Steklov Mathematical Institute of RAS Lomonosov Moscow State University P. G. Demidov Yaroslavl State University Conference in honour of Victor Buchstaber on the occasion of his 70th birthday ALGEBRAIC TOPOLOGY AND ABELIAN FUNCTIONS 18–22 June 2013 Moscow ABSTRACTS Organisers: • Steklov Mathematical Intitute of Russian Academy of Sciences (MI RAS) • Lomonosov Moscow State University (MSU), Bogolyubov Laboratory of Geometric Methods in Mathematical Physics • P.G.Demidov Yaroslavl State University (YarSU), Delone Laboratory of Discrete and Computational Geometry In cooperation with: • Kharkevich Institute for Information Transmission Problems of Russian Academy of Sciences (IITP RAS) • Laboratoire J.-V. Poncelet Organising Comittee: Sergey Novikov (Chairman, MI RAS and MSU) Vladimir Chubarikov (Vice-chairman, MSU) Alexander Kuleshov (Vice-chairman, IITP RAS) Armen Sergeev (Vice-chairman, MI RAS) Alexander Gaifullin (MI RAS, MSU, YarSU, and IITP RAS) Dmitri Millionshchikov (MSU) Taras Panov (MSU, YarSU, and IITP RAS) Supported by: Russian Academy of Sciences Russian Foundation for Basic Research (grant 13-01-06031) Grants of the Government of the Russian Federation (contracts 11.G34.31.0005 and 11.G34.31.0053) Dmitry Zimin’s Dynasty Foundation Contents Plenary lectures 7 A. Bahri (joint with M.Bendersky, F.R.Cohen, and S.Gitler), Generalizations of the Davis– Januszkiewicz construction . . . . . . . . . . . . 7 V. Z. Enolski, Periods of second kind differentials of (n,s)-curves . . . . . . . . . . . . . . . . . . . . 9 L. D. Faddeev, Zero modes in Liouville model . . . . 11 T. Januszkiewicz, Singular fibrations with fibers (Sn)k 12 A. G. Khovanskii, Co-convex bodies and multiplicities 13 I. M. Krichever, The universal Whitham hierarchy and geometry of moduli spaces of curves with punctures . . . . . . . . . . . . . . . . . . . . . . 15 M. Masuda, Toric origami manifolds in toric topology 16 I. A. Panin, A proof of the geometric case of a conjec- ture of Grothendieck and Serre concerning princi- pal bundles . . . . . . . . . . . . . . . . . . . . . 18 N. Ray, Thom complexes and stable decompositions . 19 D. Y. Suh (joint with S.Kuroki), Classification of com- plex projective towers up to dimension 8 and co- homological rigidity . . . . . . . . . . . . . . . . 21 I. A. Taimanov, Elliptic solitons and minimal tori with planar ends . . . . . . . . . . . . . . . . . . 22 A. M. Vershik, Dynamics of metrics and measures: new and old problems . . . . . . . . . . . . . . . 23 A. P. Veselov, Elliptic Dunkl operators and Calogero– Moser systems . . . . . . . . . . . . . . . . . . . 24 G. M. Ziegler (joint with P.V.M.Blagojević and W.Lück), On highly regular embeddings . . . . . 25 3 Invited lectures 27 I. V. Arzhantsev, Cox rings, universal torsors, and in- finite transitivity . . . . . . . . . . . . . . . . . . 27 I. K. Babenko, Some topological problems in systolic geometry . . . . . . . . . . . . . . . . . . . . . . 29 M. Bakuradze, On some rational and integral complex genera . . . . . . . . . . . . . . . . . . . . . . . 31 Ya. V. Bazaikin (joint with O.A.Bogoyavlenskaya), On G Holonomy Riemannian Metrics on Defor- 2 mations of Cones over S3 × S3 . . . . . . . . . . 33 C. Casacuberta, Cohomological localization towers . 35 N. P. Dolbilin, Parallelohedra and the Voronoi Con- jecture . . . . . . . . . . . . . . . . . . . . . . . 37 V. Dragović, The Sokolov system and integrable Kirchhoff elasticae via discriminantly separable polynomials and Buchstaber’s two-valued groups 39 A. N. Dranishnikov, On Gromov’s macroscopic di- mension conjecture . . . . . . . . . . . . . . . . 41 N. Yu. Erokhovets, Characterization of simplicial complexes with Buchstaber number two . . . . . 42 A. Fialowski, Deformations and Contractions of Alge- braic Structures . . . . . . . . . . . . . . . . . . 