Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


The Dolbeault dga of a formal neighborhood

Author: Shilin Yu
Journal: Trans. Amer. Math. Soc. 368 (2016), 7809-7843
MSC (2010): Primary 18D20; Secondary 14B20, 18E30, 58A20
Published electronically: June 10, 2016
MathSciNet review: 3546785
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Inspired by a work of Kapranov (1999), we define the notion of a Dolbeault complex of the formal neighborhood of a closed embedding of complex manifolds. This construction allows us to study coherent sheaves over the formal neighborhood via a complex analytic approach, as in the case of usual complex manifolds and their Dolbeault complexes. Moreover, the Dolbeault complex as a differential graded algebra can be associated with a dg-category according to Block (2010). We show that this dg-category is a dg-enhancement of the bounded derived category over the formal neighborhood under the assumption that the submanifold is compact. This generalizes a similar result of Block in the case of usual complex manifolds.

References [Enhancements On Off] (What's this?)

  • [AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39 #4129)
  • [BB] O. Ben-Bassat and J. Block, Cohesive DG Categories I: Milnor Descent,
  • [BK91] A. I. Bondal and M. M. Kapranov, Framed triangulated categories, Mat. Sb. 181 (1990), no. 5, 669-683 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 1, 93-107. MR 1055981 (91g:18010)
  • [Blo10] Jonathan Block, Duality and equivalence of module categories in noncommutative geometry, A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Amer. Math. Soc., Providence, RI, 2010, pp. 311-339. MR 2648899 (2011k:19005)
  • [Bor95] Emile Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. École Norm. Sup. (3) 12 (1895), 9-55 (French). MR 1508908
  • [BS76] Constantin Bănică and Octavian Stănăşilă, Algebraic methods in the global theory of complex spaces, Editura Academiei, Bucharest; John Wiley & Sons, London-New York-Sydney, 1976. Translated from the Romanian. MR 0463470 (57 #3420)
  • [CCT14] Damien Calaque, Andrei Căldăraru, and Junwu Tu, On the Lie algebroid of a derived self-intersection, Adv. Math. 262 (2014), 751-783. MR 3228441,
  • [DK90] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726 (92a:57036)
  • [Eis95] David Eisenbud, Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • [FM07] Domenico Fiorenza and Marco Manetti, $ L_\infty $ structures on mapping cones, Algebra Number Theory 1 (2007), no. 3, 301-330. MR 2361936 (2008k:17036),
  • [Fri67] Jacques Frisch, Points de platitude d'un morphisme d'espaces analytiques complexes, Invent. Math. 4 (1967), 118-138 (French). MR 0222336 (36 #5388)
  • [GH78] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725 (80b:14001)
  • [GPR94] H. Grauert, Th. Peternell, and R. Remmert (eds.), Several complex variables. VII, Encyclopaedia of Mathematical Sciences, vol. 74, Springer-Verlag, Berlin, 1994. Sheaf-theoretical methods in complex analysis; A reprint of Current problems in mathematics. Fundamental directions. Vol. 74 (Russian), Vseross. Inst. Nauchn. i Tekhn.Inform. (VINITI), Moscow. MR 1326617 (96k:32001)
  • [Gro60] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228. MR 0217083 (36 #177a)
  • [Gro61] A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167 (French). MR 0163910 (29 #1209)
  • [Kap99] M. Kapranov, Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71-113. MR 1671737 (2000h:57056),
  • [Kel94] Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63-102. MR 1258406 (95e:18010)
  • [Kel06] Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151-190. MR 2275593 (2008g:18015)
  • [KM58] J.-L. Koszul and B. Malgrange, Sur certaines structures fibrées complexes, Arch. Math. (Basel) 9 (1958), 102-109 (French). MR 0131882 (24 #A1729)
  • [KS90] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006 (92a:58132)
  • [Loj64] S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 449-474. MR 0173265 (30 #3478)
  • [Mal67] B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575 (35 #3446)
  • [Man07] Marco Manetti, Lie description of higher obstructions to deforming submanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 4, 631-659. MR 2394413 (2009e:17034)
  • [Mat80] Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344 (82i:13003)
  • [Ran08] Ziv Ran, Lie atoms and their deformations, Geom. Funct. Anal. 18 (2008), no. 1, 184-221. MR 2399101 (2009d:32009),
  • [Siu69] Yum-tong Siu, Noetherianness of rings of holomorphic functions on Stein compact series, Proc. Amer. Math. Soc. 21 (1969), 483-489. MR 0247135 (40 #404)
  • [Yua] Shilin Yu, The Dolbeault dga of the formal neighborhood of the diagonal, J. Noncommut. Geom. 9 (2015), no. 1, 161-184. MR 3337957,
  • [Yub] S. Yu, The $ L_\infty $-algebroid of the formal neighborhood, in preparation.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 18D20, 14B20, 18E30, 58A20

Retrieve articles in all journals with MSC (2010): 18D20, 14B20, 18E30, 58A20

Additional Information

Shilin Yu
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104

Keywords: Formal neighborhoods, derived categories, differential graded algebras, differential graded categories
Received by editor(s): March 3, 2013
Received by editor(s) in revised form: December 8, 2014
Published electronically: June 10, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society