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 hyperkählermanifolds and its applications . 4 3.2 Trianalytic subvarieties of hyperkählermanifolds ...... 5 3.3 Holomorphic Lagrangian subvarieties ...... 5 3.4 Coherent sheaves on hyperkählermanifolds ...... 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 Kähler 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 undergraduate 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 (professor), 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 (research position equivalent to American Associate Professor, paid by EPSRC personal grant of 250,000 pounds, 2001-2007). Since 2007 I have a joint appointment 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, number theory, analysis, topology). Gave lectures for first-year undergraduate students based on this curriculum (2004, 2008). Wrote problem-based undergraduate textbooks on measure theory, Galois theory and point-set topology. Gave graduate courses on algebraic geometry, Kähler manifolds, Gro- mov hyperbolic groups, Mori theory, complex surfaces, locally conformally Kählermanifolds, 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, hyperkählergeometry, calibrations on manifolds, quaternionic structures, CAT-spaces, locally conformally Kählermanifolds, ergodic theory, hyperbolic geometry, Yang-Mills theory, Hodge theory, symplectic topology.

3 Research overview

3.1 Hodge theory on hyperkählermanifolds and its applications In [V90], [V94], [V95:1], [V95:2], I studied the applications of Hodge theory for topology of hyperkählermanifolds. It was shown that cohomology of a hyperkählermanifold admit an action of the Lie group Sp(1, 1), which is similar to Lefschetz’ SL(2)-action. This was used to compute the cohomology algebra of a hyperkählermanifold, 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 hyperkählermanifolds, and better understanding of hyperkählersubvarieties of hyperkählermanifolds 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 hyperkählermanifolds 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 hyperkählermanifolds. 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 hyperkählermanifold are trianalytic. Also all deformations of trianalytic subvarieties are again trianalytic, and their deformation space is singular hyperkähler. I have studied singularities of singular hyperkählervarieties, and shown that a normalization of such variety is smooth and hyperkähler. This ap- plies to trianalytic subvarieties, which are examples of singular hyperkähler spaces. These results were applied to Hilbert schemes of points on K3 and generalized 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 symplectic manifolds. Soon after publishing this paper, we found a counterexample to one of our statements. The birational geometry of holomorphic symplectic manifolds was studied 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 symplectic 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 holomorphic symplectic forms. One of these calibrations calibrates precisely the trianalytic subvarieties. We found another such calibration, calibrating holomorphic Lagrangian subvarieties. Surprisingly, for the latter calibration to

5 be closed, a manifold needs not to be hyperkähler: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 connected 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 hyperkählermanifold is speacial Lagrangian with respect to some other complex structure, hence existence of holomorphic Lagrangian fibrations can be understand as a special 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 hyperkähler manifold which has no cohomological obstructions to existence of holomorphic Lagrangian fibrations admits, at least, a coisotropic subvariety.

3.4 Coherent sheaves on hyperkählermanifolds The papers [V92], [KV96], [V97:3], [V01:1], [MV06] are about hyperholomorphic bundles on hyperkählermanifolds. These are bundles with Hermitian connection with curvature of type (1,1) with respect to all complex structures induced by the hyperkählerstructure. It was shown that all stable bundles on generic hyperkählermanifolds 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 hyperkähler, and the deformation unobstructed (except the first obstruction, known as Yoneda product). Later, this notion was extended to coherent sheaves. Using results about trianalytic subvarieties, I have shown that a deformation of a hyperholomorphic bundle over a generic hyperkählermanifold M remains non-singular, unless M contains trianalytic subvarieties of complex codimension 2 (in the cases of a Hilbert scheme of K3 and the generalized Kummer variety, all trianalytic subvarieties have codimension > 2, except for 4-dimensional generalized Kummer). If the deformation spaces of hyperholomorphic bundles are positive- dimensional (which is unknown yet), this should lead to new examples of hyperkählermanifolds. In [KV96], a non-Hermitian version of this notion was studied. We have shown that if the Hermitian assumption is dropped, the non-Hermitian hyperholomorphic connection on M becomes essentially the same as a holomorphic structure on the lifting of the corresponding bundle to the twistor space. In [V02], [V03:3] this approach was used to study the category of coherent sheaves on generic K3 surfaces and tori. It was shown that this

6 category is independent from the choice of a generic K3 or a torus of a given dimension.

3.5 Moduli spaces of framed instanton bundles on CP 3 and the rational curves on the twistor space In [KV96], we constructed a correspondence between stable vector bundles on a twistor space of a hyperkaähler manifopld and rational curves in a twistor space of another hyperkähler manifold. This observation was used in a recent work with Marcus Jardim ([JV10], [JV11]). We have shown that the moduli space M of holomorphic vector bundles on CP 3 that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of 2 the moduli space of framed instantons on C , called the space of twistor sections. This space admits an interesting geometric structure, called a trisymplectic structure. A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has rank 2n, n or 0. We have shown that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. Then we constructed a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkählermanifold, and defined a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hy- perkählerreduction. We proved that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkählermanifold M is compatible with the hyperkählerreduction on M. As an application of these geometric ideas, we considered the ADHM construction of instantons. We have shown that the moduli space of rank r, charge c framed instanton bundles on CP 3 is a smooth, connected, trisymplectic manifold of complex dimension 4rc. In particular, it follows that the moduli space of rank 2, charge c instanton bundles on CP 3 is a smooth complex manifold dimension 8c − 3, thus settling a 30-year old conjecture of Barth and Hartshorne.

3.6 Hodge theory on hypercomplex manifolds and HKT-geometry A hypercomplex manifold is a manifold with three complex structures I, J, K satisfying quaternionic relations. It is called quaternionic Hermitian if it has a quaternionic-invariant Riemannian structure. With each quaternionic Hermitian manifold√ (M, I, J, K, g), one can associate its canonical (2, 0)-form Ω = ωJ + −1ωK . If this form is closed, M is called hyperk¨ahler(this is one of possible deﬁnitions). If ∂Ω = 0,

7 (M, I, J, K, g) is called an HKT-manifold (“hyperkählerwith torsion”). An HKT form is in many ways similar to a Kählerstructure. One can define a potential, a version of Hodge theory, Kählerclass and so on. In papers [V01:2], [V03:2], [V04:2], [V04:3], [V06:1], [BDV07] I studied hypercomplex geometry from this point of view. The Hodge theory (including Lefschetz-type SL(2)-action) was constructed for HKT manifolds with trivial canonical bundle. As an application, it was shown that a compact hypercomplex manifold which admits a Kählermetric also admits a hyperkähler structure. In another application (jointly with I. Dotti and M. L. Barbieris) it was shown that a hypercomplex nilmanifold admits an HKT structure if and only if it is abelian. This result was also useful in hyperkähler geometry, where a strong vanishing result was shown, based on this Lefschetz -type SL(2)-action. It was shown that cohomology of a holomorphic line bundle L with −c1(L) outside of a dual Kählercone vanish after the middle dimension. Moreover, form any holomorphic bunldle M, Hi(B ⊗ LN ) = 0, for N sufficiently big, and 1 i > 2 dimC M. The version of Hodge theory developed for the study of HKT manifolds, was also useful in other geometric sutuations, namely, for G2-manifolds and nearly Kählermanifolds ([V05:1], [V05:2], [V05:3]). For nearly Kählermani- folds, the Hodge relations were sufficient to obtain the Hodge decomposition. During the work on Hodge theory of G2-manifolds, many concepts of complex algebraic geometry were adapted to work on G2-manifolds. This way I obtained some basic results in the theory of calibrated plurisubharmonic functions, later rediscovered by Harvey and Lawson in a different (and more systematic) framework. This theory was used to study coherent sheaves on hyperkähler manifolds, and (jointly with Semyon Alesker) Calabi-Yau problem in HKT geometry.

3.7 Complex structures and special holonomies Since early 2000- ies, I was trying to apply complex (more precisely, Hodge-theoretic) methods to manifolds with special holonomy (G2-manifolds and their locally conformal analogues, obtained as Riemannian cones over nearly Kählermanifolds). This approach brought some interesting results. In particular, I was able to construct an analogue of Hodge theory and Kählerrelations for manifolds with special holonomy ([V01:2], [V05:1], [V05:2], [V05:3]). However, one of the main applications of the Kählerformalism still miss- ing: nobody seems to know how to prove formality for G2-manifolds. It seems that some complex structure is still required, e.g. in the form of a twistor space.

