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## Editors’ preface for the topical issue “Instantons, coherent sheaves and their moduli spaces”

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 and analytic geometries whose research interests belong to this composite area between gauge theory, moduli spaces, derived categories, vector bundles and coherent sheaves. Besides the most relevant contributions to the conference, the issue contains miscellaneous articles by other authors that fit by
subject and spirit.

Tikhomirov A. S., Markushevich D., Trautmann G., Central European Journal of Mathematics 2012 Vol. 19 No. 4 P. 1331-1355

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundled connections an in differential geometry. The new moduli spaces are algebraic spaces arising as quotients ...

Added: October 21, 2014

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

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

Bruzzo U., Markushevich D., Tikhomirov A. S., European Journal of Mathematics 2016 Vol. 2 P. 73-86

We study the moduli space $I_{n,r}$In,r of rank-2r symplectic instanton vector bundles on $\mathbb{P}^3$ℙ3 with $r\ge 2$r⩾2 and second Chern class $n\ge r+1, n-r\equiv 1(\mathrm{mod} 2)$n⩾r+1,n−r≡1(mod2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus $I_{n,r}^*$I∗n,r of tame symplectic instantons is irreducible and has the expected dimension equal to ...

Added: December 28, 2015

Bondal Alexey, Rosly Alexei, / IPMU. Series IPMU11-0117 "IPMU11-0117". 2011.

We construct a twist-closed enhancement of the derived category of coherent sheaves on a smooth compact complex-analytic manifold by means of DG-category of dbar-superconnections. ...

Added: October 30, 2013

Karasev M., Novikova E., Vybornyi E., Mathematical notes 2017 Vol. 102 No. 5-6 P. 776-786

In the model of Penning trap with a geometric asymmetry we study a resonance regime which produces a hyperbolic type algebra of integrals of motion. The algebra has qubic (non-Lie) commutation relations with creation-anihilation structure. The anharmonic part of the trap potential determines a top-like Hamiltonian over this algebra. The symmetry breaking term generates a ...

Added: October 20, 2017

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

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

Musaev E., Haupt A., Lechtenfeld O., Journal of High Energy Physics 2014 Vol. 2014 No. 11

Abstract: We consider (1+3)-dimensional domain wall solutions of heterotic supergravity on a six-dimensional warped nearly Kaehler manifold $X_6$ in the presence of gravitational and gauge instantons of tanh-kink type as constructed in [1]. We include first order alpha' corrections to the heterotic supergravity action, which imply a non-trivial Yang-Mills sector and Bianchi identity. We present ...

Added: December 8, 2014

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

Fonarev A., Kuznetsov A., 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. ...

Added: November 7, 2017

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

Jardim M., Markushevich D., Tikhomirov A. S., Moscow Mathematical Journal 2018 Vol. 18 No. 1 P. 117-148

Abstract. Let I(n) denote the moduli space of rank 2 instanton bundles of charge n on P3 . It is known that I(n) is an irreducible, nonsingular and affine variety of dimension 8n − 3. Since every rank 2 instanton bundle on P3 is stable, we may regard I(n) as an open subset of the ...

Added: August 20, 2018

Alexey Bondal, Kavli Institute for the Physics and Mathematics of the Universe News 2011 Vol. 14 P. 4-9

Дается взгляд на развитие идей гомологической алгебры и их приложений к алгебраической геометрии. Описывается связь с зеркальной симметрией и предлагается гомотопическая интерпретация категории производных категорий. ...

Added: October 14, 2013

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

Piontkovski D., Journal of Noncommutative Geometry 2016

A noncommutative Grassmanian NGr(m,n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent ...

Added: April 14, 2016

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

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

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

Jardim M., Markushevich D., Tikhomirov A. S., Annali di Matematica Pura ed Applicata 2017 Vol. 196 No. 4 P. 1573-1608

We describe new components of the Gieseker–Maruyama moduli scheme (Formula presented.) of semistable rank 2 sheaves E on (Formula presented.) with (Formula presented.), (Formula presented.) and (Formula presented.) whose generic point corresponds to nonlocally free sheaves. We show that such components grow in number as n grows, and discuss how they intersect the instanton component. ...

Added: February 18, 2017

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

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

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