44 J. Grbić, Hopf algebras and homotopy invariants . . . 45 S. M. Gusein-Zade, Higher order generalized Euler characteristics and generating series . . . . . . . 46 I. V. Izmestiev, Kokotsakis polyhedra and elliptic functions . . . . . . . . . . . . . . . . . . . . . . 47 Y. Karshon, Non-compact symplectic toric manifolds 49 H. Khudaverdian, Operator pencils on densities . . . 50 A. A. Kustarev, Equivariant almost complex qua- sitoric structures . . . . . . . . . . . . . . . . . . 51 A. Yu. Lazarev (joint with C.Braun), BV algebras ∞ in Poisson geometry . . . . . . . . . . . . . . . . 52 4 Z. Lü, Equivariant cobordism of unitary toric manifolds 53 A. E. Mironov, Periodic and rapid decay rank two self- adjoint commuting differential operators . . . . . 55 A. Murillo, Algebraic models of spaces of sections of nilpotent fibrations . . . . . . . . . . . . . . . . 56 O. R. Musin, Tucker and Fan’s lemma for manifolds . 57 A. Nakayashiki, A refined Riemann’s singularity the- orem for sigma function . . . . . . . . . . . . . . 59 E. Yu. Netay, Geometric differential equations on the universal spaces of Jacobians of elliptic and hyper- elliptic curves . . . . . . . . . . . . . . . . . . . 61 K. Pawałowski, Non-symplectic smooth group actions on symplectic manifolds . . . . . . . . . . . . . . 63 S. Terzić, Toric geometry of the action of maximal compact torus on complex Grassmannians . . . . 64 D. A. Timashev, Quotients of affine spherical varieties by unipotent subgroups . . . . . . . . . . . . . . 66 V. V. Vershinin, Brunnian and Conen braids and ho- motopy groups of spheres . . . . . . . . . . . . . 68 A. Yu. Vesnin, On hyperbolic 3-manifolds with geodesic boundary . . . . . . . . . . . . . . . . . 70 Th. Th. Voronov, Algebra of Densities . . . . . . . . 72 J. Wu (joint with R.Mikhailov), Combinatorial group theory and the homotopy groups of finite complexes 74 R. T. Živaljević, Flat polyhedral complexes . . . . . 75 Posters 77 A. A. Ayzenberg, Stanley–Reisner rings of spherical nerve-complexes . . . . . . . . . . . . . . . . . . 77 V. S. Dryuma, Homogeneous extensions of the first or- der ODE’s . . . . . . . . . . . . . . . . . . . . . 78 Yu. V. Eliyashev, Mixed Hodge structures on comple- ments of coordinate complex subspace arrangement 80 5 M. A. Gorsky, Subword complexes and 2-truncations 81 T. Horiguchi, The equivariant cohomology rings of (n − k,k) Springer varieties . . . . . . . . . . . . 83 I. Yu. Limonchenko, Subword complexes and 2-trun- cations . . . . . . . . . . . . . . . . . . . . . . . 85 E. S. Shemyakova, Factorization of Darboux trans- formations of arbitrary order for two-dimensional Schrödinger operator . . . . . . . . . . . . . . . 87 H. Tene, On the product in negative Tate cohomology for compact Lie groups . . . . . . . . . . . . . . 89 A. Yu. Volovikov, On cohomological index of free G-spaces . . . . . . . . . . . . . . . . . . . . . . 91 List of Participants 93 6 Plenary lectures Generalizations of the Davis–Januszkiewicz construction Anthony Bahri (Rider University, USA), [email protected] Martin Bendersky (City University of New York, USA), [email protected] Frederick R. Cohen (University of Rochester, USA), [email protected] Samuel Gitler (El Colegio Nacional, Mexico), [email protected] Briefly, a toric manifold M2n is a manifold covered by local charts Cn, each with the standard action of a real n-dimensional torus (cid:14) Tn, compatible in such a way that the quotient M2n Tn has the structure of a simple polytope Pn. The construction of Davis and Januszkiewicz [4, Section 1.5] realizes all toric manifolds and, in particular, all smooth projective toric varieties. The key ingredi- ent is a characteristic function λ : F −→ Zn (1) from the set of facets of the polytope Pn into an n-dimensional integer lattice, satisfying certain conditions. The construction is M2n ∼= M2n(λ) = Tn × Pn(cid:14)∼ (2) where the equivalence relation ∼ is defined in terms of the func- tion λ. Associated to Pn is a simplicial complex K(P) with m A.B. was supported in part by a Rider University Summer Research Fellowship and grant number 210386 from the Simons Foundation; F.R.C. was supported partially by DARPA grant number 2006-06918-01. 