Geometry and Analysis on Complex Algebraic Varieties

Transcription

1 Семинар Геометрия и Анализ на Комплексных Алгебраических Многообразиях (VI сессия Российско-Японского Симпозиума) Workshop Geometry and Analysis on Complex Algebraic Varieties (The 6 -th Session of Joint Russia-Japan Symposium) Moscow September 22-27, 2008 Programme & Abstracts

2 The present regular 6-th session holds at the Moscow Independent University in cooperation with Institute for Control Sciences RAS. The session is organized by A.G.Aleksandrov (Moscow) and J.Sekiguchi (Tokyo). It is a sequel of a series of meetings held in the framework of RFBR-JSPS bilateral scientific project "Geometry and Analysis on Complex Algebraic Varieties" (May, April, 2008). During this period five sessions of the Joint Symposium were organized: at Moscow Independent University (Moscow, ), at Krasnoyarsk State University (Krasnoyarsk, ), at the Research Institute for Mathematical Sciences of Kyoto University (RIMS, Kyoto University, Japan, ), at Krasnoyarsk State University (Krasnoyarsk, , together with the International Conference on Analysis and Geometry devoted to memory of B.V.Shabat), and at the Tokyo University of Agriculture and Technology (TUAT, Tokyo, Japan, ). More than 150 scientists participated in these meetings; they arrived not only from RF and Japan but also from many countries of Europe, Asia and America. A total of 124 talks has been presented; few ones were given by invited speakers, specialists in related fields. The main goal of the present Sixth session of Symposium is to present results obtained by the participants of the bilateral project, to discuss new directions and perspectives with leading specialists in related subjects from other countries, and to offer an opportunity for greater interaction and collaboration amongst researchers in Complex Analysis and Algebraic Geometry. The subject contains but not restricted by the following topics: complex analytic methods in the theory of differential equations, complete intersection singularities, logarithmic differential forms, logarithmic connections, frobenius structures, theory of residues and currents, topology of singular divisors and their complements. In particular, we hope to discuss the theory of logarithmic connections with poles along the discriminant set of the versal deformations of isolated singularities and the corresponding holonomic systems of differential equations with regular singular points, the monodromy of certain hypergeometric functions in several variables which are solutions of Picard-Fuchs systems of differential equations, the problem of classification of Saito free hypersurfaces, an approach to computing of the index of vector fields with isolated singularities on complex hypersurfaces and complete intersections with arbitrary singularities, etc. Content Programme.3 Abstracts..6 Time-Table.. 13 Index

3 Workshop "Geometry and Analysis on Complex Algebraic Varieties" (The 6-th Session of Joint Russia-Japan Symposium) PROGRAMME (updated on September 17, 2008) MOSCOW session Monday, September 22, 2008 Opening Ceremony Main Conference Hall, Independent University of Moscow 09:30-10:00 10:00 10:00-10:15 10:15 11:00 Registration All guests to be seated Opening and Welcome Addresses Reception Scientific Session I - Main Conference Hall, Independent University of Moscow 11:00 11:50 Chairing: A.G.Aleksandrov Viktor P. Palamodov (Tel Aviv University) Stokes formula for Berezin integral 11:50 12:00 12:00 12:50 Yayoi Nakamura (Kinki University, Osaka) An invariant of Reiffen's isolated singularity 12:50 13:00 13:00 14:30 Lunch Scientific Session II Main Conference Hall, Independent University of Moscow 14:30 15:20 Chairing: Jiro Sekiguchi Kiyoshi Takeuchi (Tsukuba Univeristy) Monodromy zeta functions over general toric varieties 15:20 15:30 15:30 16:20 Marko Roczen (Humboldt Universitaet, Berlin) New characterization of simple singularities 15:50 16:00 16:00 20:00 Round Table 3

