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Christian Okonek

Publications and source records attributed to Christian Okonek.

15 recordsLinked to original sources
arXiv2021

Graded tilting for gauged Landau-Ginzburg models and geometric applications

In this paper we develop a graded tilting theory for gauged Landau-Ginzburg models of regular sections in vector bundles over projective varieties. Our main theoretical result describes - under certain conditions - the bounded derived category of the zero locus $Z(s)$ of such a section $s$ as a graded singularity category of a non-commutative quotient algebra $Λ/\langle s\rangle$: $D^b(\mathrm{coh} Z(s))\simeq D^{\mathrm{gr}}_{\mathrm{sg}}(Λ/\langle s\rangle)$. Our geometric applications all come from homogeneous gauged linear sigma models. In this case $Λ$ is a non-commutative resolution of the invariant ring which defines the $\mathbb{C}^*$-equivariant affine GIT quotient of the model. We obtain purely algebraic descriptions of the derived categories of the following families of varieties: - Complete intersections. - Isotropic symplectic and orthogonal Grassmannians. - Beauville-Donagi IHS 4-folds.

math.AG↗
arXiv2014

A wall crossing formula for degrees of real central projections

The main result is a wall crossing formula for central projections defined on submanifolds of a real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a $\Z$-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the real subspace problem.

math.AG↗
arXiv2013

Kähler classes on universal moduli spaces and volumina of Quot spaces

We study the natural Kähler metrics on moduli spaces of stable oriented pairs in a very general framework, and we prove a universal formula expressing the Kähler class of such a moduli space in terms of characteristic classes of the universal bundle. We use these results to compute explicitly the volumina of certain Quot spaces.

math.DG↗
arXiv2013

Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.

math.AG↗
arXiv2012

Intrinsic signs and lower bounds in real algebraic geometry

A classical result due to Segre states that on a real cubic surface in ${\mathbb P}^3_\R$ there exists two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: - We explain a general principle which leads to lower bounds in real algebraic geometry, - We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. - We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree $2m-3$ in ${\mathbb P}^m_\R$.

math.AG↗
arXiv2011

Symmetric theta divisors of Klein surfaces

This is a slightly expanded version of the talk given by Ch.O. at the conference "Instantons in complex geometry", at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces". In this paper we compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.

math.AG↗
arXiv2011

Real determinant line bundles

This article is an expanded version of the talk given by Ch. O. at the Second Latin Congress on "Symmetries in Geometry and Physics" in Curitiba, Brazil in December 2010. In this version we explain the topological and gauge-theoretical aspects of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces".

math.AG↗
arXiv2008

Cohomotopy invariants and the universal cohomotopy invariant jump formula

Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of $S^1$-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which have clear functorial properties with respect to diffeomorphisms of 4-manifolds. Our invariants and the Bauer-Furuta classes are directly comparable for 4-manifolds with $b_1=0$; they are equivalent when $b_1=0$ and $b_+>1$, but are finer in the case $b_1=0$, $b_+=1$ (they detect the wall-crossing phenomena). We study fundamental properties of the new invariants in a very general framework. In particular we prove a universal cohomotopy invariant jump formula and a multiplicative property. The formalism applies to other gauge theoretical problems, e.g. to the theory of gauge theoretical (Hamiltonian) Gromov-Witten invariants.

math.GT↗
arXiv2003

Gauge theoretical Gromov-Witten invariants and virtual fundamental classes

This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $¶^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge theoretical Gromov-Witten type invariants of arbitrary genus associated with certain symplectic factorization problems with additional symmetry, and the computation of these invariants in terms of complex geometric objects. The main tool for describing moduli spaces associated with symplectic factorization problems is the "universal Kobayashi-Hitchin correspondence", which gives canonical isomorphisms ${\cal M}^*\to{\cal M}^{\rm st}$ between gauge theoretic moduli spaces of irreducible solutions of certain PDE's and complex geometric moduli spaces of stable framed holomorphic objects. We state a conjecture for the general situation: When the gauge theoretic problem is of Fredholm type, and the data for ${\cal M}^{\rm st}$ are algebraic, then ${\cal M}^{\rm st}$ admits a canonical perfect obstruction theory in the sense of Behrend-Fantechi, and the Kobayashi-Hitchin isomorphism ${\cal M}^*\to{\cal M}^{\rm st}$ identifies the gauge theoretic and the algebraic virtual fundamental classes. The conjecture was checked for the symplectic factorization problems which yield the toric varieties.

math.AG↗
arXiv1997

Master Spaces for stable pairs

We construct master spaces for oriented torsion free sheaves coupled with morphisms into a fixed reference sheaf. These spaces are projective varieties endowed with a natural $\C^*$-action. The fixed point set of this action contains the moduli space of semistable oriented torsion free sheaves and the quot scheme associated with the given data. In the case of curves with trivial reference sheaf, our master spaces compactify the moduli spaces constructed by Bertram, Daskalopoulos and Wentworth. In the 2-dimensional case with trivial rank 1 reference sheaf, master spaces provide algebraic analoga of compactified moduli spaces of twisted quaternionic monopoles.

alg-geom↗
arXiv1996

Seiberg-Witten invariants for manifolds with $b_+=1$

In this paper we describe the Seiberg-Witten invariants, which have been introduced by Witten, for manifolds with $b_+=1$. In this case the invariants depend on a chamber structure, and there exists a universal wall crossing formula. We take into account the contribution of the 1-homology of the base-manifold. For every Kähler surface with $p_g=0$ and $q$=0, these invariants are non-trivial for all $Spin^c(4)$-structures of non-negative index.

alg-geom↗
arXiv1996

Seiberg-Witten invariants for manifolds with $b_+=1$, and the universal wall crossing formula

In this paper we describe the Seiberg-Witten invariants, which have been introduced by Witten, for manifolds with $b_+=1$. In this case the invariants depend on a chamber structure, and there exists a universal wall crossing formula. For every Kähler surface with $p_g=0$ and $q$=0, these invariants are non-trivial for all $Spin^c(4)$-structures of non-negative index.

alg-geom↗
arXiv1995

Seiberg-Witten Invariants and Rationality of Complex Surfaces

The purpose of this paper is: 1) to explain the Seiberg-Witten invariants, 2) to show that - on a Kähler surface - the solutions of the monopole equations can be interpreted as algebraic objects, namely effective divisors, 3) to give - as an application - a short selfcontained proof for the fact that rationality of complex surfaces is a ${\cal C}^{\infty}$-property.

alg-geom↗