The Ext algebra of a quantized cycle

Calaque, Damien; Grivaux, Julien

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 31-77.

Étant donné un cycle quantifié $(X,\sigma )$ dans $Y$, nous donnons une interprétation d’une condition découverte pas Shilin Yu en termes de théorie de Lie catégorique. Cette condition, que nous appelons modération géométrique, met en jeu le second voisinage infinitésimal de $X$ dans $Y$. Sous cette hypothèse de modération, nous démontrons que l’algèbre des Ext $\mathrm{ℛ\mathscr{H}}o{m}_{{𝒪}_Y}({𝒪}_X,{𝒪}_X)$ est isomorphe à l’algèbre enveloppante du fibré normal décalé ${\mathrm{N}}_{X/Y}[-1]$, que l’on munit d’une structure de Lie catégorique bien particulière, renforçant un résultat précédent de Căldăraru, Tu et du premier auteur. Cette approche permet de fournir des démonstrations conceptuelles de plusieurs résultats majeurs du sujet : dans le cas du plongement diagonal, nous retrouvons en particulier des résultats de Kapranov, Markarian et Ramadoss à propos (a) de la structure de Lie sur le tangent décalé ${\mathrm{T}}_X[-1]$ (b) de l’algèbre enveloppante correspondante (c) du calcul des « big Chern classes » de Kapranov. Nous donnons également une nouvelle démonstration purement algébrique (basée sur les structures de Lie catégoriques) d’un résultat de Yu à propos du calcul explicite de la classe de cycle quantifiée dans le cas modéré : il s’agit de l’élément de Duflo de l’objet en algèbre de Lie ${\mathrm{N}}_{X/Y}[-1]$.

Given a quantized 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 $\mathrm{ℛ\mathscr{H}}o{m}_{{𝒪}_Y}({𝒪}_X,{𝒪}_X)$ 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 Căldăraru, Tu, and the first author. This approach enables us 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 ${\mathrm{N}}_{X/Y}[-1]$.

Classification : 14F05, 14B20, 17B35, 14C99
Keywords: Closed embeddings, formal neighborhoods, Todd class, Ext algebra, derived categories, Lie algebras, enveloping algebras, Duflo element, PBW isomorphism
Mot clés : Plongements fermés, voisinages formels, classe de Todd, algèbre des Ext, catégories dérivées, algèbres de Lie, algèbres enveloppantes, élément de Duflo, isomorphisme PBW

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