of the Atiyah class in determining the obstruction to such deformations?
• RQ3 How do noncommutative deformations of $E$ relate to the Poisson structure on $X$?
• RQ4 What is the governing algebraic structure for noncommutative deformations of $E$?
• RQ5 How does the vanishing of $H^2(Y, \mathcal{O}_Y(E))$ affect the existence of deformations?

Key findings

• A first-order deformation of $E$ exists if and only if $at_N(E \otimes L^\vee) = 0$ in $H^1(Y, N(E))$, provided $H^2(Y, \mathcal{O}_Y(E)) = 0$.
• When $X$ and $Y$ are affine, a globally split deformation $\mathcal{E}_1 \simeq E \oplus \epsilon E$ exists, and the deformation is classified by a collection of operators $\gamma_E^i: N^\vee \to \mathcal{D}^1_\heartsuit(E)$ satisfying gluing conditions.
• The curvature $c(\gamma)$ measures the failure of the deformation operator $\gamma$ to be a Lie algebra morphism, and vanishing curvature is necessary for second-order deformations.
• Noncommutative deformations of $E$ are governed by a curved dg Lie algebra that reduces to the classical relative Hochschild complex when the Poisson bivector $P$ vanishes.
• The obstruction to the existence of a first-order deformation lies in $H^0(Y, \wedge^2 N(E))$, and vanishes if and only if $Y$ is coisotropic.
• The assumption $H^2(Y, \mathcal{O}_Y(E)) = 0$ ensures that the cocycle condition for transition functions can be resolved, allowing the construction of a global deformation.