Florian Block, Lothar Göttsche,
... this author on: Oxford Academic Google Scholar Lothar Göttsche Lothar Göttsche 2International Centre for Theoretical Physics, Strada Costiera 11, ...
Tópico(s): semigroups and automata theory
2015 - Oxford University Press | International Mathematics Research Notices
Lothar Göttsche, Martijn Kool,
We conjecture a formula for the generating function of virtual $\chi_y$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. These virtual $\chi_y$-genera can be written in terms of descendent Donaldson invariants. Using T. Mochizuki's formula, the latter can be expressed in terms of Seiberg-Witten invariants ...
Tópico(s): Algebraic structures and combinatorial models
2020 - Springer Science+Business Media | Communications in Mathematical Physics
Lothar Göttsche, Martijn Kool,
We conjecture a formula for the virtual elliptic genera of moduli spaces of rank 2 sheaves on minimal surfaces S of general type.We express our conjecture in terms of the Igusa cusp form χ 10 and Borcherds type lifts of three quasi-Jacobi forms which are all related to the Weierstrass elliptic function.We also conjecture that the generating function of virtual cobordism classes of these moduli spaces depends only on χ(O S ) and K 2 S via two universal functions, one of which is determined by the cobordism ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2019 - | Communications in Number Theory and Physics
Lothar Göttsche, Martijn Kool,
We conjecture a formula for the refined SU(3) Vafa-Witten invariants of any smooth surface S satisfying H 1 (S, Z) = 0 and p g (S) > 0. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions.We prove that our formula satisfies a refined S-duality modularity transformation.We provide evidence for our formula by calculating virtual χ ygenera of moduli spaces of rank 3 stable sheaves on S in examples using Mochizuki's formula.Further ...
Tópico(s): Advanced Mathematical Identities
2018 - | Pure and Applied Mathematics Quarterly
Lothar Göttsche, Franziska Schroeter,
We introduce a tropical enumerative invariant depending on a variable y y which generalizes the tropical refined Severi degree. We show that this refined broccoli invariant is indeed independent of the point configuration, and that it specializes to a tropical descendant Gromov-Witten invariant for y = 1 y=1 and to the corresponding broccoli invariant for y = − 1 y=-1 . Furthermore, we define tropical refined descendant Gromov-Witten invariants which equal the corresponding refined broccoli ...
Tópico(s): Algebraic Geometry and Number Theory
2018 - American Mathematical Society | Journal of Algebraic Geometry
Florian Block, Lothar Göttsche,
The Severi degree is the degree of the Severi variety parametrizing plane curves of degree d with delta nodes. Recently, G\"ottsche and Shende gave two refinements of Severi degrees, polynomials in a variable y, which are conjecturally equal, for large d. At y = 1, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also ...
Tópico(s): Commutative Algebra and Its Applications
2015 - Cambridge University Press | Compositio Mathematica
Lothar Göttsche, Vivek Shende,
We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces.We also give a refinement of the Caporaso-Harris recursion, and conjecture that it produces the same invariants in the sufficiently ample setting.The refined recursion specializes at y D 1 to the Itenberg-Kharlamov-Shustin recursion for Welschinger invariants.We ...
Tópico(s): Geometric and Algebraic Topology
2014 - Mathematical Sciences Publishers | Geometry & Topology
Lothar Göttsche, Hiraku Nakajima, Kōta Yoshioka,
We propose an explicit formula connecting Donaldson invariants and Seiberg–Witten invariants of a 4-manifold of simple type via Nekrasov's deformed partition function for the N = 2 SUSY gauge theory with a single fundamental matter. This formula is derived from Mochizuki's formula, which makes sense and was proved when the 4-manifold is complex projective. Assuming our formula is true for a 4-manifold of simple type, we prove Witten's conjecture and sum rules for Seiberg–Witten invariants (superconformal ...
Tópico(s): Algebraic structures and combinatorial models
2011 - Kyoto University | Publications of the Research Institute for Mathematical Sciences
Barbara Fantechi, Lothar Göttsche,
For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2010 - Mathematical Sciences Publishers | Geometry & Topology
Lothar Göttsche, Hiraku Nakajima, Kōta Yoshioka,
In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as K-theoretic versions of the Donaldson invariants.In particular if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the ...
Tópico(s): Advanced Algebra and Geometry
2009 - | Pure and Applied Mathematics Quarterly
Lothar Göttsche, Hiraku Nakajima, Kōta Yoshioka,
For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture [33, 38, 3] and its refinement [34], we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with $b_+ = 1$ in terms of modular forms. This formula was proved earlier in [19] more generally for simply connected 4-manifolds ...
Tópico(s): Advanced Topology and Set Theory
2008 - Lehigh University | Journal of Differential Geometry
Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli,
Tópico(s): Geometric and Algebraic Topology
2006 - American Mathematical Society | Mathematical surveys
Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli,
Grothendieck topologies, fibered categories and descent theory: Introduction Preliminary notions Contravariant functors Fibered categories Stacks Construction of Hilbert and Quot schemes: Construction of Hilbert and Quot schemes Local properties and Hilbert schemes of points: Introduction Elementary deformation theory Hilbert schemes of points Grothendieck's existence theorem in formal geometry with a letter of Jean-Pierre Serre: Grothendieck's existence theorem in formal geometry The Picard scheme: ...
