Memorial Misha Verbitsky, January 20141 Contents 1 Personal details 2 2 Brief Description of Academic Activities 2 2.1 Undergraduate . 2 2.2 Doctorate . 3 2.3 Activities Since Doctorate. 3 2.3.1 Positions held . 3 2.3.2 Visits to International Centres: . 3 2.3.3 Research interests . 4 2.3.4 Teaching . 4 2.3.5 Publications . 4 2.3.6 Research interests . 4 3 Research overview 4 3.1 Hodge theory on hyperk¨ahlermanifolds and its applications . 4 3.2 Trianalytic subvarieties of hyperk¨ahlermanifolds . 5 3.3 Holomorphic Lagrangian subvarieties . 5 3.4 Coherent sheaves on hyperk¨ahlermanifolds . 6 3.5 Moduli spaces of framed instanton bundles on CP 3 and the rational curves on the twistor space . 7 3.6 Hodge theory on hypercomplex manifolds and HKT-geometry . 7 3.7 Complex structures and special holonomies . 8 3.8 Plurisubharmonic functions in hypercomplex geometry . 9 3.9 Locally conformally K¨ahler geometry . 9 3.10 Toric and elliptic fibrations . 10 3.11 Moduli spaces and the mapping class group . 11 4 Published Works 17 4.1 Books . 17 4.2 Journal articles . 17 4.3 Book chapters, conference proceedings . 22 [email protected] 1 5 Citations of Published Works 23 6 Conferences organized 24 7 Awards and distinctions 25 8 Grants obtained 25 9 Selected talks at international conferences 26 10 Teaching activities 30 10.1 Courses taught . 30 10.2 Supervising . 33 11 Administrative duties 33 12 Other skills 34 1 Personal details Born: June 20, 1969, Moscow, USSR Citizenship: Russia Homepage: http://verbit.ru/ Telephone Number (work): +7 (495) 772-95-90 *44160 E-mail: verbit[]mccme.ru, verbit[]verbit.ru 2 Brief Description of Academic Activities 2.1 Undergraduate I never got an undergraduate degree, getting my undergraduate education at the mathematical high school 57 in Moscow. While in high school, I learned algebraic geometry and (just after my grad- uation from 57) wrote a paper \On the action of a Lie algebra SO(5) on the cohomology of a hyperk¨ahlermanifold," (Func. Analysis and Appl. 24(2) p. 70-71). Based on this paper and several talks on Yau's seminar, I was admitted to graduate school at Harvard University in 1991. I spent some time in Tver University and in Moscow University to con- form with Soviet social policy, without intention to complete the undergrad- uate degree here. 2 2.2 Doctorate I defended my Ph. D. thesis \Cohomology of compact hyperk¨ahlermanifolds", at Harvard under supervision of David Kazhdan (1995). 2.3 Activities Since Doctorate. 2.3.1 Positions held • 1990-91 Visiting Scholar, MIT • 1991-95 Graduate Student, Harvard University • 1996-97 A member of Institute of Advanced Study, Princeton • 1997-99 A member of European Post-Doctoral Institute • 1996{ Moscow Independent University (professor) • 2003-10 ITEP (Moscow) • 2002{07 Glasgow University, EPSRC Advanced Fellow • 2008{ University of Tokyo, IPMU (joint appointment) • 2010{ Faculty of Mathematics, Higher School of Economics (pro- fessor), Laboratory of algebraic geometry and its applications (chair, vice-chair). • Visitor at IHES (1997, 2006), MPI, Bonn (1999-2000, 2006). 2.3.2 Visits to International Centres: After my doctorate, I spent a year at IAS, Princeton (1996-97), a year at IHES, Bures sur Yvette as a member of European Post-Doctoral Institute (1997-98), half a year as a visitor of Max-Planck, Bonn (1999-2000) and 7 years as an EPSRC Advanced Research Fellow at Glasgow University (re- search position equivalent to American Associate Professor, paid by EPSRC personal grant of 250,000 pounds, 2001-2007). Since 2007 I have a joint ap- pointment at Tokyo University (Kavli Institute of Physics and Mathematics of the Universe), upgraded to Senior Joint Appointment after I became a full professor at my university in 2012. I held brief visiting positions at IHES (2006), Max-Planck Institute, Bonn (2006), MSRI (2003), and Fields Institute, Toronto (2006). 3 2.3.3 Research interests 2.3.4 Teaching Designed a curriculum for first- and second-year math majors (algebra, num- ber theory, analysis, topology). Gave lectures for first-year undergraduate students based on this curriculum (2004, 2008). Wrote problem-based un- dergraduate textbooks on measure theory, Galois theory and point-set topol- ogy. Gave graduate courses on algebraic geometry, K¨ahler manifolds, Gro- mov hyperbolic groups, Mori theory, complex surfaces, locally conformally K¨ahlermanifolds, differential geometry, spinors and elliptic equations. Gave undergraduate courses on topology, analysis on manifolds, measure theory, Galois theory, algebraic geometry and geometric group theory. 2.3.5 Publications I wrote 77 papers, 42 of them in collaboration, a monograph and a textbook. 68 of these papers were published in peer-refereed publications, the rest is available from arxiv.org. According to Google Scholar,2 my Hirsch index is 20, and there are 1212 people citing my papers. 2.3.6 Research interests Algebraic geometry, differential geometry, hyperk¨ahlergeometry, calibra- tions on manifolds, quaternionic structures, CAT-spaces, locally conformally K¨ahlermanifolds, ergodic theory, hyperbolic geometry, Yang-Mills theory, Hodge theory, symplectic topology. 