Three contributions to topology, algebraic geometry and representation theory: homological finiteness of abelian covers, algebraic elliptic cohomology theory, and monodromy theorems in the elliptic setting by Yaping Yang B.S. in Mathematics, Zhejiang University M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy April 18, 2014 Dissertation directed by Valerio Toledano Laredo Professor of Mathematics Acknowledgments I would like to express my deepest gratitude to my advisor, Prof. Valerio Toledano Laredo, and unofficial advisors, Prof. Alex Suciu and Prof. Marc Levine. During the past 5 years, I have received tremendous support and encouragement from them. They showed me the road and helped to get me started on Mathematics. They were always available for my questions and were very positive, patient and gave generously of their time and knowledge. I wrote my first paper with Prof. Alex Suciu, and gave my first talk under the help of Prof. Alex Suciu. I am still benefit from his way of thinking and his attitude to mathematics. I am very grateful to Prof. Valerio Toledano Laredo for his enormous help, who treats everything seriously to make it perfect and always reminds me the seminars, workshops, talks, deadlines and etc. He has done me a great favor (even spent more time than me) in the process of finding me a postdoctoral position. I would like to thank Prof. Pavel Etingof and Prof. Ivan Losev for being members of the committee. Their enthusiasm and passion for mathematics always inspires me. Their responsible, humor and pizza make the seminar enjoyable and memorable. I am particularly grateful to Prof. Pavel Etingof for encouragement and insights with admiration, who provides valuable advices to the work and generously shares his ideas, and always ready to help. I would also like to thank my professors at Northeastern University, Prof. Venkatraman Lakshmibai, Prof. Alina Marian, Prof. Jonathan Weitsman, Prof. Ben Webster, Prof. Jerzy Weyman and Prof. Andrei Zelevinsky for offering excellent courses and seminars over the years, for their guidance and help. ii In addition, I would like to give many thanks to my friends and colleagues, Gufang, Andrea, Sachin, Salvo, Shih-Wei, Yinbang, Andy, Kavita, Federico, Thomas, Jason, Andras, Nate, Jeremy, Barbara, Simone, Adina, Chen, Brian, Rueven, Floran, Undine, Nicholas, Ryan and Ryan, Gouri, Anupam, Jose, Rahul, Liang, He, Ruoran, Huijun, Sasha, Saif, Andrew, Peng, Yong, Thi, Toan, Lei, Annabelle, Tian, Jiuzu, Yi, John, Lee-Peng, and many others. Many years with them, full of happiness and joy. Especially, thanks Gufang and Sasha who are the potentially speakers whenever I couldn't find enough speakers of the graduate student seminar, thanks Andrea and Hanai for sharing their wonderful templates, thanks Shih-Wei for liking every post on my facebook. And finally, I would like to thank my family, my parents, my sister, my brother in-law, my lovely niece for their love and support. I am particularly grateful to my fiance, and collaborator Gufang, for cooking up good math and delicious food for us over the past 5 years. Yaping Yang Northeastern University April 182014 iii Abstract of Dissertation My doctoral work consists of three projects. The first project is joint work with A. Suciu and G. Zhao described in more details in Chapter 2 and 3 of this dissertation. Chapter 2 is based on the paper [82], where we exploit the classical correspondence between finitely generated abelian groups and abelian complex algebraic reductive groups to study the intersection theory of translated subgroups in an abelian complex algebraic reductive group, with special emphasis on intersections of (torsion) translated subtori in an algebraic torus. Chapter 3 is based on the paper [83], we present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of rank r coincides with the Grassmanian of r-planes in H1(X; Q). Inside this parameter space, there i is a subset ΩA(X) consisting of all the covers with finite Betti numbers up to degree i. Building on work of Dwyer and Fried, we show how to compute these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. For certain spaces, such as smooth, quasi-projective varieties, the generalized Dwyer{Fried invariants that we introduce here can be computed in terms of intersections of algebraic subtori in the character group. For many spaces of interest, the homological finiteness of abelian covers can be tested through the corresponding free abelian covers. Yet in general, abelian covers exhibit different homological