7 vertices corresponding to the m facets of Pn. In the moment- angle complex formalism established by Buchstaber and Panov [3], the map λ determines a subtorus ker(λ) = Tm−n ⊂ Tm and a homeomorphism (cid:46) Tn × Pn(cid:14)∼ ∼= Z(K ;(D2,S1)) Tm−n. (3) P A reinterpretation of this construction allows for a generalization in such a way that it can be used to construct the family of mani- folds M(J) described in [2]. These are determined by a sequence of positive integers J = (j ,j ,...,j ). Associated to the new 1 2 m construction are polyhedral products Z(K ;(CP∞,CPJ−1)) that P play the role which the spaces Z(K ;(CP∞,∗)) do in the stan- P dard case. The construction is generalized further to yield spaces associated to the composed simplicial complexes of Ayzenberg [1]. References [1] A. Ayzenberg, Composition of simplicail complexes, poly- topes and multigraded Betti numbers, Available online at: http://arxiv.org/abs/1301.4459 [2] A. Bahri, M. Bendersky, F. Cohen and S. Gitler, Opera- tions on polyhedral products and a new topological construc- tion of infinite families of toric manifolds. Available online at: http://arxiv.org/abs/1011.0094 [3] V. Buchstaber and T. Panov, Torus actions and their applica- tions in topology and combinatorics, AMS University Lecture Series, 24, (2002). [4] M. Davis, and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 62 (1991), no. 2, 417–451. 8 Periods of second kind differentials of (n, s)-curves Victor Enolski (University of Edinburgh, UK), [email protected] The following problem is discussed and solved in particular cases here: Given curve C of genus g > 1 and its a and b-periods of holomorphic differentials 2ω,2ω(cid:48). Let 2η,2η(cid:48) are periods of the second kind differentials conjugated to 2ω,2ω(cid:48) according to the generalized Legendre relations, ıπ ηTω = ωTη, ηTω(cid:48) − ωTη(cid:48) = , η(cid:48)Tω(cid:48) = ω(cid:48)Tη(cid:48). 2 Express periods of the second kind differentials in terms these data including θ-constants, depending on Riemann period matrix τ = ω(cid:48)/ω and coefficients of polynomial defining the curve C In the case of elliptic curve y2 = 4x3 − g x − g this question 2 3 is answered by the Weierstrass formulae 4 1 (cid:88) ϑ(cid:48)(cid:48)(0) 1 ϑ(cid:48)(cid:48)(cid:48)(0) η = − k and equivalently η = − 1 . 12ω ϑ (0) 12ω ϑ(cid:48) (0) k 1 k=2 The θ-constant representation of periods of the second kind dif- ferentials is important for defining of the multi-dimensional σ- function which theory attract many attention now and was in- tensively developed during the last decade by V.M.Buchstaber with co-workers, see the recent book project V. M. Buchstaber and V. Z. Enolski and D. V. Leykin, Multi-Dimensional Sigma- Functions, arXiv:1208.0990 [math-ph], 267, pp. 2012. We present here an approach to solve the formulated above problem for the family of (n,s)-curves that represent a natural 9 generalization of Weierstrass elliptic cubic to higher genera, (cid:88) f(x,y) = yn − xs − λ xαyβ αn+βs α,β C with λ ∈ and 0 ≤ α < s − 1, , 0 ≤ β < n − 1. The k case of hyperelliphic curves and especially genus two curve were considered in details. A number of new (to the best knowledge of the authors) θ-constant relations is derived as consequence of the formulae obtained and a generalization of the Jacobi derivative formula is one from them. 10