The Ext algebra of a quantized cycle

TL;DR

This paper provides a categorical Lie-theoretic interpretation of the Ext algebra of a quantized cycle, establishing its isomorphism to a universal enveloping algebra under tameness, and offers conceptual proofs of related geometric results.

Contribution

It introduces a new Lie-theoretic framework for understanding the Ext algebra of quantized cycles, generalizing previous results and simplifying proofs of key geometric properties.

Findings

Ext algebra is isomorphic to the universal enveloping algebra of the shifted normal bundle under tameness.

Recovers and generalizes results on Lie structures on shifted tangent bundles.

Provides a new proof of the quantized cycle class as the Duflo element.

Abstract

Given a quantized analytic cycle $(X, \sigma)$ in $Y$, we give a categorical Lie-theoretic interpretation of a geometric condition, discovered by Shilin Yu, that involves the second formal neighbourhood of $X$ in $Y$. If this condition (that we call tameness) is satisfied, we prove that the derived Ext algebra $\mathcal{RH}om_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)$ is isomorphic to the universal enveloping algebra of the shifted normal bundle $\mathrm{N}_{X/Y}[-1]$ endowed with a specific Lie structure, strengthening an earlier result of C\u{a}ld\u{a}raru, Tu, and the first author This approach allows to get some conceptual proofs of many important results in the theory: in the case of the diagonal embedding, we recover former results of Kapranov, Markarian, and Ramadoss about (a) the Lie structure on the shifted tangent bundle $\mathrm{T}_X[-1]$ (b) the corresponding universal enveloping algebra (c) the calculation of Kapranov's big Chern classes. We also give a new Lie-theoretic proof of Yu's result for the explicit calculation of the quantized cycle class in the tame case: it is the Duflo element of the Lie algebra object $\mathrm{N}_{X/Y}[-1]$.