Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

Reconstruction theorems for Gromov-Witten invariants

We prove a Reconstruction Theorem for (ordinary) Gromov-Witten invariants which improves the First Reconstruction Theorem of Kontsevich and Manin for manifolds whose Picard number is not one. In some cases our Reconstruction Theorem gives 1-point reconstruction.We discuss some interesting examples in detail, and finally we describe four applications: rational surfaces, Fano threefolds, the blow-up of the projective space along a linear subspace, and the non-Fano moduli space of curves .     

The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry

Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso–Harris formula in terms of both tropical geometry and lattice paths. H. Markwig has been funded by the DFG grant Ga 636/2.     

Hamiltonian / Gauge theoretical Gromov-Witten invariants of toric varieties

We present a purely algebraic approach to the Hamiltonian / Gauge theoretical invariants associated to torus actions on affine spaces. Secondly, we address the issue of computing the invariants: a localization and a genus recursion formula are deduced. Partially supported by: EAGER - European Algebraic Geometry Research Training Network, contract No. HPRN-CT-2000-00099 (BBW).     

On Grothendieck transformations in Fulton-MacPherson’s bivariant theory

W. Fulton and R. MacPherson have introduced a notion unifying both covariant and contravariant theories, which they called a Bivariant Theory. A transformation between two bivariant theories is called a Grothendieck transformation. The Grothendieck transformation induces natural transformations for covariant theories and contravariant theories. In this paper we show some general uniqueness and existence theorems on Grothendieck transformations associated to given natural transformations of covariant theories. Our guiding or typical model is MacPherson’s Chern class transformation c_∗:F→H_∗. The existence of a corresponding bivariant Chern class γ:F→H was conjectured by W. Fulton and R. MacPherson, and was proved by J.-P. Brasselet under certain conditions.     

A genus-4 topological recursion relation for Gromov-Witten invariants

In this paper,we give a new genus-4 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds via Pixton’s relations on the moduli space of curves.As an application,we prove that Pixton’s relations imply a known topological recursion relation on Mg,1 for genus g≤4.     

To the theory of viscosity solutions for uniformly elliptic Isaacs equations

We show how a theorem about the solvability in C^1,1$C^{1,1}$ of special Isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate Isaacs equations. We apply it also to establish the C^1+χ$C^{1+\chi }$ regularity of viscosity solutions and show that finite-difference approximations have an algebraic rate of convergence. The main coefficients of the Isaacs equations are supposed to be in C^γ$C^{\gamma }$ with γ slightly less than 1/2.     

A Riemann–Roch theorem in tropical geometry

Recently, Baker and Norine have proven a Riemann–Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann–Roch theorem for divisors on (abstract) tropical curves.     

Quantum D-modules and equivariant Floer theory for free loop spaces

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry'. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work of Guest, we formulate the notion of ``abstract quantum D-module' which generalizes the D-module defined by the small quantum cohomology algebra. Second, we define the equivariant Floer cohomology of toric complete intersections rigorously as a D-module, using Givental's model. This is shown to satisfy the axioms of abstract quantum D-module. By Givental's mirror theorem [Giv3], it follows that equivariant Floer cohomology defined here is isomorphic to the quantum cohomology D-module.     

On Waring's problem for many forms and Grassmann defective varieties

This paper is devoted to the classical Waring type problem for several algebraic forms. Its geometric translation in terms of Grassmann defectivity for projective varieties yields the best possible solution whenever the number of polynomials is greater than the number of variables. In particular, together with a classical theorem of Alessandro Terracini, this result gives a complete answer to Waring's problem for several ternary forms.     

A topological central point theorem

In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.     

Secant varieties of toric varieties

Let X_P be a smooth projective toric variety of dimension n embedded in P^r using all of the lattice points of the polytope P. We compute the dimension and degree of the secant variety . We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties X_A embedded using a set of lattice points A⊂P∩Zⁿ containing the vertices of P and their nearest neighbors.     

On Waring’s problem for several algebraic forms

We reconsider the classical problem of representing a finite number of forms of degree d in the polynomial ring over n + 1 variables as scalar combinations of powers of linear forms. We define a geometric construct called a grove, which, in a number of cases, allows us to determine the dimension of the space of forms which can be so represented for a fixed number of summands. We also present two new examples, where this dimension turns out to be less than what a naïve parameter count would predict.     

Enumeration of one-nodal rational curves in projective spaces

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. The formula involves intersections of tautological classes on moduli spaces of stable rational maps. We combine the methods and results from three different papers.     

A unifying theory of a posteriori finite element error control

Summary Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm |||| of a linear functional of the formin the variable of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker of . As a consequence, any residual estimator that is a computable bound of |||| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.Supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.     

Residual intersection theory with reducible schemes

We develop a formula (Theorem 5.2) which allows to compute top Chern classes of vector bundles on the vanishing locus V(s) of a section of this bundle. This formula particularly applies in the case when V(s) is the union of local complete intersections giving the individual contribution of each component and their mutual intersections. We conclude with applications to the enumeration of rational curves in complete intersections in projective space.     

The local Gromov-Witten theory of curves

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a complete and effective solution.

The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of , and the orbifold quantum cohomology of the symmetric product of . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.