The Dolbeault dga of a formal neighborhood

Yu, Shilin

doi:10.1090/tran6646

The Dolbeault dga of a formal neighborhood
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Trans. Amer. Math. Soc. 368 (2016), 7809-7843 Request permission

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

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
O. Ben-Bassat and J. Block, Cohesive DG Categories I: Milnor Descent, arXiv:math.AG/1201.6118v1.
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, DOI 10.1070/SM1991v070n01ABEH001253
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, DOI 10.1090/crmp/050/24
Emile Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. École Norm. Sup. (3) 12 (1895), 9–55 (French). MR 1508908
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
Damien Calaque, Andrei Căldăraru, and Junwu Tu, On the Lie algebroid of a derived self-intersection, Adv. Math. 262 (2014), 751–783. MR 3228441, DOI 10.1016/j.aim.2014.06.002
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
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Domenico Fiorenza and Marco Manetti, $L_\infty$ structures on mapping cones, Algebra Number Theory 1 (2007), no. 3, 301–330. MR 2361936, DOI 10.2140/ant.2007.1.301
Jacques Frisch, Points de platitude d’un morphisme d’espaces analytiques complexes, Invent. Math. 4 (1967), 118–138 (French). MR 222336, DOI 10.1007/BF01425245
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
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, DOI 10.1007/978-3-662-09873-8
A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
M. Kapranov, Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71–113. MR 1671737, DOI 10.1023/A:1000664527238
Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406
Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151–190. MR 2275593
J.-L. Koszul and B. Malgrange, Sur certaines structures fibrées complexes, Arch. Math. (Basel) 9 (1958), 102–109 (French). MR 131882, DOI 10.1007/BF02287068
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, DOI 10.1007/978-3-662-02661-8
S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 449–474. MR 173265
B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
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
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Ziv Ran, Lie atoms and their deformations, Geom. Funct. Anal. 18 (2008), no. 1, 184–221. MR 2399101, DOI 10.1007/s00039-008-0655-x
Yum-tong Siu, Noetherianness of rings of holomorphic functions on Stein compact series, Proc. Amer. Math. Soc. 21 (1969), 483–489. MR 247135, DOI 10.1090/S0002-9939-1969-0247135-7
Shilin Yu, The Dolbeault dga of the formal neighborhood of the diagonal, J. Noncommut. Geom. 9 (2015), no. 1, 161–184. MR 3337957, DOI 10.4171/JNCG/190
S. Yu, The $L_\infty$-algebroid of the formal neighborhood, in preparation.