8 During the course of this work, I realized that a knot space of a G2- manifold has a formally integrable Kählerstructure; also, a CR-holomorphic twistor space was constructed ([V10:1], [V10:2]). This structure is analogous to the Brylinski’s formally integrable Kählerstructure on a know space of a Riemannian 3-manifold, but in the G2-case the formal integrability of this Kählerstructure depends on differential-geometric properties of a manifolds (and, in fact, it is equivalent to the holonomy condition). It seems that these structures should play in G2-geometry the same role as the usual twistor spaces play in quaternionic geometry.

3.8 Plurisubharmonic functions in hypercomplex geometry In [AV05], [AV08] we studied the plurisubharmonic function on a hypercom- n plex manifold M. If M is H , these are functions which are subharmonic on all 1-dimensional quaternionic planes. The theory of quaternionic plurisubharmonic functions is deeply related with HKT-geometry, because HKT- potentials are precisely the C2-functions which are strictly plurisubharmonic. We formulated a version of Calabi conjecture for HKT manifolds and proved uniqueness of its solution and C0-estimates. In [V08:3], it was shown that an HKT metric is Calabi-Yau HKT if and only if it is balanced dim M−1 (satisﬁes dω C = 0). The appropriate notion of positivity (called K-positivity there) origi- nates in [V01:1], where it was used to study direct image of hyperholomorphic sheaves. In order to prove stability, an L2-estimation of singularities was required. It was obtained by a clumsy approximation argument. In [V08:1] and [V08:2] the theory of calibrated plurisubharmonic function was developed, in parallel with the usual complex analysis, and the stability of higher direct images of hyperholomorphic sheaves was obtained in a clean way as an application of this theory. Another application of this formalism was obtained in a joint paper with Grantcharov, [GV10], where new calibrations on hyperk¨ahlerand HKT- manifolds were constructed.

3.9 Locally conformally Kählergeometry A locally conformally Kählermanifold (LCK-manifold) is a complex manifold which is covered by a Kähler,with the deck transform acting by holomorphic homotheties. An impotrant special case is so-called Vaisman manifolds, which are covered by a Kählermanifold where R acts by holomorphic homotheties. Sim- ilarly one defines a locally conformally hyperkählermanifold. In [V03:1], I obtained a structure theorem for locally conformally hyperkählermanifolds, reducing their classification to classification of 3-Sasakian manifolds, which is due to Boyer, Galicki, Demailly and Kollar.

9 Since then, I collaborated with Liviu Ornea in a series of papers ([OV03:1], [OV03:12], [OV04], [OV06:1], [OV06:2], [OV06:3], [OV07:1], [OV09:1], [OV09:2], [OV10:1]) on locally conformally K¨ahler geometry. We have established a structure theorem for Vaisman manifolds, reducing the Vaisman geometry to Sasakian geometry, and proved that a Vais- man manifold admits a holomorphic immersion in a linear Hopf manifold. This was used to obtain similar immersion results for Sasakian manifold, proving that they always admit a CR-holomorphic embedding to a contact sphere. These results were also used to characterize CR-manifolds admitting Sasakian structure in terms of their automorphism group. In attempt to understand the immersion theorem, we invented a new class of LCK-manifold, called LCK-manifolds with automorphic potential. This is an intermediate class between the Vaisman manifolds and LCK-manifolds. Unlike the Vaisman and LCK-manifolds, LCK-manifolds with automorphic potential are stable under small complex deformations. Also, such manifolds admit holomorphic embedding to linear Hopf manifold. This result can be understood as a locally conformally K¨ahlerversion of a Kodaira embedding theorem. Later on, we found that LCK-manifolds with automorphic potential can be characterized in terms of vanishing of a certain cohomology class, which can be understood as a holomorphic version of Morse-Novikov cohomology. This was used to characterize such manifolds in terms of their automorphism groups, and to obtain important results about topology, describing their topology in terms of topology of certain algebraic varieties.

3.10 Toric and elliptic fibrations In [V04:1], I studied vector bundles and coherent sheaves on compact complex non-Kähler manifolds admitting a principal elliptic fibration with a Kählerbase. There are many examples of such manifolds coming from physics, hypercomplex geometry and LCK- geometry. Positive elliptic fibrations were defined (a big class including quasi-regular Vaisman and Calabi-Eckmann manifolds). I have shown that any stable sheaf on a positive elliptic fibration of dimC ≥ 3 is lifted from a base (up to a product with line bundle). This implies, in particular, that all coherent sheaves are filtrable (in constract to the case of non-Kähler surfaces, where coherent sheaves are rarely filtrable). In [V04:4], filtrability was generalized to Hopf manifolds of dimC ≥ 3. In [V07:1], the notion of positive toric fibration was discussed, including the positive elliptic fibrations and invariant complex structures on compact Lie groups. It was shown that all connected subvarieties in a positive toric fibration are contained in a fiber, or lifted from a base. In [OV10:2], this approach was used to show that the Oeljeklaus-Toma manifolds (multi-dimensional generalizations of Inoue surfaces) have no non-

10 trivial complex subvarieties.

3.11 Moduli spaces and the mapping class group Let Comp be an infinite-dimensional space of all complex structures on a manifold M, and Diff the group of diffeomorphisms. The quotient Comp / Diff parametrizes the set of equivalence classes of complex structures. It is natural to think of Comp / Diff as of the moduli space of complex structures on M. Following Teichmüller,we split the process of taking the quotient Comp / Diff in two parts. Consider the connected component Diff0 of the group of diffeomorphisms. We define the Teichmüllerspace as a quotient Teich := Comp / Diff0. Unlike the quotient Comp / Diff, which can be very ugly, the Teichmüllerspace is a complex manifold (at least when M is Calabi- Yau), possibly non-Hausdorff. Then, the moduli space is obtained as a quotient Teich /Γ, where Γ : Diff / Diff0 is a mapping class group. It was observed in [V:13] that the mapping class group action on Teich might have dense orbits. In fact, it is ergodic when M is a complex torus of dimension at least 2 or a hyperkählermanifold. This leads to many surprising results and a proof of several long-standing conjectures. From ergodicity theorem, it follows that any quantity continuously de- pending on complex structure is constant on families admitting ergodic complex structures. Similarly, the semi-continuity of a quantity implies that it is constant on complex structures having dense Γ-orbits (such complex structures are called ergodic). For example, one can take a pseudo-metric structure defined on any compact manifold and called Kobayashi pseudo-metric. Kovayashi pseudo- metric is known to be semi-continuous in families of complex structures, and continuous when non-degenerate. In the later case a manifold is called Kobayashi hyperbolic. It was conjectured that any Calabi-Yau manifold has vanishing Kobayashi metric, or at least non-hyperbolic. Using ergodic approach as above, the second conjecture was proven for all hyperkähler manifolds, and the first one for all known examples of hyperkählermanifolds ([V:13, KLV]). However, there are many more possible applications of ergodicity than those mentioned above. One of the natural quantities which are semi-continuous in families is Lelong numbers of currents. Any cohomology class on a boundary of a Kählercone can be represented by a positive current. If this class is rational, vanishing of its Lelong number can be translated to Q-effectivity of the corresponding line bundle ([V08:4, V09:1]). If successful, this argument might lead to construction of Lagrangian fibrations on hyperkählermanifolds. This would advance the conjectural

