The Pennsylvania State University The Graduate School THE DOLBEAULT DGA OF A FORMAL NEIGHBORHOOD A Dissertation in Mathematics by Shilin Yu © 2013 Shilin Yu Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2013 The dissertation of Shilin Yu was reviewed and approved∗ by the following: Nigel Higson Evan Pugh Professor of Mathematics Dissertation Advisor, Chair of Committee Paul Baum Evan Pugh Professor of Mathematics Ping Xu Distinguished Professor of Mathematics Radu Roiban Professor of Physics Yuxi Zheng Professor and Department Head of Mathematics ∗Signatures are on file in the Graduate School. Abstract The derived category of coherent sheaves has been widely accepted as an interest- ing invariant of varieties and played a key role in Kontsevich’s homological mirror symmetry conjecture. On the other hand, however, it is known that derived category has several defects, which can be overcome by introducing the notion of differential graded category (dg-category). Recently, Jonathan Block defined a dg-category PA for any given differential graded algebra (dga) A = (A•, d). When A is the Dolbeault complex of a compact complex manifold X, he showed that the homotopy category of b PA is equivalent to the bounded derived category D (X) of X. Hence PA serves as an explicit model for Db(X). The same construction can be carried out for curved dgas and was used by Block to study noncommutative T-duality. This dissertation proves a generalized version of Block’s theorem for the coherent sheaves over the formal neighborhood Yˆ of a closed embedding i : X ,! Y of com- plex manifolds. We define a dga (A• , d), which is the analogue to the usual Dolbeaut Yˆ complex, for Yˆ and prove that the homotopy category of PA in this situation is equiv- alent to the bounded derived category Db(Yˆ ) of Yˆ . This dissertation also provides a geometric discription of the dga (A• , d), which generalizes the result of Kapranov on Yˆ the diagonal embedding ∆ : X ,! X × X. iii Table of Contents Acknowledgments vii Chapter 1 Introduction 1 1.1 Cohesive modules . 1 1.2 Ext groups in algebraic geometry . 6 1.3 Main results of the dissertation . 8 Chapter 2 Preliminaries 12 2.1 Derived categories . 12 2.1.1 Derived category of an abelian category . 12 2.1.2 Triangulated structure . 23 2.1.3 Derived functors . 24 2.2 Derived categories of coherent sheaves . 28 2.2.1 Derived categories in algebraic geometry . 28 2.2.2 Derived categories in analytic geometry . 30 2.2.3 Derived functors in algebraic and analytic geometry . 31 2.2.3.1 Direct image . 31 2.2.3.2 Sheaf cohomology . 33 2.2.3.3 Tensor product . 33 2.2.3.4 Inverse image . 35 2.3 DGAs and dg-categories . 35 2.4 The perfect category of cohesive modules . 38 2.4.1 The perfect category PA ........................ 38 2.4.2 Triangulated structure . 39 2.4.3 Homotopy equivalences . 40 iv 2.4.4 The case of complex manifolds . 41 Chapter 3 Dolbeault dga of formal neighborhoods 43 3.1 Definition and basic properties . 43 3.2 Diagonal embeddings and jet bundles . 54 Chapter 4 Dolbeault resolutions on formal neighborhood 58 4.1 Dolbeault resolutions of coherent sheaves . 58 4.2 Holomorphic vector bundles over a formal neighborhood . 60 4.3 Descent of PA .................................. 66 4.3.1 Inverse image functor . 66 4.3.2 Homotopy fiber products of dg-categories . 67 4.3.3 Descent . 69 4.3.4 The case of formal neighborhoods . 70 4.4 Cohesive modules over a formal neighborhood . 71 4.A Flatness of A • over O ............................. 83 Yˆ Yˆ 4.A.1 Preliminaries on completions . 83 4.A.1.1 Completions of modules and sheaves . 83 4.A.1.2 Completed tensor products . 85 4.A.1.3 Two useful lemmas . 87 4.A.2 Some Stein theory . 89 4.A.2.1 Semi-analytic subsets . 90 4.A.2.2 Cartan’s Theorem A & B for Yˆ . 91 4.A.3 Proof of Theorem 4.1.4 . 96 Chapter 5 Diagonal embedding and formal geometry 101 5.1 Differential geometry on formal discs . 102 5.2 Formal geometry . 107 5.3 The Kahler¨ case: Kapranov’s result revisited . 114 Chapter 6 General case of arbitrary embeddings 119 6.1 Differential geometry of complex submanifolds . 119 6.1.1 Splitting of normal exact sequence and Kodaira-Spencer class . 119 6.1.2 Shape operator . 121 6.2 Taylor expansions in normal direction . 122 6.2.1 General discussions . 122 v ∗ 6.2.2 Description of πe ............................126 6.2.3 Description of the derivation D . 