finiteness properties than their free abelian counterparts. iv The second project is joint work with M. Levine and G. Zhao described in more details in Chapter 4 of this dissertation, which is based on the paper [91]. We define the algebraic elliptic cohomology theory coming from Krichever's elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic cobordism ring with rational coefficients, the ideal generated by differences of classical flops coincides with the kernel of Krichever's elliptic genus. This generalizes a theorem of B. Totaro in the complex analytic setting. The third project consists of two parts. Chapter 5 is joint work with N. Guay, where we give a double loop presentation of the deformed double current algebras, which are de- formations of the central extension of the double current algebras g[u; v], for a simple Lie algebra g. We prove some nice properties of the algebras using the double loop presentation. Especially, we construct a central element of the deformed double current algebra. Chapter 6 is joint work with V. Toledano Laredo. In [10], Calaque-Enriquez-Etingof constructed the universal KZB equation, which is a flat connection on the configuration space of n points on an elliptic curve. They show that its monodromy yields an isomorphism between the completions of the group algebra of the elliptic braid group of type An−1 and the holonomy algebra of coefficients of the KZB connection. We generalized this connection and the corresponding formality result to an arbitrary root system in [91]. We also gave two concrete incarnations of the connection: one valued in the rational Cherednik algebra of the corresponding Weyl group, the other in the double deformed current algebra D(g) of the corresponding Lie algebra g. The latter is a deformation of the double current algebra g[u; v] recently defined by Guay in [34, 35], and gives rise to an elliptic version of the Casimir connection. v Table of Contents Acknowledgments ii Abstract of Dissertation iv Table of Contents vi Disclaimer xiv Chapter 1 Introduction1 1.1 Brief Overview..................................1 1.2 Characteristic varieties and Betti numbers of abelian covers..........2 1.2.1 Motivations................................2 1.2.2 Results achieved.............................3 1.2.3 Organization of Chapter 3........................4 1.3 The algebraic elliptic cohomology theory....................5 1.3.1 Motivations................................5 1.3.2 Results achieved.............................7 1.3.3 Future directions.............................9 1.3.4 Organization of Chapter 4........................ 11 1.4 Universal KZB equations and the elliptic Casimir connection......... 12 1.4.1 Motivations................................ 12 vi 1.4.2 Results achieved............................. 16 1.4.3 Future directions............................. 19 Chapter 2 Intersections of translated algebraic subtori 22 2.1 Finitely generated abelian groups and abelian reductive groups........ 22 2.1.1 Abelian reductive groups......................... 22 2.1.2 The lattice of subgroups of a finitely generated abelian group..... 24 2.1.3 The lattice of algebraic subgroups of a complex algebraic torus.... 25 2.1.4 Counting algebraic subtori........................ 27 2.2 Primitive lattices and connected subgroups................... 28 2.2.1 Primitive subgroups........................... 28 2.2.2 The dual lattice.............................. 30 2.3 Categorical reformulation............................. 33 2.3.1 Fibered categories............................. 33 2.3.2 A lattice over AbRed ........................... 34 2.3.3 An equivalence of fibered categories................... 35 2.4 Intersections of translated algebraic subgroups................. 35 2.4.1 Two morphisms.............................. 35 2.4.2 Translated algebraic subgroups..................... 36 2.4.3 Some consequences............................ 39 2.4.4 Abelian covers............................... 40 2.5 Exponential interpretation............................ 41 2.5.1 The exponential map........................... 41 2.5.2 Exponential map and Pontrjagin duality................ 42 2.5.3 Exponentials and determinant groups.................. 43 2.5.4 Some applications............................. 44 2.6 Intersections of torsion-translated subtori.................... 45 vii 2.6.1 Virtual belonging............................. 45 2.6.2 Torsion-translated tori.......................... 47 Chapter 3 Homological finiteness of abelian covers 48