11 approach to the Mirror Symmetry known as SYZ conjecture, as well as the hyperkählergeometry as a while. It seems that the ergodicity phenomenon is not restricted to the hy- perkählergeometry. It was conjectured by M. Kontsevich that the mapping class group acts with dense orbits on the Teichmüllerspace of rational surfaces with b2 ≥ 19. Also, ergodicity was conjectured for Calabi-Yau manifolds. Another interesting phenomenon appearing in conjunction with ergodic actions is the way that the Gromov-Hausdorff limits of Ricci-flat metrics behave. Let NI be the space of all Ricci-flat Kählermetric on a holomorphically symplectic manifold (M,I). We consider NI as a metric space equipped with the Gromov-Hausdorff metric. This space was much studied recently, because its degenerations (“collapse”) reveal much information about Mirror Symmetry (the Strominger-Yau-Zaslow special Lagrangian fibrations) and the birational geometry of (M,I); for some of the literature, please see [GTZ1], [GTZ2] and [KS]. Clearly, NI is identified with the quotient of the Kählercone of (M,I) by holomorphic isometries. The group of holomorphic isometries of a simple hyperkählermanifold stabilising a point of the Kählercone is finite, hence NI is an orbifold. Let N¯I denote the completion of NI with respect to the Gromov-Hausdorff metric. The boundary ∂NI := N¯I \NI is of considerable interest to geometry, because its points classify the collapsing limits of Ricci- ≥0 flat metrics on (M,I). Consider the diameter function diam : N¯ −→ R , mapping a metric space into its diameter. By Gromov’s compactness theorem, diam is proper ([Gr]), that is, preim- −1 ≥0 age diam (K) of a compact subset K ⊂ R is compact. This observation follows from Gromov’s compactness theorem, because non-collapsed metrics g ∈ N are Ricci-flat, and the space of Ricci-flat metrics of bounded diameter on M is precompact in Gromov-Hausdorff topology. Suppose now that I is an ergodic complex structure, that is, a complex structure with Γ-orbit dense in Teich. The space NI is a metric orbifold, however, its Gromov boundary ∂NI = N¯I \NI is quite bizzarre. For each complex structure I0 ∈ Teich, there exists a sequence of diffeomorphisms 0 νi such that limi νi(I) = I . By Kodaira stability of Kählercones, any 0 0 ∗ hyperkählermetric g on (M,I ) is obtained as a limit νi (gi) of the metrics gi ∈ N. This implies that the space ∂NI of collapsing metrics on (M,I) contains hyperkählermetrics on M, for all complex structures. This latter space has real dimension at least 2 dimR N, hence the boundary behaviour of collapsing hyperkählermetrics is very chaotic and counterintuitive: the dimension of a boundary ∂NI is at least 2 dim NI .

12 References

[A] Apostol Apostolov, Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected, arXiv:1109.0175. [BS] Samuel Boissiere, Alessandra Sarti, A note on automorphisms and birational transformations of holomorphic symplectic manifolds, arXiv:0905.4370. [Gr] Gromov, Misha, Metric structures for Riemannian and non-Riemannian spaces, Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. BirkhäuserBoston, Inc., Boston, MA, 1999. xx+585 pp. [GTZ1] M. Gross, V. Tosatti, Y. Zhang, Collapsing of Abelian Fibered Calabi-Yau Manifolds, Duke Math. J. 162 (2013), no. 3, 517-551. [GTZ2] Mark Gross, Valentino Tosatti, Yuguang Zhang, Gromov-Hausdorff collapsing of Calabi-Yau manifolds, arXiv:1304.1820. [H] Daniel Huybrechts, A global Torelli theorem for hyperkählermanifolds (after Verbitsky), arXiv:1106.5573. [KS] M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fibrations, in Symplectic geometry and mirror symmetry, 203-263, World Sci. Publishing 2001. [Ma] Markman, E. A survey of Torelli and monodromy results for holomorphic- symplectic varieties, Proceedings of the conference ”Complex and Differential Geometry”, Springer Proceedings in Mathematics, 2011, Volume 8, 257–322, arXiv:math/0601304. [Mn] Giovanni Mongardi, Symplectic involutions on deformations of K3[2], arXiv:1107.2854. [V90] Verbitsky M. On the action of a Lie algebra SO(5) on the cohomology of a hyperkählermanifold, Func. Analysis and Appl. 24(2) p 70-71 (1990). [V92] Verbitsky M., Hyperholomorphic bundles over the hyperkählermanifolds, alg- geom 9307008, Journ. of Alg. Geom., 5 no. 4 (1996) pp. 633-669. [V93] Verbitsky M., Hyperkählerand holomorphic symplectic geometry I. alg-geom 9307009, also in Journ. of Alg. Geom., vol. 5 no. 3 pp. 401-415 (1996). [V94] Verbitsky, M., Hyperkählerembeddings and holomorphic symplectic geometry II. alg-geom 9403006, 14 pages, also in GAFA vol. 5 no. 1 (1995) pp. 92-104 [V95:1] Verbitsky M., Cohomology of compact hyperkählermanifolds and its applications alg-geom 9511009, also in: GAFA vol. 6 no. 4 pp. 601–612 (1996) [V95:2] Verbitsky M., Mirror Symmetry for hyperkähler manifolds, alg-geom 9512195 (published in Mirror Symmetry III, International Press). [KV96] Kaledin, D., Verbitsky, M., Non-Hermitian Yang-Mills connections, alg- geom 9606019, also in Selecta Math. 4 (1998) 279-320

13 [V96:1] Verbitsky M., Deformations of trianalytic subvarieties of hyperkählerman- ifolds, alg-geom 9610010 (Selecta Math., New ser. 4 (1998) 447-490) [V96:2] Verbitsky M., Desingularization of singular hyperkählervarieties I, alg- geom 9611015, also Math. Res. Lett. 4 (1997), no. 2-3, pp. 259–271. [V97:1] Verbitsky M., Hypercomplex Varieties, alg-geom 9703016 (Comm. Anal. Geom. 7 (1999), no. 2, 355–396.) [V97:2] Verbitsky M., Trianalytic subvarieties of the Hilbert scheme of points on a K3 surface, alg-geom 9705004, GAFA, Vol. 8 (1998) 732-782. [V97:3] Verbitsky M., Hyperholomorphic sheaves and new examples of hyperkähler manifolds, alg-geom 9712012 (published as a part of a book “Hyperkähler manifolds”, by M. Verbitsky and D. Kaledin, Mathematical Physics, 12, In- ternational Press, 1999.) [KV98:1] Kaledin D., Verbitsky M., Trianalytic subvarieties of generalized Kummer varieties, math.AG 9801038, published in Int. Math. Res. Not. 1998, No. 9, 439–461. [V98] Verbitsky M., Wirtinger numbers and holomorphic symplectic immersions, math.AG:9812077, Selecta Math. (N.S.) 10 (2004), no. 4, 551–559. [KV98:2] Kaledin D., Verbitsky M., Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties, math.AG 9812078 (33 pages). [V99] M. Verbitsky, Holomorphic symplectic geometry and orbifold singularities, math.AG 9903175 (17 pages, LaTeX 2e), published in Asian Journ. of Math., Vol. 4 (2000), No. 3, pp. 553-564 [KV00] D. Kaledin, M. Verbitsky, Period map for non-compact holomorphically symplectic manifolds, math.AG:0005007, GAFA 12 (2002), no. 6, 1265–1295. [V01:1] M. Verbitsky, Hyperholomorpic connections on coherent sheaves and stability, 43 pages, math.AG:0107182 [V01:2] M. Verbitsky, Hyperkähler manifolds with torsion, supersymmetry and Hodge theory, math.AG:0112215, Asian J. of Math., Vol. 6, No. 4, pp. 679-712 (2002) [V02] M. Verbitsky, Coherent sheaves on general K3 surfaces and tori, math.AG:0205210, 63 pages, Pure Appl. Math. Q. 4 (2008), no. 3, part 2, 651–714. [V03:1] M. Verbitsky, Vanishing theorems for locally conformal hyperkählermani- folds, math.DG:0302219, 41 pages, Proc. of Steklov Institute, vol. 246, 2004, pp. 54-79. [V03:2] M. Verbitsky, Hyperkählermanifolds with torsion obtained from hyperholomorphic bundles math.DG:0303129, Math. Res. Lett. 10 (2003), no. 4, 501– 513.