130 Bibliography 136 vi Acknowledgments I owe my deepest gratitude to my advisor, Nigel Higson, for his patient guidance and support over the years. His encouragement helped me get over my difficult times of the past two years. I would like to thank Jonathan Block, Damien Calaque, Andrei Cald˘ araru,˘ Math- ieu Stienon, Junwu Tu and Ping Xu for their inspriational conversations and generous sharing of ideas. It is also a pleasure to thank all my committe members for their help- ful comments. It is always enjoyable to talk to the students and postdocs in the Noncommuta- tive Geometry and Mathematical Physics group, including, but not limited to, Tyrone Crisp, Calder Daenzer, Paul Siegel, Yanli Song, Qijun Tan, Tianyu Tan, Rufus Willett, Yijun Yao and Allan Yashinski. It is also a lot fun for me to talk to students from other areas of mathematics, such as Jianyu Chen and Jingjing Huang. I am deeply grateful to Junyan Cao, Zhechao Ruan and Zhijie Sun for their friendships over so many years. No words can express my thankfulness to my family, especilly my parents and my stepmother, for their love and support. I could never have made it to this point without them. vii Dedication Dedicated to my father, my stepmother and to the memory of my mother. viii Chapter 1 Introduction 1.1 Cohesive modules In algebraic geometry, the notion of coherent (algebraic) sheaves on algebraic vari- eties is fundamental and lies at the heart of the subject. It is well known that coher- ent sheaves over non-singular, quasi-projective varieties always admit resolutions by locally free sheaves of finite type. In contrast, however, not every coherent analytic sheaf over an analtyic space has a resolution by holomorphic vector bundles, although locally such resolutions always exist. This is one of the main differences between al- gebraic and analytical geometry and it causes problems when one wants to translate concepts from the algebraic side to the analytical side. For example, Grothendieck [Gro58] defined Chern classes for coherent sheaves using locally free resolutions. For coherent analytic sheaves, the situation is more subtle due to the lack of global resolu- tions and one needs to get around this issue by implementing alternative approaches (see, e.g., [Gri10]). Thus the meaning of ‘resolutions’ has to be adjusted to be fitted into the analytical world. One effort in this direction goes back to the papers of Toledo and Tong, [TT76], 2 [TT78], in which they introduced the concept of twisted complexes. The construction was of Czech-type.ˇ Recently, a corresponding Dolbeault version was carried out by Block, [Blo10], [Blo06]. To any compact complex manifold X, he assigned a differen- tial graded (dg) category PA that he called the perfect category of cohesive modules over the Dolbeault differential graded algebra (dga) A = (A•(X), ¶), where Aq(X) are the spaces of smooth (0, q)-forms on X and ¶ is the (0, 1)-part of the De Rham differen- tial. The significance of this notion is embodied in (but not limited to) the following theorem: Theorem 1.1.1 ([Blo10]). Let X be a compact complex manifold and A = (A•(X), ¶) the b Dolbeault dga. The homotopy category HoPA of the dg-category PA is equivalent to Dcoh(X), the bounded derived category of complexes of sheaves of OX-modules with coherent cohomology. In fact, the notion of cohesive modules was defined in [Blo10] for an arbitrary (non- negatively graded) dga A = (A•, d) (and even more generally, for a curved dga). We write A = A0 for the degree 0 part of A• and think of it as “functions” over some • (noncommutative) space. A cohesive module (E , E) over A consists of two pieces of • data: a bounded Z-graded (left) A-module E which is finitely generated and projec- tive, together with a Z-connection of total degree one • • • • E : A ⊗A E !A ⊗A E that satisfies the Leibniz rule for the action of A• and also the integrability condition • • • E ◦ E = 0. Thus E is determined by its value on E ⊆ A ⊗A E and has a canonical decomposition E = E0 + E1 + E2 + ··· k • k •−k+1 1 such that E : E !A ⊗A E . It is not hard to see that E is a connection on each 3 component En in the ordinary sense and that Ek is A-linear for k 6= 1. Also note that • the integrability condition for E implies that E0 ◦ E0 = 0, so (E , E0) is a complex of A- • modules in the usual sense. A morphism between two cohesive modules E1 = (E1, E1) • • • • • • and E2 = (E2, E2) is an A -linear homomorphism φ : A ⊗A E1 !A ⊗A E2. The • set of all morphisms forms a complex PA(E1, E2) with the differential defined in a standard way jφj d(φ) = E2 ◦ φ − (−1) φ ◦ E1. • In the case when A is the Dolbeault dga of a compact complex manifold X, (E , E0) corresponds to a complex of smooth complex vector bundles over X and E1 : En ! 1 n n A ⊗A E gives a (0, 1)-connection on each E .