4 Tuesday, September 23, 2008 Scientific Session III Main Conference Hall, Independent University of Moscow 10:00 10:50 Chairing: Yayoi Nakamura Shinichi Tajima (Niigata Univeristy) Algebraic local cohomology classes and holonomic D-modules attached to Bayer-Hefez plane curve singularities 10:50 11:00 11:00 11:50 Vjacheslav P. Krivokolesko (Siberian Federal State University, Krasnoyarsk) Integral representations and combinatorial identities 11:50 12:00 12:00 12:50 Antonio Campillo (University of Valladolid) Poincaré series, induced filtrations and Toric varieties 12:50 13:00 13:00 14:30 Lunch 14:30 17:00 Joint work 17:00 20:00 Cultural program & Round table Wednesday, September 24, 2008 Scientific Session IV Main Conference Hall, Independent University of Moscow 10:00 10:50 Chairing: Timur Sadykov Alexander I. Esterov (Independent University of Moscow) Newton polyhedra of discriminants of complete intersections 10:50 11:00 11:00 11:50 Yutaka Matsui (Kinki University, Osaka) Topological Radon transforms and their application to A-discriminants 11:50 12:00 12:00 12:50 Leonid A. Gutnik (Moscow Institute of Electronics and Mathematics, Moscow) On some systems of difference equations and their applications to diophantine approximations 12:50 13:00 13:00 14:30 Lunch 14:30 17:00 Joint work 17:00 20:00 Cultural program & Round table 4

5 Thursday, September 25, 2008 Scientific Session V Main Conference Hall, Independent University of Moscow 10:00 10:50 Chairing: Shinichi Tajima Timur M. Sadykov (Siberian Federal State University, Krasnoyarsk) Hypergeometric systems with maximally reducible monodromy 10:50 11:00 11:00 11:50 Susumu Tanabe (Kumamoto University) Monodromy groups for complete intersections related to Calabi-Yau complete intersections in projective spaces 11:50 12:00 12:00 12:50 Alexey N. Kuznetsov (Moscow State Technical University "MAMI") Holonomic D-modules of Fuchsian and logarithmic types 12:50 13:00 13:00 14:30 Lunch 14:30 17:00 Joint work 17:00 20:00 Cultural program & Round table Friday, September 26, 2008 Scientific Session VI Main Conference Hall, Independent University of Moscow 10:00 10:50 Chairing: Susumu Tanabe Jiro Sekiguchi (Tokyo University of Agriculture and Technology) Unitary reflection groups of rank three and Saito Free Divisors 10:50 11:00 11:00 11:50 Sabir M. Gusein-Zade (Lomonosov State University, Moscow) Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors 11:50 12:00 12:00 12:50 12:50 13:00 13:00 14:30 14:30 16:30 Lunch Round Table/Closing Alexander G. Aleksandrov (Institute for Control Sciences RAS, Moscow) Enumeration theory of Saito free divisors 5