14 [V03:3] Verbitsky, M., Coherent sheaves on generic compact tori, math.AG:0310329, CRM Proc. and Lecture Notices vol. 38 (2004), 229-249 [V03:4] Verbitsky, M., Subvarieties in non-compact hyperkähler manifolds, math.AG:0312520, Math. Res. Lett. vol. 11 (2004), no. 4, pp. 413-418 [OV03:1] L. Ornea, M. Verbitsky, Structure theorem for compact Vaisman manifolds, math.DG:0305259, Math. Res. Lett, 10(2003), no. 5-6, 799-805 [OV03:12] L. Ornea, M. Verbitsky, Immersion theorem for Vaisman manifolds, math.AG:0306077, 28 pages, Math. Ann. 332 (2005), no. 1, 121–143. [V04:1] Verbitsky, M., Stable bundles on positive principal elliptic fibrations, math.AG:0403430, 17 pages, Math. Res. Lett. 12 (2005), no. 2-3, 251–264. [V04:2] Verbitsky, M., Hypercomplex structures on Kähler manifolds, math.AG:0406390, 10 pages, GAFA 15 (2005), no. 6, pp. 1275-1283. [V04:3] Verbitsky, M., Hypercomplex manifolds with trivial canonical bundle and their holonomy, math.DG:0406537, 13 pages, “Moscow Seminar on Mathe- matical Physics II”, Translations of AMS, vol. 221 (2007). [OV04] L. Ornea, M. Verbitsky, Locally conformal Kählermanifolds with potential, math.AG:0407231, Mathematische Annalen, Vol. 248 (1), 2010, pp. 25-33. [V04:4] Verbitsky, M., Holomorphic bundles on diagonal Hopf manifolds, 20 pages, math.AG:0408391, Izvestiya Math., 2006, 70, no. 5, pp. 13-31. [V05:1] Verbitsky, M., Manifolds with parallel differential forms and Kähleridenti- ties for G2-manifolds, math.DG:0502540, Journal of Geometry and Physics, vol. 61 (6), pp. 1001-1016 (2011). [V05:2] Verbitsky, M., An intrinsic volume functional on almost complex 6- manifolds and nearly Kählergeometry, arXiv:math:0507179, Pacific J. of Math., Vol. 235, No. 2 (2008), 323-344 [AV05] Semyon Alesker, Misha Verbitsky, Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry, 34 pages, Journal of Geometric Anal- ysis, 16, 3, Sep. 2006, pp. 375-400 [V05:3] Verbitsky, M., Hodge theory on nearly Kählermanifolds, Geometry and Topology, 2011, No. 15. pp. 2111-2133. [V06:1] Verbitsky, M., Quaternionic Dolbeault complex and vanishing theorems on hyperkählermanifolds, math.AG:0604303, 30 pages, Compos. Math. 143 (2007), no. 6, 1576–1592. [OV06:1] L. Ornea, M. Verbitsky, Sasakian structures on CR-manifolds, Geom. Dedicata 125 (2007), 159–173. [OV06:2] Liviu Ornea, Misha Verbitsky, Einstein-Weyl structures on complex manifolds and conformal version of Monge-Ampere equation, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 51(99) (2008), no. 4, 339–353.

15 [OV06:3] Liviu Ornea, Misha Verbitsky, Embeddings of compact Sasakian manifolds, math.DG:0609617, 10 pages, Math. Res. Lett. 14 (2007), no. 4, 703– 710. [MV06] Ruxandra Moraru, Misha Verbitsky, Stable bundles on hypercomplex surfaces, Cent. Eur. J. Math. 8 (2010), no. 2, 327–337. [V07:1] Misha Verbitsky, Positive toric fibrations, J. Lond. Math. Soc. (2) 79 (2009), no. 2, 294–308. [OV07:1] Liviu Ornea, Misha Verbitsky, Morse-Novikov cohomology of locally conformally Kähler manifolds, arXiv:0712.0107, J. Geom. Phys. 59 (2009), no. 3, 295–305. [BDV07] Maria Laura Barberis, Isabel G. Dotti, Misha Verbitsky, Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry, arXiv:0712.3863, Math. Res. Lett. 16 (2009), no. 2, 331–347. [V08:1] Misha Verbitsky, Plurisubharmonic functions in calibrated geometry and q-convexity, arXiv:0712.4036, Math. Z., Vol. 264, No. 4, pp. 939-957 (2010). [V08:2] Misha Verbitsky, Positive forms on hyperkähler manifolds, Osaka J. Math. Volume 47, Number 2 (2010), 353-384. [AV08] Semyon Alesker, Misha Verbitsky, Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds, arXiv:0802.4202, Israel Journal of Mathematics, Vol. 176, Number 1 (2010), 109-138. [V08:3] Misha Verbitsky, Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds, arXiv:0808.3218, Math. Res. Lett. 16 (2009), no. 4, 735–752. [V08:4] Misha Verbitsky, HyperkählerSYZ conjecture and semipositive line bundles, arXiv:0811.0639, GAFA 19, No. 5, 1481-1493 (2010). [OV09:1] Liviu Ornea, Misha Verbitsky, Topology of locally conformally Kähler manifolds with potential, arXiv:0904.3362, Int. Math. Res. Not. 2010, No. 4, 717-726 (2010). [OV09:2] Liviu Ornea, Misha Verbitsky, Automorphisms of locally conformally Kählermanifolds, arXiv:0906.2836, 9 pages [V09:1] M. Verbitsky, Parabolic nef currents on hyperkähler manifolds, arXiv:0907.4217. [V09:2] M. Verbitsky, A global Torelli theorem for hyperkähler manifolds, arXiv:0908.4121. [GV10] Gueo Grantcharov, Misha Verbitsky, Calibrations in hyperkählergeometry, arXiv:1009.1178 [OV10:1] Liviu Ornea, Misha Verbitsky Locally conformally Kählermanifolds admitting a holomorphic conformal flow, arXiv:1004.4645.

16 [OV10:2] Ornea L., Verbitsky, M., Oeljeklaus-Toma manifolds admitting no complex subvarieties, Mathematical Research Letters, 2011, No. 04 (18). pp. 747- 754. [V10:1] M. Verbitsky, A CR twistor space of a G2-manifold, Differential Geometry and its Applications, 2011, No. 29 (apr), pp. 101-107 [V10:2] M. Verbitsky, A formally Kähler structure on a knot space of a G2- manifold, arXiv:1003.3174, accepted by Selecta Math. [AV10] Sasha Anan’in, Misha Verbitsky, Any component of moduli of polarized hyperkählermanifolds is dense in its deformation space, arXiv:1008.2480. [JV10] Jardim M., Verbitsky, M. Moduli spaces of framed instanton bundles on CP 3 and twistor sections of moduli spaces of instantons on C2, Adv. Math., 2011, No. 227. pp. 1526-1538. [JV11] Jardim M., Verbitsky, M. Trihyperkählerreduction and instanton bundles on CP 3, arXiv:1103.4431. [KLV] Ljudmila Kamenova, Steven Lu, Misha Verbitsky, Kobayashi pseudometric on hyperkähler manifolds, arXiv:1308.5667. [V:13] Verbitsky, M., Ergodic complex structures on hyperkähler manifolds, arXiv:1306.1498, 22 pages.

4 Published Works

4.1 Books

• Verbitsky, Misha; Kaledin, Dmitri Hyperk¨ahlermanifolds. Mathemat- ical Physics (Somerville), 12. International Press, Somerville, MA, 1999. iv+257 pp. ISBN: 1-57146-071-3

• “Topology for ﬁrst-year students”, by M. Verbitsky (in Russian), 370 pp, to appear in Independent University of Moscow Press, http://verbit.ru/MATH/UCHEBNIK/top-book.pdf

4.2 Journal articles

[1] Verbitsky M. On the action of a Lie algebra SO(5) on the cohomology of a hyperk¨ahlermanifold, Func. Analysis and Appl. 24(2) pp. 70-71 (1990).

[2] Verbitsky M., Hyperholomorphic bundles over the hyperk¨ahlermanifolds, Journ. of Alg. Geom., 5 no. 4 (1996) pp. 633-669.

[3] Verbitsky M., Hyperk¨ahlerand holomorphic symplectic geometry I, Journ. of Alg. Geom., vol. 5 no. 3 pp. 401-415 (1996).

17 [4] Verbitsky, M., Trianalytic subvarieties of hyperk¨ahlermanifolds, GAFA vol. 5 no. 1 (1995) pp. 92-104

[5] Misha Verbitsky, Cohomology of compact hyperk¨ahlermanifolds and its applications, GAFA vol. 6 no. 4 pp. 601–612 (1996)

[6] Kaledin, D., Verbitsky, M., Non-Hermitian Yang-Mills connections, Selecta Math. 4 (1998) 279-320

[7] Verbitsky M., Algebraic structures on a hyperk¨ahlermanifold, Math. Res. Lett. 3 763-767, 1996

[8] Misha Verbitsky, Deformations of trianalytic subvarieties of hyperk¨ahler manifolds, Selecta Math., New ser. 4 (1998) 447-490

[9] Verbitsky M., Desingularization of singular hyperk¨ahlervarieties I, Math. Res. Lett. 4 (1997), no. 2-3, pp. 259–271.

