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A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Ursula Ludwig (2011)

Annales de l’institut Fourier

The Witten deformation is an analytic method proposed by Witten which, given a Morse function f : M R on a smooth compact manifold M , allows to prove the Morse inequalities. The aim of this article is to generalise the Witten deformation to stratified Morse functions (in the sense of stratified Morse theory as developed by Goresky and MacPherson) on a singular complex algebraic curve. In a previous article the author developed the Witten deformation for the model of an algebraic curve with cone-like singularities...

Algebraic and symplectic Gromov-Witten invariants coincide

Bernd Siebert (1999)

Annales de l'institut Fourier

For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the symplectic side, we prove that both points of view are equivalent

Amibes de variétés algébriques et dénombrement de courbes

Ilia Itenberg (2002/2003)

Séminaire Bourbaki

Les amibesdes variétés algébriques dans ( * ) n sont les images de ces variétés par l’application des moments Log : ( * ) n n , Log : ( z 1 , ... , z n ) ( log | z 1 | , ... , log | z n | ) . Des résultats obtenus par G. Mikhalkin montrent l’utilité des amibes pour l’étude des variétés algébriques réelles et complexes. Les amibes peuvent être déformées en des complexes polyédraux appelésvariétés algébriques tropicales. Cette déformation permet, en particulier, de calculer les invariants de Gromov-Witten du plan projectif et d’autres surfaces toriques en dénombrant des courbes...

An introduction to quantum sheaf cohomology

Eric Sharpe (2011)

Annales de l’institut Fourier

In this note we review “quantum sheaf cohomology,” a deformation of sheaf cohomology that arises in a fashion closely akin to (and sometimes generalizing) ordinary quantum cohomology. Quantum sheaf cohomology arises in the study of (0,2) mirror symmetry, which we review. We then review standard topological field theories and the A/2, B/2 models, in which quantum sheaf cohomology arises, and outline basic definitions and computations. We then discuss (2,2) and (0,2) supersymmetric Landau-Ginzburg...

Chen–Ruan Cohomology of 1 , n and ¯ 1 , n

Nicola Pagani (2013)

Annales de l’institut Fourier

In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable n -pointed curves of genus 1 . In the first part of the paper we study and describe stack theoretically the twisted sectors of 1 , n and ¯ 1 , n . In the second part, we study the orbifold intersection theory of ¯ 1 , n . We suggest a definition for an orbifold tautological ring in genus 1 , which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations

Dan Abramovich, Steffen Marcus, Jonathan Wise (2014)

Annales de l’institut Fourier

We consider four approaches to relative Gromov–Witten theory and Gromov–Witten theory of degenerations: J. Li’s original approach, B. Kim’s logarithmic expansions, Abramovich–Fantechi’s orbifold expansions, and a logarithmic theory without expansions due to Gross–Siebert and Abramovich–Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov–Witten invariants associated...

Computing the quantum cohomology of some Fano threefolds and its semisimplicity

Gianni Ciolli (2004)

Bollettino dell'Unione Matematica Italiana

We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from P 3 or the smooth quadric. Systematic usage of the associativity property of quantum product implies that only a very small and enumerative subset of Gromov- Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing...

Elliptic curves on spinor varieties

Nicolas Perrin (2012)

Open Mathematics

We prove irreducibility of the scheme of morphisms, of degree large enough, from a smooth elliptic curve to spinor varieties. We give an explicit bound on the degree.

Enumeration of real conics and maximal configurations

Erwan Brugallé, Nicolas Puignau (2013)

Journal of the European Mathematical Society

We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in P n is maximal. That is, there exist generic configurations of real linear spaces such that all complex conics passing through these constraints are actually real.

Finiteness of cominuscule quantum K -theory

Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, Nicolas Perrin (2013)

Annales scientifiques de l'École Normale Supérieure

The product of two Schubert classes in the quantum K -theory ring of a homogeneous space X = G / P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on  X . We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to  X that take the marked points to general Schubert varieties and whose domains...

Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities

Ikuo Satake, Atsushi Takahashi (2011)

Annales de l’institut Fourier

We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.

Invariance of tautological equations I: conjectures and applications

Y.-P. Lee (2008)

Journal of the European Mathematical Society

The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, this framework gives an efficient algorithm to calculate all tautological equations using only finite-dimensional linear algebra. Other applications include the proofs of Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy and Virasoro conjecture for target manifolds with conformal semisimple quantum cohomology, both for genus up to...

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