6 Abstracts Enumeration theory of Saito free divisors A.G.Aleksandrov A wide range of problems involving classification and enumeration of singularities is a significant part of modern singularity theory. In general, one may consider various equivalence relations between singularities such as the right equivalence (change of coordinates in the source of defining mapping), contact equivalence (change of coordinates and multiplication with a unit, that is, preserving the isomorphism class of the corresponding germ), and others. Anyway the initial stage of the solving of classification problems is a description of simple singularities. As a rule, one can write out a finite or at least perceptible list of normal forms or similar data. However a scheme of classification of more complicated isolated singularities seems to be a rather nontrivial and hard problem since such phenomenon as moduli or parametric families may occur. Furthermore, essential and serious difficulties arise in the theory of non-isolated singularities. Among different approaches to further classification it seems methods of the deformation theory are very fruitful and useful. The following observation is also very important: some types of non-isolated singularities appear as degenerate fibers in parametric families or deformations of isolated singularities, other ones (for example, divisors with normal crossing) are natural ingredients of compactifications of algebraic varieties in the Hodge theory and in related questions, and so on. Historically, the theory of singularities originated in studies of quasihomogeneous functions with isolated critical points or, in other terms, hypersurfaces with isolated singularities given by quasihomogeneous functions. Unfortunately, in contrast with the theory of isolated singularities the type of homogeneity in non-isolated case does not determine neither topological nor analytical structure of singularities. Moreover, there are types of homogeneity associated with non-isolated singularities that can not be realized as types of isolated singularities at all. There are few different ways of classification or enumeration singularities; they occur naturally in the context of the deformation theory. Thus in order to classify nonisolated singularities one can endeavor to create a list of them ordered by some rules or numerical invariants: Milnor numbers, types of homogeneity, weights of variables or vector fields, etc. It is also very important to choose a suitable representation for members of the list. In the standard theory one usually takes generators of the defining ideal, functions or polynomials, in other terms, normal forms of singularities. However in the non-isolated case any classification depends on types of singular loci of singularities themselves. So it is necessary to analyze singular loci. Further, it is possible to classify all pairs of singular hypersurfaces and their singular loci. Another way is to obtain a classification of local algebras associated with singularities, Lie algebras of differentiations, and so on. The purpose of this talk is to discuss an approach to the study of classification problem for quasi-homogeneous non-isolated Saito singularities making use of deformation theory of varieties with Gm-action. Among other things we describe an approach for computation of free deformations of quasi-cones over quasi-smooth varieties. We also discuss some useful applications. 6

7 Poincaré series, induced filtrations and Toric varieties A.Campillo Poincaré series for multi-index filtrations of ideals given by valuative type conditions can be studied by means of integrals with respect to the Euler characteristic. Nevertheless, in general, the induced filtrations of those ones on subspaces are not given by valuative conditions. We show that this is not the case for monomial valuative conditions on toric varieties. We also show formulae for such embedded Poincaré series and show their relations with zeta functions and topology. In the toric case, the theory of Poincaré seies becomes special, with rather interesting features. Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors S.M.Gusein-Zade To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, there was obtained a formula for the Poincaré series of the corresponding multiindex filtration similar to the one associated to plane germs. It is shown that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines "the topology of the set of the divisors" in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. The used methods also permit to simplify the proof of the statement by Yamamoto that the Alexander polynomial of a plane curve singularity determines its embedded topology. This ia a joint work with A.Campillo and F.Delgado. Newton polyhedra of discriminants of complete intersections A.I.Esterov Let P be the restriction of the standard projection Ck Cn to a complete intersection V contained in Ck. We describe the Newton polyhedron of the discriminant of P in terms of the Newton polyhedra of equations of V, provided that the coefficients of the equations are in general position. We discuss a method of computation of the index of a differential 1form on a complete intersection in terms of the Newton polyhedra of the components of the form and of the equations of V under the assumption that the leading coefficients of the corresponding defining functions are generic. Monodromy groups for complete intersections related to Calabi-Yau complete intersections in projective spaces Susumu Tanabe Let Y be a smooth Calabi--Yau complete intersection in a projective space. Consider the mirror complete intersection manifold X of Y in the sense of Batyrev and Borisov. We show that the space of quadratic invariants of the monodromy group associated with the affine part of X is one-dimensional and spanned by the Gram matrix of a classical generator of the derived category of coherent sheaves on Y with respect to the Euler form. This is a joint work with Kazushi Ueda. 7