3 Research overview 3.1 Hodge theory on hyperk¨ahlermanifolds and its applications In [V90], [V94], [V95:1], [V95:2], I studied the applications of Hodge theory for topology of hyperk¨ahlermanifolds. It was shown that cohomology of a hyperk¨ahlermanifold admit an action of the Lie group Sp(1; 1), which is similar to Lefschetz' SL(2)-action. This was used to compute the cohomol- ogy algebra of a hyperk¨ahlermanifold, showing that its part generated by H2(M) is symmetric, up to the middle dimension. Among applications of these results, a proof of Mirror Conjecture for hyperk¨ahlermanifolds, and better understanding of hyperk¨ahlersubvarieties of hyperk¨ahlermanifolds and coherent sheaves. 2http://scholar.google.com/citations?user=8KhODVoAAAAJ&hl=en 4 The latest achievement in this direction was a proof of global Torelli theorem, [V09:2], which was a subject of Bourbaki seminar talk by Dan Huybrechts ([H]); this result already has many applications ([AV10], [A], [BS], [Mn], most notably { a proof of Kawamata-Morrison's cone conjecture for hyperk¨ahlermanifolds by Eyal Markman ([Ma]). 3.2 Trianalytic subvarieties of hyperk¨ahlermanifolds The papers [V93], [V94], [V96:1], [V96:2], [V97:1], [V97:2], [KV98:1], [V98], [KV98:2] and [V03:4] deal with trianalytic subvarieties in hyperk¨ahlermanifolds. These are subvarieties which are complex analytic with respect to three complex structures I; J; K. It was shown that all complex subvarieties of a generic hyperk¨ahlermanifold are trianalytic. Also all deformations of trianalytic subvarieties are again trianalytic, and their deformation space is singular hyperk¨ahler. I have studied singularities of singular hyperk¨ahlervarieties, and shown that a normalization of such variety is smooth and hyperk¨ahler. This ap- plies to trianalytic subvarieties, which are examples of singular hyperk¨ahler spaces. These results were applied to Hilbert schemes of points on K3 and gen- eralized Kummer varieties. For Hilbert schemes of points on K3, it was shown that its generic deformation has no subvarieties. For generalized Kummer varieties, a similar attempt (joint with D. Kaledin) was foiled by our imperfect understanding of birational geometry of holomorphic symplec- tic manifolds. Soon after publishing this paper, we found a counterexample to one of our statements. The birational geometry of holomorphic symplectic manifolds was stud- ied by D. Kaledin in several papers in much detail. Our joint work in this direction resulted in [KV00], where a deformation theorem, analoguous to Bogomolov-Tian-Todorov, was obtained for non-compact holomorphic sym- plectic manifolds. Also, in [V99] I have studied quotient singularities of holomorphic symplectic manifolds admitting a holomorphic symplectic res- olution, and proved that such singularities are always quotients by groups generated by symplectic reflections. 3.3 Holomorphic Lagrangian subvarieties In a joint paper with Geo Grantcharov [GV10], we put the results on trianalytic subvarieties into a more general framework of calibrated geometry. We considered calibrations on hyperk¨ahlermanifolds which can be expressed as polynomials of holo- morphic symplectic forms. One of these calibrations calibrates precisely the trianalytic subvarieties. We found another such calibration, calibrating holo- morphic Lagrangian subvarieties. Surprisingly, for the latter calibration to 5 be closed, a manifold needs not to be hyperk¨ahler:we have shown that such a calibration exists on a hypercomplex manifold, provided that its holonomy lies in SL(n; H). The holomorphic Lagrangian subvarieties appear in another context con- nected to the Mirror Symmetry. A Strominger-Yau-Zaslow (SYZ) conjecture postulates existence of special Lagrangian fibrations pm Calabi-Yaus; the duality of the fibers of these fibrations is responsible for Mirror Symmetry. Any holomorphic Lagrangian subvariety on a hyperk¨ahlermanifold is speacial Lagrangian with respect to some other complex structure, hence existence of holomorphic Lagrangian fibrations can be understand as a spe- cial case of the SYZ conjecture. I wrote a couple of papers ([V08:4], [V09:1]) dealing with this conjecture, and proved, in particular, that a hyperk¨ahler manifold which has no cohomological obstructions to existence of holomor- phic Lagrangian fibrations admits, at least, a coisotropic subvariety. 3.4 Coherent sheaves on hyperk¨ahlermanifolds The papers [V92], [KV96], [V97:3], [V01:1], [MV06] are about hyperholomorphic bundles on hyperk¨ahlermanifolds. These are bundles with Hermitian connection with curvature of type (1,1) with respect to all complex structures induced by the hyperk¨ahlerstructure. It was shown that all stable bundles on generic hyperk¨ahlermanifolds admit a hyperholomorphic connection, which is unique. Conversely, every hyperholomorphic bundle is a direct sum of stable bundles. The moduli spaces of such bundles wre shown to be singular hyperk¨ahler, and the deformation unobstructed (except the first obstruction, known as Yoneda product).