[10] Verbitsky M., Hypercomplex Varieties, Comm. Anal. Geom. 7 (1999), no. 2, 355–396.

[11] Verbitsky M., Trianalytic subvarieties of the Hilbert scheme of points on a K3 surface, GAFA, Vol. 8 (1998) 732-782.

[12] Kaledin D., Verbitsky M., Trianalytic subvarieties of generalized Kum- mer varieties, Int. Math. Res. Not. 1998, No. 9, 439–461.

[13] Verbitsky M., Wirtinger numbers and holomorphic symplectic immersions, Selecta Math. (N.S.) 10 (2004), no. 4, 551–559.

[14] M. Verbitsky, Holomorphic symplectic geometry and orbifold singularities, math.AG 9903175 (17 pages, LaTeX 2e), published in Asian Journ. of Math., Vol. 4 (2000), No. 3, pp. 553-564

[15] D. Kaledin, M. Verbitsky, Period map for non-compact holomorphically symplectic manifolds, math.AG/0005007, GAFA 12 (2002), no. 6, 1265–1295.

[16] M. Verbitsky, Hyperk¨ahlermanifolds with torsion, supersymmetry and Hodge theory, math.AG/0112215, Asian J. of Math., Vol. 6, No. 4, pp. 679-712 (2002)

[17] M. Verbitsky, Coherent sheaves on general K3 surfaces and tori, math.AG/0205210, Pure Appl. Math. Q. 4 (2008), no. 3, part 2, 651–714.

18 [18] M. Verbitsky, Vanishing theorems for locally conformal hyperk¨ahler manifolds, math.DG/0302219, 41 pages, Proc. of Steklov Institute, vol. 246, 2004, pp. 54-79

[19] M. Verbitsky, Hyperk¨ahlermanifolds with torsion obtained from hyperholomorphic bundles math.DG/0303129, Math. Res. Lett. 10 (2003), no. 4, 501–513.

[20] L. Ornea, M. Verbitsky, Structure theorem for compact Vaisman manifolds, math.DG/0305259, Math. Res. Lett, 10(2003), no. 5-6, 799-805

[21] L. Ornea, M. Verbitsky, Immersion theorem for Vaisman manifolds, 28 pages, Math. Ann. 332 (2005), no. 1, 121–143.

[22] Verbitsky, M., Subvarieties in non-compact hyperk¨ahlermanifolds, math.AG/0312520, Math. Res. Lett. vol. 11 (2004), no. 4, pp. 413-418

[23] Verbitsky, M., Stable bundles on positive principal elliptic ﬁbrations, 17 pages, Math. Res. Lett. 12 (2005), no. 2-3, 251–264.

[24] Verbitsky, M., Hypercomplex structures on K¨ahlermanifolds, math.AG/0406390, 10 pages, GAFA 15 (2005), no. 6, pp. 1275-1283

[25] Verbitsky, M., Holomorphic bundles on diagonal Hopf manifolds, 20 pages, math.AG/0408391, Izvestiya Math., 2006, 70, no. 5, pp. 13-31.

[26] Misha Verbitsky, An intrinsic volume functional on almost complex 6- manifolds and nearly K¨ahlergeometry, arXiv:math/0507179, Paciﬁc J. of Math., Vol. 235, No. 2 (2008), 323-344

[27] Semyon Alesker, Misha Verbitsky, Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry, 34 pages, Journal of Geo- metric Analysis, 16, 3, Sep. 2006, pp. 375-400

[28] Verbitsky, M., Quaternionic Dolbeault complex and vanishing theorems on hyperk¨ahler manifolds, math.AG/0604303, Compos. Math. 143 (2007), no. 6, 1576–1592.

[29] L. Ornea, M. Verbitsky, Sasakian structures on CR-manifolds, math.DG/0606136, 23 pages, Geom. Dedicata 125 (2007), 159–173.

[30] Liviu Ornea, Misha Verbitsky, Einstein-Weyl structures on complex manifolds and conformal version of Monge-Ampere equation, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 51(99) (2008), no. 4, 339–353.

19 [31] Liviu Ornea, Misha Verbitsky, Embeddings of compact Sasakian manifolds, math.DG/0609617, 10 pages, Math. Res. Lett. 14 (2007), no. 4, 703–710.

[32] Misha Verbitsky, Positive toric ﬁbrations, J. Lond. Math. Soc. (2) 79 (2009), no. 2, 294–308.

[33] Liviu Ornea, Misha Verbitsky, Morse-Novikov cohomology of locally conformally K¨ahlermanifolds, arXiv:0712.0107, J. Geom. Phys. 59 (2009), no. 3, 295–305.

[34] Maria Laura Barberis, Isabel G. Dotti, Misha Verbitsky, Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry, Math. Res. Lett. 16 (2009), no. 2, 331–347.

[35] Misha Verbitsky, Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds, Math. Res. Lett. 16 (2009), no. 4, 735–752.

[36] L. Ornea, M. Verbitsky, Locally conformal K¨ahlermanifolds with potential, math.AG/0407231, Mathematische Annalen, Vol. 248 (1), 2010, pp. 25-33.

[37] Ruxandra Moraru, Misha Verbitsky, Stable bundles on hypercomplex surfaces, math.DG/0611714, Cent. Eur. J. Math. 8 (2010), no. 2, 327–337

[38] Misha Verbitsky, Plurisubharmonic functions in calibrated geometry and q-convexity, arXiv:0712.4036, Math. Z., Vol. 264, No. 4, pp. 939-957 (2010)

[39] Misha Verbitsky, Positive forms on hyperk¨ahlermanifolds, arXiv:0801.1899, Osaka J. Math. Volume 47, Number 2 (2010), 353- 384

[40] Semyon Alesker, Misha Verbitsky, Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds, arXiv:0802.4202, Israel Journal of Mathematics, Vol. 176, Number 1 (2010), 109-138

[41] Misha Verbitsky, Hyperk¨ahlerSYZ conjecture and semipositive line bundles, arXiv:0811.0639, GAFA 19, No. 5, 1481-1493 (2010)

[42] Liviu Ornea, Misha Verbitsky, Topology of locally conformally K¨ahler manifolds with potential, Int. Math. Res. Not. 2010, No. 4, 717-726 (2010)

20 [43] Verbitsky, M., Manifolds with parallel diﬀerential forms and K¨ahler identities for G2-manifolds, Journal of Geometry and Physics, vol. 61 (6), pp. 1001-1016 (2011).

[44] Verbitsky, M., A CR twistor space of a G2-manifold, Differential Ge- ometry and its Applications, Vol. 29, pp. 101-107 (2011). [45] Misha Verbitsky, Hyperholomorpic connections on coherent sheaves and stability, Central European Journal of Mathematics, vol. 9 (3). pp. 535-557 (2011). [46] M. Verbitsky Hodge theory on nearly Kählermanifolds, Geometry and Topology, 2011, No. 15. pp. 2111-2133 [47] Jardim M., Verbitsky, M. Moduli spaces of framed instanton bundles on CP 3 and twistor sections of moduli spaces of instantons on C2, Adv. Math., 2011, No. 227. pp. 1526-1538. [48] Ornea L., Verbitsky, M., Oeljeklaus-Toma manifolds admitting no complex subvarieties, Mathematical Research Letters, 2011, No. 04 (18). pp. 747-754. [49] Liviu Ornea, Misha Verbitsky, Automorphisms of locally conformally Kählermanifolds, Int. Math. Res. Not. 2012, no. 4, 894-903. [50] Andrey Soldatenkov, Misha Verbitsky, Subvarieties of hypercomplex manifolds with holonomy in SL(n,H), Journal of Geometry and Physics, Volume 62, Issue 11, November 2012, Pages 2234-2240. [51] Misha Verbitsky, A formally Kählerstructure on a knot space of a G2-manifold, Sel. Math. New. Ser. (2012) 18:539-555 [52] Liviu Ornea, Misha Verbitsky, Victor Vuletescu, Blow-ups of locally conformally Kählermanifolds, Int. Math. Res. Not. IMRN 2013, no. 12, 2809-2821. [53] Gueo Grantcharov, Misha Verbitsky, Calibrations in hyperkählerge- ometry, Commun. Contemp. Math. 15 (2013), no. 2, 1250060, 27 pp. [54] Misha Verbitsky, Pseudoholomorphic curves on nearly Kählermani- folds, Communications in Mathematical Physics November 2013, Vol- ume 324, Issue 1, pp 173-177. [55] Liviu Ornea, Misha Verbitsky, Locally conformally Kählermanifolds admitting a holomorphic conformal flow, Mathematische Zeitschrift, Volume 273, Issue 3 (2013), Page 605-611.

