The Ext algebra of a quantized cycle
Damien Calaque; Julien Grivaux
Journal de l'École polytechnique — Mathématiques, Volume 6  (2019), p. 31-77

Given a quantized cycle (X,σ) 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 ℛℋom 𝒪 Y (𝒪 X ,𝒪 X ) is isomorphic to the universal enveloping algebra of the shifted normal bundle 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 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 N X/Y [-1].

Étant donné un cycle quantifié (X,σ) 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 ℛℋom 𝒪 Y (𝒪 X ,𝒪 X ) est isomorphe à l’algèbre enveloppante du fibré normal décalé 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 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 N X/Y [-1].

Received : 2017-12-03
Accepted : 2018-12-04
Published online : 2019-01-21
DOI : https://doi.org/10.5802/jep.87
Classification:  14F05,  14B20,  17B35,  14C99
Keywords: Closed embeddings, formal neighborhoods, Todd class, Ext algebra, derived categories, Lie algebras, enveloping algebras, Duflo element, PBW isomorphism
@article{JEP_2019__6__31_0,
     author = {Damien Calaque and Julien Grivaux},
     title = {The Ext algebra of a quantized cycle},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {6},
     year = {2019},
     pages = {31-77},
     doi = {10.5802/jep.87},
     mrnumber = {3909113},
     zbl = {1408.14066},
     language = {en},
     url = {https://jep.centre-mersenne.org/item/JEP_2019__6__31_0}
}
Calaque, Damien; Grivaux, Julien. The Ext algebra of a quantized cycle. Journal de l'École polytechnique — Mathématiques, Volume 6 (2019) , pp. 31-77. doi : 10.5802/jep.87. https://jep.centre-mersenne.org/item/JEP_2019__6__31_0/

[1] D. Arinkin & A. Căldăraru - “When is the self-intersection of a subvariety a fibration?”, Adv. Math. 231 (2012) no. 2, p. 815-842 | Article | MR 2955193 | Zbl 1250.14006

[2] F. A. Berezin - “Some remarks about the associated envelope of a Lie algebra”, Funct. Anal. Appl. 1 (1967) no. 2, p. 91-102 | Article | Zbl 0227.22020

[3] N. Bourbaki - Algebra I. Chapters 1–3, Elements of Mathematics, Springer-Verlag, Berlin, 1998 | Zbl 0904.00001

[4] D. Calaque & M. Van den Bergh - “Hochschild cohomology and Atiyah classes”, Adv. Math. 224 (2010) no. 5, p. 1839-1889 | Article | MR 2646112 | Zbl 1197.14017

[5] D. Calaque, A. Căldăraru & J. Tu - “PBW for an inclusion of Lie algebras”, J. Algebra 378 (2013), p. 64-79 | Article | MR 3017014 | Zbl 1316.17008

[6] D. Calaque, A. Căldăraru & J. Tu - “On the Lie algebroid of a derived self-intersection”, Adv. Math. 262 (2014), p. 751-783 | Article | MR 3228441 | Zbl 1297.14018

[7] D. Calaque & C. A. Rossi - Lectures on Duflo isomorphisms in Lie algebra and complex geometry, EMS Lect. in Math., European Mathematical Society (EMS), Zürich, 2011 | Article | Zbl 1220.53006

[8] P. Deligne & J. W. Morgan - “Notes on supersymmetry (following Joseph Bernstein)”, in Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), American Mathematical Society, Providence, RI, 1999, p. 41-97 | Zbl 1170.58302

[9] M. Duflo - “Opérateurs différentiels bi-invariants sur un groupe de Lie”, Ann. Sci. École Norm. Sup. (4) 10 (1977) no. 2, p. 265-288 | Article | Zbl 0353.22009

[10] M. Duflo & T. Oshima - “Open problems in representation theory of Lie groups”, in Conference on Analysis on homogeneous spaces (Katata, Japan), 1986, p. 1-5

[11] J. Grivaux - “On a conjecture of Kashiwara relating Chern and Euler classes of O-modules”, J. Differential Geom. 90 (2012) no. 2, p. 267-275 | Article | MR 2899876 | Zbl 1247.32013

[12] J. Grivaux - “The Hochschild-Kostant-Rosenberg isomorphism for quantized analytic cycles”, Internat. Math. Res. Notices (2014) no. 4, p. 865-913 | Article | MR 3168398 | Zbl 1312.14027

[13] J. Grivaux - “Derived geometry of the first formal neighborhood of a smooth analytic cycle” (2015), arXiv:1505.04414

[14] G. Hochschild, B. Kostant & A. Rosenberg - “Differential forms on regular affine algebras”, Trans. Amer. Math. Soc. 102 (1962), p. 383-408 | Article | MR 142598 | Zbl 0102.27701

[15] M. Kapranov - “Rozansky-Witten invariants via Atiyah classes”, Compositio Math. 115 (1999) no. 1, p. 71-113 | Article | MR 1671737 | Zbl 0993.53026

[16] M. Kashiwara & P. Schapira - Deformation quantization modules, Astérisque, vol. 345, Société Mathématique de France, Paris, 2012 | Zbl 1260.32001

[17] M. Kontsevich - “Deformation quantization of Poisson manifolds”, Lett. Math. Phys. 66 (2003) no. 3, p. 157-216 | Article | MR 2062626 | Zbl 1058.53065

[18] J.-L. Loday & M. Ronco - “Combinatorial Hopf algebras”, in Quanta of maths, Clay Math. Proc., vol. 11, American Mathematical Society, Providence, RI, 2010, p. 347-383 | MR 2732058 | Zbl 1217.16033

[19] J.-L. Loday & B. Vallette - Algebraic operads, Grundlehren Math. Wiss., vol. 346, Springer-Verlag, Berlin Heidelberg, 2012 | MR 2954392 | Zbl 1260.18001

[20] N. Markarian - “The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem”, J. London Math. Soc. (2) 79 (2009) no. 1, p. 129-143 | Article | MR 2472137 | Zbl 1167.14005

[21] A. C. Ramadoss - “The big Chern classes and the Chern character”, Internat. J. Math. 19 (2008) no. 6, p. 699-746 | Article | MR 2431634 | Zbl 1165.13011

[22] A. C. Ramadoss - “The relative Riemann-Roch theorem from Hochschild homology”, New York J. Math. 14 (2008), p. 643-717 | MR 2465798 | Zbl 1158.19002

[23] D. Toledo & Y. L. L. Tong - “Duality and intersection theory in complex manifolds. I”, Math. Ann. 237 (1978) no. 1, p. 41-77 | Article | MR 506654 | Zbl 0391.32008

[24] S. Yu - “Todd class via homotopy perturbation theory” (2015), arXiv:1510.07936