Papers by Alexei Gorodentsev
Uspehi matematičeskih nauk, 2003
Cambridge University Press eBooks, Jul 12, 1990
Aspects of mathematics. E, 1994
ABSTRACT In this paper we will discuss the connections between the description of the combinatori... more ABSTRACT In this paper we will discuss the connections between the description of the combinatorial structure of the set of exceptional vector bundles and some natural questions about the unimodular (non-symmetrical) bilinear forms. For investigation of the set of exceptional vector bundles (such bundles E that Ext i (E,E)=0 for i≠0 and dimExt 0 (E,E)=1) there was developed some time ago [by the author and A. N. Rudakov in Duke Math. J. 54, 115-130 (1987; Zbl 0646.14014) and by the author in Math. USSR, Izv. 32, No. 1, 1-13 and 33, No. 1, 67-83 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 1, 3-15 and No. 4, 747-757 (1988; Zbl 0664.14010 and Zbl 0736.14003)] the theory of helices. In section 1 we give a review of the main result of this theory. In section 2 we formulate some questions about classes of exceptional vector bundles in the Grothendieck group K 0 in terms of a lattice with non-symmetrical non-degenerate integer bilinear form. Arithmetical problems of helix theory are equivalent to the description of the orbits of isometries and orbits of the natural action of braid group on semiorthogonal bases of a lattice. In section 3 we speak about the general properties of non-symmetrical forms and their isometries and then give some examples in section 4.
American Mathematical Society translations, Oct 30, 2007
We consider the graded space R of syzygies for the coordinate algebra A of projective variety X =... more We consider the graded space R of syzygies for the coordinate algebra A of projective variety X = G/P embedded into projective space as an orbit of the highest weight vector of an irreducible representation of semisimple complex Lie group G. We show that R is isomorphic to the Lie algebra cohomology H = H • (L 2 , C), where L 2 is graded Lie subalgebra of the graded Lie s-algebra L = L1 ⊕ L 2 Koszul dual to A. We prove that the isomorphism identifies the natural associative algebra structures on R and H coming from their Koszul and Chevalley DGA resolutions respectively. For subcanonically embedded X a Frobenius algebra structure on the syzygies is constructed. We illustrate the results by several examples including the computation of syzygies for the Plücker embeddings of grassmannians Gr(2, N).
Cambridge University Press eBooks, Jul 12, 1990
Mathematics of the USSR, Feb 28, 1989
The author studies the properties of exceptional coherent sheaves on algebraic surfaces S with \o... more The author studies the properties of exceptional coherent sheaves on algebraic surfaces S with \operatorname{dim}H^0(-\mathscr K_S)\geqslant2 and describes a method for constructing infinite families of such sheaves, based on the technique developed here of reconstructions of the exceptional objects of a derived category. Bibliography: 12 titles.
Mathematics of the USSR, Feb 28, 1989
ABSTRACT The author gives a method of constructing a large family of exceptional bundles on Pn wi... more ABSTRACT The author gives a method of constructing a large family of exceptional bundles on Pn with the aid of canonical transformations. These bundles axe then put into infinite ordered sets to which a theorem of A. A. Beĭlinson can be generalized. Bibliography: 3 titles.
Izvestiâ Akademii nauk SSSR. Seriâ matematičeskaâ, 2001
Duke Mathematical Journal, 1987
arXiv (Cornell University), Sep 27, 1994
In this paper we consider orthogonal geometry of the free Z-module K 0 (P n) with respect to the ... more In this paper we consider orthogonal geometry of the free Z-module K 0 (P n) with respect to the non-symmetric unimodular bilinear form χ(E, F) = (−1) ν dim Ext ν (E, F). We calculate the isometry group of this form and describe invariants of its natural action on K 0 (P n). Also we consider some general constructions with non-symmetric unimodular forms. In particular, we discuss orthogonal decomposition of such forms and the action of the braid group on a set of semiorthonormal bases. We formulate a list of natural arithmetical conjectures about semiorthogonal bases of the form χ.
Russian Mathematical Surveys
Uspekhi Matematicheskikh Nauk
American Mathematical Society Translations: Series 2, 2007
Beiträge Algebra Geom., 1998
In this note we introduce a family Σ i , i = 0,. .. , n − 2 of discriminants in the space P n of ... more In this note we introduce a family Σ i , i = 0,. .. , n − 2 of discriminants in the space P n of polynomials of degree n in one variable and study some of their algebraic and topological properties following [Ar]-[Va] and [GKZ]. The discriminant Σ i consists of all polynomials p such that some nontrivial linear combination α 0 p+α 1 p ′ +• • •+α i p (i) has a zero of multiplicity greater or equal i+2. In particular, using the inversion of differential operators with constant coefficients (which induces the nonlinear involution on P n) we obtain the algebraic isomorphism of Σ i and Σ n−2−i for all i.
Journal of Mathematical Sciences, 1996
Izvestiya: Mathematics, 2001
This paper begins a detailed exposition of a geometric approach to quantization, which is present... more This paper begins a detailed exposition of a geometric approach to quantization, which is presented in a series of preprints ([23], [24], ...) and which combines the methods of algebraic and Lagrangian geometry. Given a prequantization -bundle on a symplectic manifold , we introduce an infinite-dimensional Kähler manifold of half-weighted Planck cycles. With every Kähler polarization on we canonically associate
Успехи математических наук, 2003
We explain how to calculate the correlation function for SYM N=2 SO(3)-gauge QFT with 4 flavors i... more We explain how to calculate the correlation function for SYM N=2 SO(3)-gauge QFT with 4 flavors in terms of top Chern classes of the universal bundles over the moduli spaces of rank 2 stable torsion free coherent sheaves with det=-1 on the complex projective plane. We give a direct geometrical description of these moduli spaces, study a 2-dimensional torus action on them, and calculate several initial coefficients of the correlation function via Bott residue formula.
arXiv preprint alg-geom/9604011, 1996
Abstract: We explain how to calculate the correlation function for SYM N= 2 SO (3)-gauge QFT with... more Abstract: We explain how to calculate the correlation function for SYM N= 2 SO (3)-gauge QFT with 4 flavors in terms of top Chern classes of the universal bundles over the moduli spaces of rank 2 stable torsion free coherent sheaves with det=-1 on the complex ...
Известия Российской академии наук. Серия математическая, 2001
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Papers by Alexei Gorodentsev