Invited Speakers

Nov.11th (FRI), 11:00–12:00 •H. Tsuji (Sophia University) “Variation of Bergman Kernels and its applications” We prove semipositivity of the curvature of the Narashimhan-Simha metric on an arbitrary flat projective family of varieties with only canonical singularities. This method gives a simple analytic proof and a generalization of Viehweg’s result which proves the quasiprojectivity of the moduli space of canonical models of varieties of general type with the fixed global index and the Hilbert polynomial with respect to the canonical polarization. Nov. 11th (FRI), 13:30–14:30 •R. Bielawski (Edinburgh University) “Kähler metrics on complexified symmetric spaces” We develop the relation between G-invariant Kähler metrics on the complexification of a symmmetric space of compact type G/K and K-invariant Riemannian metrics on the dual symmetric space G ∗ /K. In particular, we show that the complexification of G/K admits a G-invariant Kähler metric having a prescribed G-invariant exact Ricci form. Nov. 11th (FRI), 15:00–16:00 •A. Sergeev (Steklov Mathematical Institute, Moscow) “Harmonic maps into loop spaces of compact Lie groups” The twistor approach allows to reduce the ”real” problem of constructing harmonic maps φ : M → N from a Riemann surface M to a Riemannian manifold N to a ”complex” problem of constructing pseudoholomorphic maps ψ : M → Z from M to the twistor bundle π : Z → N . Here Z is an almost complex manifold, having the following characteristic property: the projection π◦ψ : M → N of any pseudoholomorphic map ψ : M → Z is a harmonic map. We apply this approach to the study of harmonic maps from Riemann surfaces to the loop spaces ΩG of compact Lie groups G. These harmonic maps are of special interest because of their relation to the Yang–Mills equations on R. Nov. 12th (SAT), 9:30–10:30 •K. Sugiyama (Chiba University) “On a geometric non-abelian class field theory and an application to threefolds” Let K be a number field. Let ClK be its idele class group and Clo K its identity component. Then the classical class field theory of the number theory tells us that there is an isomorphism between the group of connected components π0(ClK) = ClK/Cl K of ClK and the Galois group Gal(K/K) where K is the maximal abelian extension of K. This has the following geometric analogue. Let S be a compact Riemmanian surface. Then a local system χ over S naturally defines a local system of rank one χ̌ on its Jacobian Jac(S) and vice versa. We will call this correspondence as “a geometric abelian class field theory”. We will discuss an extension of such a correspondence to a non-abelian case. More precisely we will discuss a correspondence between flat PSL2(C) bundles over S which has parabolic reductions at certain points {Pi} and perverse sheaves over the modular stack of principal SL2(C) bundles over S which have paradolic reduction at {Pi}. Using a relative version of our geometric non-abelian class field theory, we will also discuss a relation between a characteristic polynomial of a monodromy representation of the KZ-connection and the Alexander polynomial of a certain threefold. Nov. 12th (SAT), 11:00–12:00 •O. Sheinman (Steklov Mathematical Institute, Moscow) “Krichever-Novikov algebras and Knizhnik-Zamolodchikov connection for positive genera” Krichever-Novikov algebras appeared in 1987 as a counterpart of the Virasoro and Kac-Moody algebras, related to Riemann surfaces with punctures and fixed (jets of) coordinates in their neighborhoods. They possess a remarkable almost graded structure which enables us to construct modules generated by vacuum vectors. For zero genus, Krichever-Novikov algebras coinside with affine Kac-Moody and Virasoro algebras, respectively. For positive genera, Krichever-Novikov algebras bring numerous new phenomena of algebraic-geometrical nature into the theory of Lie algebras and their representations. For example, fermionic representations of those algebras are parameterized by the holomorphic vector bundles on the Riemann surface. There is a fundamental relation, based on the KodairaSpencer theory, between the Krichever-Novikov algebras and the moduli spaces of Riemann surfaces with punctures which is a starting point for the applications of Krichever-Novikov algebras to the Conformal Field Theory. In our talk we are going to introduce Krichever-Novikov algebras and their fermionic representations, give the corresponding modification of the Sugawara construction, describe the Kuranishi tangent space to the moduli space of curves (for a generic point) in terms of those algebras. After that we intend to present a certain development, based on using Krichever-Novikov algebras, of the A.TsuchiyaK.Ueno-Y.Yamada approach to the CFT, and construct Knizhnik-Zamolodchikov connection on the moduli space of curves of a positive genus. Nov. 12th (SAT), 13:30–14:30 •R. Miyaoka (Kyushu University) “Special geometry in connenction with hypersurface geometry” There have been known many metrics with holonomy group G2 on both complete and compact 7-manifold M . But examples with explicitly described metric are only those given by Bryant-Salamon in 1989. These cohomogeneity one complete metrics turn out to have relations with hypersurface geometry, especially with the isoparametric hypersurface theory. Using this, we know the topology of M , (co)associative submanifold of M , which are intensively discussed by AtiyahWitten from the physical view point. By using the hypersurface geometry, we also give examples of special Lagrangian submanifolds in T S equipped with a Ricci-flat K?”ahler metric called the Stenzel metric. Nov. 12th (SAT), 15:00–16:00 •A. L. Gorodentsev (ITEP, Moscow) “Abelian Lagrangian Algebraic Geometry (after A.N.Tyurin)” We will discuss classic-style algebraic-geometric approach to the geometry of lagrangian cycles in a presence of abelian gauge group U(1). It was developed by A. N. Tyurin during the last years of his life. We give a review of basic concepts of ALAG (spaces of half-weighted Bohr-Sommerfeld cycles, BortwickPaul-Uribe maps, and ALAG-quantization) and explane how it can be applied to undestand geometrically famious recent developments in math. mhysics, such as projective flat connections on the bundles of conformal blocks and non-abelian theta-functions.