21 [56] Verbitsky, M., A global Torelli theorem for hyperk¨ahlermanifolds, Duke Math. J. Volume 162, Number 15 (2013), 2929-2986.

[57] Anan0in, Sasha; Verbitsky, Misha, Any component of moduli of polarized hyperk¨ahlermanifolds is dense in its deformation space, J. Math. Pures Appl. (9) 101 (2014), no. 2, 188-197.

[58] Verbitsky, Misha, Rational curves and special metrics on twistor spaces, Geom. Topol. 18 (2014), no. 2, 897-909.

[59] Kamenova, Ljudmila; Verbitsky, Misha, Families of Lagrangian ﬁbra- tions on hyperk¨ahlermanifolds, Adv. Math. 260 (2014), 401-413.

[60] Campana, Frédéric;Demailly, Jean-Pierre; Verbitsky, Misha, Compact Kähler3-manifolds without nontrivial subvarieties, Algebr. Geom. 1 (2014), no. 2, 131-139.

[61] Kamenova, Ljudmila; Lu, Steven; Verbitsky, Misha, Kobayashi pseudometric on hyperk¨ahler manifolds, J. Lond. Math. Soc. (2) 90 (2014), no. 2, 436-450.

[62] Verbitsky, Misha, Holography principle for twistor spaces, Pure Appl. Math. Q. 10 (2014), no. 2, 325-354.

4.3 Book chapters, conference proceedings

[1] Kazhdan D., Verbitsky M., Cohomology of restricted quantized universal enveloping algebras, Quantum Deformations of Algebras and Their Representations, vol. 7 of Isr. Math. Conf. Proc. (1993)

[2] M. Verbitsky, Mirror Symmetry for hyperk¨ahlermanifolds, AMS/IP Stud. Adv. Math. 10 (1999) 115-156

[3] M. Verbitsky, Singularities in hyperk¨ahlergeometry, Proceedings of the Second Meeting on Quaternionic Structures in Mathematics and Physics, Roma, Italy September 6-10, 1999, http://www.emis.de/ proceedings/QSMP99/

[4] Verbitsky, M., Coherent sheaves on generic compact tori, math.AG/0310329, CRM Proc. and Lecture Notices vol. 38 (2004), 229-249

[5] Misha Verbitsky, Hypercomplex manifolds with trivial canonical bundle and their holonomy, math.DG/0406537, 13 pages, “Moscow Seminar on Mathematical Physics II”, Translations of AMS, vol. 221 (2007).

22 [6] Liviu Ornea, Misha Verbitsky, A report on locally conformally K¨ahler manifolds, Harmonic Maps and Diﬀerential Geometry, E. Loubeau and S. Montaldo eds., Contemporary Mathematics, 2011. Vol. 542. 284 pp.

5 Citations of Published Works

According to Google Scholar, http://scholar.google.com/citations? user=8KhODVoAAAAJ, I am cited 1212 times, with Hirsch index 20 (January 10, 2015). The list of most cited papers:

Paper Cited by year Cohomology of compact hyperkählermanifolds and its 74 1996 applications M Verbitsky Geometric and Functional Analysis GAFA 6 (4), 601-611 Hyperholomorphic bundles MS Verbitsky J. Algebr. 73 1996 Geom. 5 (alg-geom/9307008), 633-669 Mapping class group and a global Torelli theorem for 58 2013 hyperkählermanifolds M Verbitsky Duke Mathemati- cal Journal 162 (15), 2929-2986 Hyperkähler manifolds with torsion, supersymme- 54 2001 try and Hodge theory M Verbitsky arXiv preprint math/0112215 Action of the Lie algebra SO (5) on the cohomology of 49 1990 a hyperkählermanifold MS Verbitskii Functional Anal- ysis and Its Applications 24 (3), 229-230 Holomorphic symplectic geometry and orbifold singu- 45 1999 larities M Verbitsky arXiv preprint math/9903175 Tri-analytic subvarieties of hyperkähler manifolds M 42 1995 Verbitsky Geometric And Functional Analysis 5 (1), 92-104 Structure theorem for compact Vaisman manifolds L 37 2003 Ornea, M Verbitsky arXiv preprint math/0305259 Canonical bundles of complex nilmanifolds, with ap- 33 2007 plications to hypercomplex geometry ML Barberis, IG Dotti, M Verbitsky arXiv preprint arXiv:0712.3863 Plurisubharmonic functions on hypercomplex mani- 32 2006 folds and HKT-geometry S Alesker, M Verbitsky The Journal of Geometric Analysis 16 (3), 375-399 Vanishing theorems for locally conformal hyperkähler 31 2003 manifolds M Verbitsky arXiv preprint math/0302219

23 An immersion theorem for Vaisman manifolds L Ornea, 30 2005 M Verbitsky Mathematische Annalen 332 (1), 121-143 Mirror symmetry for hyper-Kählermanifolds M Ver- 29 1999 bitsky Mirror symmetry 3, 115-156 Hypercomplex varieties M Verbitsky arXiv preprint 29 1997 alg-geom/9703016 Hyperkählermanifolds M Verbitsky, D Kaledin arXiv 27 1999 preprint alg-geom/9712012 Locally conformal Kählermanifolds with potential L 26 2010 Ornea, M Verbitsky Mathematische Annalen 348 (1), 25-33 Period map for non-compact holomorphically symplec- 25 2002 tic manifolds D Kaledin, M Verbitsky Geometric and Functional Analysis GAFA 12 (6), 1265-1295 Non-Hermitian Yang-Mills connections D Kaledin, M 22 1998 Verbitsky Selecta Mathematica 4 (2), 279-320 Cohomology of compact hyperkähler manifolds M Ver- 21 1995 bitsky arXiv preprint alg-geom/9501001 Morse-Novikov cohomology of locally conformally 20 2009 Kählermanifolds L Ornea, M Verbitsky Journal of Ge- ometry and Physics 59 (3), 295-305

6 Conferences organized

• Quaternionic structures in algebraic geometry, 16-18 November 2007, University of Glasgow. • Supersymmetry in complex geometry: generalized complex structures and generalized K¨ahlerstructures on complex manifolds, 4-9 January 2009, IPMU, University of Tokyo, Japan. • Instantons in complex geometry, 14-18 March, 2011, Laboratory of Algebraic Geometry, HSE, Moscow. • Geometric structures on complex manifolds, 3-7 October 2011, Labo- ratory of Algebraic Geometry, HSE, Moscow. • Summer School “Algebra and geometry” (Yaroslavl. Russia), 2011, 2013, 2014. • Workshop on complex geometry and foliations, dedicated to the memory of Marco Brunella, September 17-21, 2012, Laboratory of Alge- braic Geometry, HSE, Moscow.

24 • Geometry of K¨ahlermanifolds, 21-25 May 2012, Laboratoire de Math´ematiquesJean Leray, Nantes (an event dedicated to Bogomolov’s 65 anniversary).

• The second workshop on complex geometry and foliations (February 25-March 1, 2013, Moscow, HSE).

• May 19-23, 2014: A workshop on the Chow group of holomorphically symplectic manifolds (Moscow, HSE).

• November 10-14, 2014, a workshop ”Complex manifolds, dynamics and birational geometry” (Moscow, HSE).

7 Awards and distinctions

• Simons-IUM Fellowship (2011, 2013)

• ICM section talk (Seoul, 2014)

8 Grants obtained

• CRDF grant RM1-2087 (as a principal co-investigator)

• CRDF grant RM1-2354-MO02.

• EPSRC grant GR/R77773/01 (Advanced Research Fellow grant, 250,000 pounds).

• RFBR grants: 09-01-00242-a, 10-01-93113-NCNIL-a.

• FAPESP grant 2009/12576-9.

