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## Derived Categories of Curves as Components of Fano Manifolds

Journal of London Mathematical Society. 2018. Vol. 97. No. 2. P. 24-46.

We prove that the derived category D(C) of a generic curve of genus greater than one embeds into the derived category D(M) of the moduli space M of rank two stable bundles on C with fixed determinant of odd degree.

Publication based on the results of:

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 2 P. 337-361

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of U(gln), and formulate a conjectural answer for the small quantum cohomology rings in terms of ...

Added: October 9, 2012

Costa A., Gusein-Zade S., Natanzon S. M., Indiana University Mathematics Journal 2011 Vol. 60 No. 3 P. 985-995

Klein foams are analogues of Riemann and Klein surfaces with one-dimensional singularities. We prove that the field of dianalytic functions on a Klein foam Ω coincides with the field of dianalytic functions on a Klein surface K Ω We construct the moduli space of Klein foams, and we prove that the set of classes of ...

Added: November 24, 2012

Fonarev A., Известия РАН. Серия математическая 2013 Т. 77 № 5 С. 203-224

We construct two Lefschetz decompositions of the derived category of coherent sheaves on the Grassmannian of k-dimensional subspaces in a vector space of dimension n. Both of them admit a Lefschetz basis consisting of equivariant vector bundles. We prove fullness of the first decomposition and conjecture it for the second one. In the case when ...

Added: December 3, 2012

Sergey Natanzon, Pratoussevitch A., Journal of Singularities 2013 Vol. 7 P. 61-87

We describe all connected components of the space of hyperbolic Gorenstein quasi-homogeneous surface singularities. We prove that any connected component is homeomorphic to a quotient of R^d by a discrete group. ...

Added: August 19, 2013

Shevchishin V., Complex Variables and Elliptic Equations 2013 Vol. 58 No. 11 P. 1527-1548

We show that under mild boundary conditions the moduli space of non-compact curves on a complex surface is (locally) an analytic subset of a ball in a Banach manifold, defined by finitely many holomorphic functions. ...

Added: March 18, 2013

Tikhomirov A. S., Markushevich D., Verbitsky M., Central European Journal of Mathematics 2012 Vol. 10 No. 4 P. 1185-1187

In this preface we give a short description of the current issue of the Central European Journal of Mathematics containing 22 papers which spin around the topics of the conference “Instantons in complex geometry”, held on March 14–18, 2011 in Moscow. The main goal of the conference was to bring together specialists in complex algebraic ...

Added: October 21, 2014

Galkin S., Katzarkov L. V., Mellit A. et al., Advances in Mathematics 2015 Vol. 278 P. 238-253

We conjecture that derived categories of coherent sheaves on fake projective n-spaces have a semi-orthogonal decomposition into a collection of n+1 exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group (such as Keum’s surface). Then by passing to equivariant categories we construct new examples ...

Added: October 20, 2014

Gorsky E., Advances in Mathematics 2014 Vol. 250 P. 588-595

We derive a formula for the Sn-equivariant Euler characteristic of the moduli space Mg,n of genus g curves with n marked points. ...

Added: December 9, 2014

Jardim M., Maican M., Tikhomirov A. S., Pacific Journal of Mathematics 2017 Vol. 291 No. 2 P. 399-424

We study the irreducible components of the moduli space of instanton sheaves on P^3, that is, µ-semistable rank 2 torsion-free sheaves E with c_1(E)= c_3(E)=0 satisfying h^1(E(−2))= h^2(E(−2))=0. In particular, we classify all instanton sheaves with c_2(E) ≤4, describing all the irreducible components of their moduli space. A key ingredient for our argument is the ...

Added: September 20, 2017

Tikhomirov A. S., Matemática Contemporânea 2020 Vol. 47 P. 301-316

In this article, we will give a review of recent results on the geography and geometry of the Gieseker-Maruyama moduli scheme M = M (c1 , c2 ) of rank 2 semi-stable coherent sheaves with first Chern class c1 = 0 or −1, second Chern class c2 , and third Chern class 0 on the ...