8 Hypergeometric systems with maximally reducible monodromy T.M.Sadykov A hypergeometric system of partial differential equations is said to have maximally reducible monodromy if its space of holomorphic solutions splits into the direct sum of one-dimensional invariant subspaces. In the talk, we give a necessary and sufficient condition for a bivariate nonconfluent system of hypergeometric type to have maximally reducible monodromy. In particular, we will prove that any hypergeometric configuration defined by a plane zonotope admits this property. Algebraic local cohomology classes and holonomic D-modules attached to Bayer-Hefez plane curve singularities Shinichi Tajima In 2001, V. Bayer and A. Hefez extended a result of O. Zariski and gave an explicit description of germs of singular plane curves having the property that the difference of these Milnor number and Tyurina number is equal to one or two. In this talk we consider Bayer-Hefez plane curves in the context of computational algebraic analysis. We give an explicit description of algebraic local cohomology classes and holonomic systems attached to these plane curves. Holonomic D-modules of Fuchsian and logarithmic types A.N.Kuznetsov The aim of the talk is to discuss the theory of differential holonomic systems with regular singularities. The main result is the following: an arbitrary integrable Fuchsian Dmodule is meromorphically isomorphic to integrable logarithmic one. The proof is based on certain properties of eigenvalues of the monodromy determined under the assumption of integrability of the D-module. As a consequence we obtain a simple classification of integrable Fuchsian modules at non-singular points of their singular loci. Another corollary is a complete description of regular singular integrable systems with one unknown. Among other things this result delivers an explicit representation of closed logarithmic forms in terms of eigenvalues of the monodromy associated with the corresponding logarithmic holonomic D-module. We develop a method of computation of eigenvalues of the monodromy, the most important invariants of regular singular holonomic integrable systems. In conclusion, some examples and clear interpretations of the obtained results are constructed. New Characterization of Simple Singularities Marko Roczen Algebraic or geometric classification problems with certain finiteness conditions tend to be related to the classification of simple singularities. In a recent paper [Adv. Math. 218, 2008], Christensen, Piepmeyer, Striuli and Takahashi gave a new characterization of simple singularities by finiteness of the category G(R), of Gorenstein modules of dimension 0, which was introduced by Auslander and Bridger in Starting with early origins of the related classification problem, the above result is discussed here. Finally, a Brauer-Thrall type conjecture will be formulated for the category G(R). 8

9 Unitary reflection groups of rank three and Saito free divisors Jiro Sekiguchi The discriminants of unitary reflection groups of rank three are studied in detail based on the theory of Saito free divisors. Such discriminants give us, in fact, nice examples of Saito free divisors; they can be obtained by restrictions of variables of discriminants of polynomials of lower degree. As an application we also describe and investigate integrable connections whose singularities are contained in the zero sets of the discriminants. Topological Radon transforms and their application to A-discriminants Yutaka Matsui As an application of topological Radon transforms we give explicit formulas for the dimensions and the degree of A-discriminant varieties introduced by Gelfand-KapranovZelevinsky. We give also combinatorial formulas for Euler obstructions of general toric varieties. These formulas are described by the geometry of the configurations A. This is a joint work with Kiyoshi Takeuchi. Monodromy zeta functions over general toric varieties Kiyoshi Takeuchi Making use of sheaf-theoretical (functorial) methods we prove some explicit formulas for the monodromy zeta functions of non-degenerate polynomials over general (not necessarily normal) toric varieties. The techniques of constructible sheaves used in the proof whould be of interest in their own rights. This is a joint work with Y. Matsui. Stokes formula for Berezin integral V.P.Palamodov We initiate a version of Stokes theorem that engages superdomains whose boundary is the union of smooth parts of codimension (1,0) and (0,1).This theorem is extended for domains whose boundary has normal crossing singularities. 9

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15 Index Aleksandrov A.G. 2, 3, 5, 6, 14 Campillo A. 4, 7, 13 Esterov A.I. 4, 7, 14 Gusein-Zade S.M. 5, 7, 14 Gutnik L.A. 4, 10, 14 Krivokolesko V.P. 4, 11, 13 Kuznetsov A.N. 5, 8, 14 Matsui Y. 3, 4, 9, 14 Nakamura Y. 3, 4, 12, 13 Palamodov V.P. 3, 9, 13 Polunin V.A. 11 Roczen M. 3, 8, 13 Sadykov T.M. 4, 5, 8, 14 Sekiguchi J. 2, 3, 5, 9, 14 Soldatov A.P. 11 Tajima Sh. 4, 5, 8, 12, 13 Takeuchi K. 3, 4, 9, 13 Tanabe S. 5, 7, 14 15

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