• Science Foundation of the SU-HSE award No. 10-09-0015.

• AG Laboratory NRU-HSE, RF government grant, ag. 11.G34.31.0023 (150,000,000 million roubles, Fedor Bogomolov was a principal investigator, Verbitsky was responsible for writing the grant application).

• “Algebraic geometry on manifolds with special holonomy”, HSE Aca- demic Fund Program, 2013-2014, research grant 12-01-0179.

• RSCF grant 14-21-00053 for fundamental research performed by a col- lective of an established laboratory (60,000,000 million roubles, principal investigator - Misha Verbitsky)

25 9 Selected talks at international conferences

1. 17.03.2006: Hodge theory on nearly K¨ahlermanifolds, Kuhlungsborn, Germany, (Workshop on ”Special Geometries in Mathematical Physics”).

2. 08.08.2007: Hypercomplex structures on K¨ahlermanifolds, La Falda, Argentina, (Third Workshop on Diﬀerential Geometry).

3. 28.08.2007: Principal Toric Fibrations, CRM, Universite de Mon- treal, (Workshop on Non-linear integral transforms: Fourier-Mukai and Nahm.)

4. 05.11.2007: Quaternionic Monge-Ampere equation, Imperial College of London.

5. 26.11.2007: Algebraic geometry over quaternions, Durham University, UK.

6. 03.04.2008: Hypercomplex manifolds with holonomy SL(n,H), Kuh- lungsborn, Germany, (Second workshop on ”Special Geometries in Mathematical Physics”).

7. 05.06.2008: Sasakian manifolds, Technion, Haifa, Israel.

8. 18.12.2008: Hyperk¨ahlerSYZ conjecture, Havana, Cuba (First Cuban Congress on Symmetries in Geometry and Physics).

9. 16.01.2009: Topology of locally conformally K¨ahlermanifolds, Tokyo Metropolitan University.

10. June 2009: a conference at the University of Lille 1 ”Holomorphi- cally symplectic varieties and moduli spaces”. A mini-course on ”Hy- perk¨ahlerSYZ conjecture and multiplier ideal sheaves” (3 lectures).

11. November 2009: a trimester and a workshop on K¨ahlerand related geometries at Nantes University. A mini-course on Sasakian and locally conformally K¨ahlergeometry (3 lectures).

12. 13.11.2009: Global Torelli theorem for hyperk¨ahlermanifolds

13. 07.01.2010: History of Monge-Ampere equation, University of Delhi (National Meet on History of Mathematical Sciences).

14. 29.08.2010: Global Torelli theorem for hyperk¨ahlermanifolds, Ober- wolfach (”Komplexe Analysis” conference).

26 3 15. 26.10.2010: Stable bundles on CP and special holonomies, CIRM, Luminy (Geometry of complex manifolds IV).

16. 08.02.2011: Extremal metrics in quaternionic geometry, CIRM, Lu- miny, a conference ”Extremal metrics: evolution equations and stability”.

17. 21.02.2011: Generalization of Inoue surfaces by Oeljeklaus-Toma and number theory, CIRM, Luminy, a conference ”Non-K¨ahlerianaspects of complex geometry”.

18. June 4-5 Holomorphic symplectic varieties, Courant Institute, New York University, June 4-5, 2011, FRG Workshop.

• Global Torelli theorem for hyperk¨ahlermanifolds (June 4). • Subtwistor metric on the moduli of hyperk¨ahlermanifolds and its applications (June 5).

19. 16.06.2011: An intrinsic volume functional on almost complex 6-manifolds and nearly K¨ahlergeometry, (Oberseminar Inst. fur Algebraische Ge- ometrie, Leibniz Universitat, Hanover).

3 20. 01.07.2011: Instanton bundles on CP and special holonomies (The Seventh Congress of Romanian Mathematicians, Brasov).

21. 21.09.2011: Formally K¨ahlerstructure on a knot space of a G2-manifold, (”Geometric structures in mathematical physics,” Golden Sands, Bul- garia).

22. 11.10.2011: Any component of moduli of polarized hyperk¨ahlermanifolds is dense in its deformation space, (”Moduli spaces and automorphic forms”, Luminy, CIRM, France).

23. 21.10.2011: Morse-Novikov cohomology and Kodaira-type embedding theorem for locally conformally K¨ahlermanifolds, (”Complex geometry and uniformisation”, Luminy, CIRM, France).

24. 23.01.2012: Twistor correspondence for hyperk¨ahlermanifolds and the space of instantons, and 24.01.2012, Trihyperk¨ahlerreduction, Cavli IPMU, University of Tokyo, Japan, MS Seminar: Mathematics and String Theory.

25. 16.03.2012: Trisymplectic manifolds, Advances in hyperk¨ahlerand holomorphic symplectic geometry (BIRS, Canada, March 11-16, 2012).

27 26. 14.06.2012: Stable bundles on non-K¨ahlermanifolds with transversally K¨ahlerfoliations, (”Holomophic foliations and complex dynamics”, 11- 15 June 2012, Poncelet Laboratory, Moscow, Russia)

27. 07.08.2012: ”Global Torelli theorem for hyperk¨ahlermanifolds,” Au- gust 6 - August 10, Kyoto University, Japan, 7th Paciﬁc Rim Complex Geometry Conference.

28. 24.08.2012: ”Holomorphic connections on the space of quasilines”, Ge- ometry Seminar, Florida International University.

29. 07.09.2012: Rational curves on non-K¨ahler manifolds, a conference ”Komplexe Analysis”, 2-8 September 2012, Oberwolfach.

30. 28.10.2012: ”Global Torelli theorem for hyperk¨ahlermanifolds”, at ”International Conference on Cycles, Calibrations and Nonlinear Par- tial Diﬀerential Equations Celebrating Blaine Lawson’s 70th Birthday” (October 22-28, 2012, Stony Brook University).

31. 29.10.2012: ”Local structure of twistor spaces”, at ”Hyper-K¨ahlerGe- ometry Workshop”, Simons Center for Geometry and Physics, October 29 - November 2, 2012.

32. 07.12.2012: Non-hyperbolicity of hyperk¨ahlermanifolds, at ”Victor Kulikov’s 60-th Birthday” conference (December 3-7, 2012, Steklov Institute, Moscow).

3 33. 09.03.2013: Instanton bundles on CP and rational curves on twistor spaces, TIFR, Mumbai, India, International Conference on Analytic and algebraic geometry related to bundles, March 18-22, 2013.

34. 27.03.2013: Ratner’s theorem and ergodic complex structures,, KIAS, Seoul, Workshop on Deformation and Moduli in Complex Geometry, March 25-29, 2013.

35. 03.067.2013: Holography principle and Moishezon twistor spaces, Work- shop “Moduli Spaces and their Invariants in Mathematical Physics”, CRM, Montreal, June 3-14, 2013.

36. 03.10.2013: Ergodic complex structures and Kobayashi metric, ”Sym- plectic algebraic geometry”, 30/09/2013 to 04/10/2013, Kansai Semi- nar House, Kyoto.

37. 8.11.2013: Towards the cone conjecture for hyperk¨ahlermanifolds, a talk at Quiver Varieties Program Seminar, Simons Center, Stony Brook.

28 38. 6.12.2013: Subtwistor metric on the moduli of hyperk¨ahlermanifolds: International conference “Geometry and analysis on metric structrures”, Geometric control theory laboratory, Novosibirsk, December 4-7, 2013.

39. 26.03.2014: Hyperk¨ahlermanifolds are non-hyperbolic, a conference ”Moduli spaces of irreducible symplectic varieties, cubics and Enriques surfaces”, Laboratoire Painleve, Lille, March 24-28, 2014.

40. 16.03.2014: Hypercomplex manifolds of quaternionic dimension 2 and HKT-structures, workshop “Complex Geometry and Lie Groups”, To- rino, 16-20 June 2014.

41. 05.08.2014: Holography principle and Moishezon twistor spaces, Con- ference on The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles, National University of Singapore, 4-8 Aug 2014.

42. 10.08.2014: Complex subvarieties in homogeneous complex manifolds, Topology of Torus Actions and Applications to Geometry and Combi- natorics (ICM Satellite conference) Aug. 7-11, 2014, Daejeon, Korea.

43. 18.08.2014: Teichmuller spaces, ergodic theory and global Torelli theorem, SEOUL ICM 2014, August 13-August 22, 2014.