Tópico(s): Mathematics and Applications
2006 - American Mathematical Society | Mathematical surveys
Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli,
Tópico(s): Matrix Theory and Algorithms
2006 - American Mathematical Society | Mathematical surveys
AbstractThe main aim of this chapter is to prove the following result of Kumar-Thomsen. For any nonsingular split surface X, the Hilbert scheme X[n] (parametrizing length-n subschemes of X) is split as well. Here, as earlier in this book, by split we mean Frobenius split. The proof relies on some results of Fogarty on the geometry of X[n] and a study of the Hilbert-Chow morphism γ : X[n] → X(n), where X(n) denotes the n-fold symmetric product of X (parametrizing effective 0-cycles of degree n), and γ ...
Tópico(s): Algebraic structures and combinatorial models
2005 - Birkhäuser | Progress in mathematics
Barbara Fantechi, Lothar Göttsche,
Let $X$ be an orbifold that is a global quotient of a manifold $Y$ by a finite group $G$. We construct a noncommutative ring $H\sp \ast(Y, G)$ with a $G$-action such that $H\sp*(Y, G)\sp G$ is the orbifold cohomology ring of $X$ defined by W. Chen and Y. Ruan [CR]. When $Y=S\sp n$, with $S$ a surface with trivial canonical class and $G = \mathfrak {S}\sb n$, we prove that (a small modification of) the orbifold cohomology of $X$ is naturally isomorphic to the cohomology ring of the Hilbert scheme $S\sp {[n]}$, computed ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2003 - Duke University Press | Duke Mathematical Journal
Tópico(s): Electromagnetic Scattering and Analysis
2001 - International Press of Boston | Mathematical Research Letters
We compute generating functions for the Hodge numbers of the moduli spaces of H-stable rank 2 sheaves on a rational surface S in terms of theta functions for indefinite lattices. If H lies in the closure of the ample cone and has self-intersection 0, it follows that the generating functions are Jacobi forms. In particular the generating functions for the Euler numbers can be expressed in terms of modular forms, and their transformation behaviour is compatible with the predictions of S-duality. We ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
1999 - Springer Science+Business Media | Communications in Mathematical Physics
Tópico(s): Advanced Algebra and Geometry
1998 - Birkhäuser | Selecta Mathematica
Geir Ellingsrud, Lothar Göttsche,
Tópico(s): Advanced Algebra and Geometry
1998 - Oxford University Press | The Quarterly Journal of Mathematics
I give a conjectural generating function for the numbers of δ-nodal curves in a linear system of dimension δ on an algebraic surface. It reproduces the results of Vainsencher [V] for the case δ &\le; 6 and Kleiman–Piene [K-P] for the case δ &\le; 8. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular ...
Tópico(s): Commutative Algebra and Its Applications
1998 - Springer Science+Business Media | Communications in Mathematical Physics
Lothar Göttsche, Rahul Pandharipande,
We compute the Gromov-Witten invariants of the projective plane blown up in r general points. These are determined by associativity from r+1 intial values. Applications are given to the enumeration of rational plane curves with prescribed multiplicities at fixed general points. We show that the numbers are enumerative if at least one of the prescribed multiplicities is 1 or 2. In particular, all the invariants for r<=8 (the Del Pezzo case) are enumerative.
Tópico(s): Algebraic structures and combinatorial models
1998 - Lehigh University | Journal of Differential Geometry
Tópico(s): Homotopy and Cohomology in Algebraic Topology
1996 - Springer Science+Business Media | Mathematische Zeitschrift
Tópico(s): Algebraic Geometry and Number Theory
1996 - American Mathematical Society | Journal of the American Mathematical Society
Lothar Göttsche, Daniel Huybrechts,
International Journal of MathematicsVol. 07, No. 03, pp. 359-372 (1996) No AccessHODGE NUMBERS OF MODULI SPACES OF STABLE BUNDLES ON K3 SURFACESL. GÖTTSCHE and D. HUYBRECHTSL. GÖTTSCHEMax-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany and D. HUYBRECHTSMax-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germanyhttps://doi.org/10.1142/S0129167X96000219Cited by:5 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend ...
Tópico(s): Geometry and complex manifolds
1996 - World Scientific | International Journal of Mathematics
Tópico(s): Homotopy and Cohomology in Algebraic Topology
1996 - Springer Science+Business Media | Mathematische Zeitschrift
Article Variation of moduli spaces and Donaldson invariants under change of polarization. was published on January 1, 1995 in the journal Journal für die reine und angewandte Mathematik (volume 1995, issue 467).
Tópico(s): Advanced Algebra and Geometry
1995 - De Gruyter | Journal für die reine und angewandte Mathematik (Crelles Journal)
In this book we study Hilbert schemes of zero-dimensional subschemes of smooth varieties and several related parameter varieties of interest in enumerative geometry. The main aim here is to describe t
Tópico(s): Commutative Algebra and Its Applications
1994 - Springer Nature | Lecture notes in mathematics
Barbara Fantechi, Lothar Göttsche,
Article The cohomology ring of the Hilbert scheme of 3 points on a smooth projective variety. was published on January 1, 1993 in the journal Journal für die reine und angewandte Mathematik (volume 1993, issue 439).
Tópico(s): Algebraic structures and combinatorial models
1993 - De Gruyter | Journal für die reine und angewandte Mathematik (Crelles Journal)
Lothar Göttsche, Wolfgang Soergel,
Tópico(s): Commutative Algebra and Its Applications
1993 - Springer Nature | Mathematische Annalen