Added: December 22, 2020

Tikhomirov A. S., Bruzzo U., Markushevich D., Mathematische Zeitschrift 2013 Vol. 275 No. 3-4 P. 1073-1093

We construct a compactification $M^{μss}$ of the Uhlenbeck–Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $\gamma: M^{ss}\to M^{μss}$, where $M^{μss}$ is the moduli space of $S$-equivalence classes of Gieseker-semistable framed sheaves. ...

Added: October 20, 2014

Ananʼin S., Verbitsky M., Journal de Mathématiques Pures et Appliquées 2014 Vol. 101 No. 2 P. 188-197

Let M be a compact hyperkähler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in H^2(M) defines a divisor Dv in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W. ...

Added: January 28, 2015

Natanzon S. M., Pratoussevitch А. А., / Institut des Hautes 'Etudes Scientifiques, Institut des Hautes ´Etudes Scientifiques (IHES). Series IHES "09/2010". 2012.

Added: January 27, 2013

Tyurin N. A., / Cornell University. Series arXiv "math". 2018.

We present an example of modified moduli space of special Bohr-Sommerfeld lagrangian submanifolds for the case when the given algebraic variety is the full flag F3 for C3 and the very ample bundle is K^{-1/2}_{F3} ...

Added: October 15, 2018

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 3 P. 573-607

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal ...

Added: October 9, 2012

Felikson А. A., Natanzon S. M., Differential Geometry and its Application 2012 Vol. 30 No. 5 P. 490-508

We consider (local) parameterizations of Teichmüller space Tg,n (of genus g hyperbolic surfaces with n boundary components) by lengths of 6 g- 6 + 3 n geodesics. We find a large family of suitable sets of 6 g- 6 + 3. n geodesics, each set forming a special structure called "admissible double pants decomposition". For ...

Added: February 5, 2013

Tyurin N. A., / Cornell University. Series arXiv "math". 2018.

In the previous papers we present a construction of the set U_SBS in the direct product B_S×PΓ(M, L) of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle L→M over a given compact simply connected symplectic manifold M. Canonical projections ...

Added: October 15, 2018

Natanzon S. M., Pratoussevitch A., Russian Mathematical Surveys 2016 Vol. 71 No. 2 P. 382-384

In this paper, we present all higher spinor structures on Klein surfaces. We present also topological invariants that describe the connected components of moduli of Klein surfaces with higher spinor structure. Each connected component is represented as a cell factorable by a discrete group . ...

Added: March 25, 2016

Springer, 2020

This volume collects contributions from speakers at the INdAM Workshop “Birational Geometry and Moduli Spaces”, which was held in Rome on 11–15 June 2018. The workshop was devoted to the interplay between birational geometry and moduli spaces and the contributions of the volume reflect the same idea, focusing on both these areas and their interaction. ...

Added: August 13, 2020

Gorinov A., / Cornell University. Series math "arxiv.org". 2014. No. 1402.5946.

We present a modification of the method of conical resolutions \cite{quintics,tom}. We apply our construction to compute the rational cohomology of the spaces of equations of nodal cubics in CP2, nodal quartics in CP2 and nodal cubics in CP3. In the last two cases we also compute the cohomology of the corresponding moduli spaces. ...

Added: February 26, 2014

Kochetkov Y., / Cornell University Library. 2013. No. 1301.6059.

We consider the space $\mathcal{M}_{2,1}$ --- the open moduli space of complex curves of genus 2 with one marked point. Using language of chord diagrams we describe the cell structure of $\mathcal{M}_{2,1}$ and cell adjacency. This allows one to construct matrices of boundary operators and compute Betty numbers of $\mathcal{M}_{2,1}$ over $\mathbb{Q}$. ...

Added: February 24, 2013

Kazaryan M., Lando S., Prasolov V., Switzerland : Springer, 2018

This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well.
The book begins by studying individual smooth algebraic curves, including the most beautiful ones, ...

Added: November 19, 2018

Finkelberg M. V., Rybnikov L. G., Algebraic Geometry 2014 Vol. 1 No. 2 P. 166-180

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on ...

Added: October 25, 2013

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2013. No. 52/13.

We give an w\explicit example of non-regular behaviour of fuchsian systems moduli space in the case of resonant singular points. Tha set of systems with same singularities, asymptotics and monodromy but still not globally equivalent is constructed. ...

Added: March 31, 2014