44. 05.09.2014: K¨ahlerstructure on the knot space of a G2-manifold, Si- mons Center for Geometry and Physics, Stony Brook, G2 manifolds: September 2 - 5, 2014.

45. 11.09.2014: K¨ahler threefolds without subvarieties, a talk at REAL and COMPLEX DIFFERENTIAL GEOMETRY 8-12 September 2014, Faculty of Mathematics and Computer Science, University of Bucharest.

46. 19.09.2014: Holography principle and Moishezon twistor spaces, a section “Quaternion-K¨ahlermanifolds and related structures in Rieman- nian and algebraic geometry,” Joint Meeting of the German Mathe- matical Society (DMV) and the Polish Mathematical Society (PTM), Poznan, 17-20 September 2014.

47. 01.10.2014: ”K¨ahlerthreefolds without subvarieties” at Complex Ge- ometry, Analysis and Foliations: a conference dedicated to the memory of M. Brunella, ICTP, Trieste, 29.09.2014-3.10.2014.

48. 21.10.2014: Symplectic packing for simple K¨ahlermanifolds, hyperk¨ahler manifolds and tori, Skolkovo Institute of Science and Technology, In- ternational conference ”Geometry, Topology and Integrability”, Skolkovo, 20-25 October 2014.

29 49. 04.12.2014: ”Pseudoholomorphic curves with boundaries on holomorphic Lagrangian subvarieties”, a conferemce ”Lagrangian submanifolds and related topics”, University of Milan, 4-5 December 2014.

10 Teaching activities

10.1 Courses taught Lecture notes and exercises for some of my courses are available from my personal homepage, http://verbit.ru/. Here is a short list of recent courses with additional supplementary materials. In English: • 2010 (Fall): A minicourse on K¨ahlergeometry (Tel-Aviv University, December 2010). Slides:

– Lecture 1: K¨ahlergeometry and holonomy – Lecture 2: Calabi-Yau theorem – Lecture 3: Bochner’s vanishing and Bogomolov’s decomposition for Calabi-Yau manifolds – Lecture 4: Supersymmetry and K¨ahleridentities

• 2013 (Spring): Geometry of manifolds (Math in Moscow and HSE). An advanced undergraduate course.

– Lecture 1: Smooth manifolds: introduction – Lecture 2: Sheaves, smooth manifolds and locally finite covers – Lecture 3: Partition of unity. – Lecture 4: Hausdorff dimension and Hausdorff measure. – Lecture 5: Vector fields and derivations. – Lecture 6: Germs and the sheaf of derivations. – Lecture 7: Locally free fibrations and categories. – Lecture 8: Locally free fibrations and vector bundles. – Lecture 9: Serre-Swan theorem. – Lecture 10: De Rham algebra. – Lecture 11: Lie derivative. – Lecture 12: Poincare lemma.

In Russian: • 2001 (Fall): Algebraic geometry over C (an advanced undergraduate course). Exam problems.

30 • 2004 (Fall): Algebra and geometry for ﬁrst-year students (an integrated course)

• 2005 (Spring): Measure theory for ﬁrst-year students.

• 2006 (Fall): Hodge theory and its applications (an advanced undergraduate course).

• 2008 (Spring): Foundations of K¨ahlergeometry (an advanced undergraduate course).

• 2008 (Spring): Topology for ﬁrst-year students.

– Lecture 0, Zorn’s lemma and Axiom of Choice. – Lectures 1 and 2, metric spaces, completion, p-adic numbers. – Lectures 3 and 4, compacts in metric spaces, Heine-Borel theorem, Hopf-Rinow theorem. – Lecture 5, Hausdorff axioms. – Lecture 6, products topology. – Lectures 7 and 8 , metrization theorem and compactness. – Lecture 9, product of compact spaces, Tychonoff’s theorem and ultrafilters. – Lecture 10, Banach spaces, Frechet spaces. – Lectures 11 and 12, the space of continuous maps, connected and path connected spaces. – Lectures 13 and 14, totally disconnected spaces, Boolean algebras, Stone’s representation theorem. – Lecture 15, fundamental group. – Lecture 16, universal covering and fundamental group. – Lectures 17 and 18, Seifert-van Kampen theorem, free groups, Nielsen-Schreier theorem.

• A book is written, based on this lectures (PDF, 4.5 Mb, to appear in Independent University of Moscow press). Its annotation:

A beginner’s course of topology, suitable for a ﬁrst-year student. The author covers metric geometry (completions, compactness, Heine-Borel and Hopf-Rinow theorem), point- set topology (up to Tychonoﬀ theorem, Stone’s representation theorem for Boolean algebras, and Urysohn’s metrization theorem), some category theory is included. The set-up

31 is very basic (set theory is explained from the beginnings and up to Axiom of Choice and its applications). The book ends with an introduction to homotopy theory; Galois coverings, fundamental group, Nielsen-Schreier theorem and Seifert- van Kampen theorem. There ara two expositions (of equal length) going in parallel, one purely lecture-based, another a problem based course, suitable for a more advanced student; in the second part, all theorems are split into series of problems for the student to solve.

• 2009 (Spring): Geometry of complex surfaces.

– Lecture 1: Kodaira-Enriques classiﬁcation. – Lecture 2: positive currents and Hahn-Banach theorem. – Lecture 3: Elliptic operators, Gauduchon metric and strong max- imum principle.

– Lecture 4: Surfaces with even b1 and Gauduchon metrics. – Lectures 5 and 6: Montel spaces, K¨ahlercurrents and the theorem of Lamari. – Lecture 7: The Kobayashi-Hitchin correspondence and Bogo- molov’s theorem on surfaces of class VII.

• 2010 (Spring): The Ricci ﬂow and topology (an advanced undergraduate course). Exam problems.

• 2010 (Spring): Analysis on manifolds (for second-year students).

• 2010 (Fall): Measure theory (for second-year students).

– Lecture 1: Triangulation of polyhedral bodies. – Lectures 2-3: Measure theory for polyhedra. Dehn’s invariant. – Lecture 4: Boolean algebras and measurable sets – Lecture 5: Measurable functions. – Lecture 6: Hahn’s decomposition, Radon-Nikodym’s and Fubini’s theorems. – Lecture 7: Caratheodory’s extension theorem. – Lecture 8: Haar measure.

• Spring 2010-Fall 2011: K¨ahlermanifolds and complex algebraic geometry (a graduate course).

32 • 2011 (Summer): Geometric group theory: amenable groups and poly- nomial growth.

• 2011 (Fall): Commutative algebra and algebraic geometry (basic undergraduate course).

• 2011 (Fall): Mori program (an advanced graduate course). Exam problems.

• 2012 (Spring): Complex surfaces (an advanced graduate course).

• 2012 (Spring): Basic topology (ﬁrst year undergraduate).

• 2012 (Fall): Gromov hyperbolic groups (advanced undergraduate).

• 2013 (Spring): Galois theory (a half-semester course; second year undergraduate).

• 2013 (Summer): Symplectic capacity and pseudoholomorphic curves

• Assorted notes:

– C0-estimates in Calabi-Yau theorem (2006) – C2-estimates in Calabi-Yau theorem (2008) – Derived brackets and generalized complex manifolds (2013; in English). – An integrated math curriculum.

10.2 Supervising My graduate student Andrey Soldatenkov defended his Ph. D. thesis titled “Geometry of hypercomplex manifolds” (June 2014). I am currently supervising research and study of two graduate students, two master students, and 7 undergraduate students.

11 Administrative duties

In 2010 I organized “Laboratory of Algebraic Geometry” at my university, with staﬀ of 40 researchers, by applying for a government grant of 5 million dollars. The Laboratory was supervised by Fedor Bogomolov, who agreed to serve as its principal investigator and research head. I was its director for a year, responsible for day-to-day administrative work. Currently I am its vice-director, and Bogomolov continues to serve as a principal investigator. Currently my duties include overseeing scientiﬁc reports, supporting the website, and running the weekly Seminar of Laboratory.

33 12 Other skills

In my free time, I worked as a freelance journalist for several opposition newspapers and Internet publications, designed books and record sleeves for an underground record label, administered a number of web-servers and a blog server. I am a fairly good system administrator (Linux) and an OK programmer (mostly Perl and C).