The Ext algebra of a quantized cycle

Damien Calaque (1, 2) , Julien Grivaux (3, 4, 5, 6)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 IUF - Institut universitaire de France
3 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
4 IHES - Institut des Hautes Études Scientifiques
5 I2M - Institut de Mathématiques de Marseille
6 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

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 $\mathrm{N}_{X/Y}[-1]$.

Domaines: Géométrie algébrique [math.AG]

Damien Calaque, Julien Grivaux. The Ext algebra of a quantized cycle. Journal de l'École polytechnique — Mathématiques, 2019, 6, pp.31--77. ⟨10.5802/jep.87⟩. ⟨hal-01649952⟩