Enumerative Geometry Research Articles

Deformation of tropical Hirzebruch surfaces and enumerative geometry

We illustrate the use of tropical methods by generalizing a formula due to Abramovich and Bertram, extended later by Vakil. Namely, we exhibit relations between enumerative invariants of the Hirzebruch surfaces Σ n \Sigma _n and Σ n + 2 \Sigma _{n+2} , obtained by deforming the first surface to the latter. Our strategy involves a tropical counterpart of deformations of Hirzebruch surfaces and tropical enumerative geometry on a tropical surface in three-space.

Exact quantization conditions for cluster integrable systems

We propose exact quantization conditions for the quantum integrable systems of Goncharov and Kenyon, based on the enumerative geometry of the corresponding toric Calabi–Yau manifolds. Our conjecture builds upon recent results on the quantization of mirror curves, and generalizes a previous proposal for the quantization of the relativistic Toda lattice. We present explicit tests of our conjecture for the integrable systems associated to the resolved and orbifolds.

Membranes and sheaves

Our goal in this paper is to discuss a conjectural correspondence between the enumerative geometry of curves in Calabi–Yau 5-folds Z and 1-dimensional sheaves on 3-folds X that are embedded in Z as fixed points of certain C×-actions. In both cases, the enumerative information is taken in equivariant K-theory, where the equivariance is with respect to all automorphisms of the problem. In Donaldson–Thomas theory, one sums over all Euler characteristics with a weight (−q)χ, where q is a parameter, informally referred to as the boxcounting parameter. The main feature of the correspondence is that the 3-dimensional boxcounting parameter q becomes in dimension 5 the equivariant parameter for the C×-action that defines X inside Z. The 5-dimensional theory effectively sums up the q-expansion in the Donaldson–Thomas theory. In particular, it gives a natural explanation of the rationality (in q) of the DT partition functions. Other expected as well as unexpected symmetries of the DT counts follow naturally from the 5-dimensional perspective. These involve choosing different C×-actions on the same Z, and thus relating the same 5-dimensional theory to different DT problems. The important special case Z = X×C2 is considered in detail in Sections 7 and 8. If X is a toric Calabi–Yau 3-fold, we compute the theory in terms of a certain index vertex. We show that the refined vertex found combinatorially by Iqbal, Kozcaz, and Vafa is a special case of the index vertex. 1. A brief introduction 1.1 Overview Our goal in this paper is to discuss a conjectural correspondence between the enumerative geometry of curves in Calabi–Yau 5-folds Z and 1-dimensional sheaves on 3-folds X that are embedded in Z as fixed points of certain C×-actions. In both cases, the enumerative information is taken in equivariantK-theory, where the equivariance is with respect to all automorphisms of the problem. In Donaldson–Thomas theory, one sums over all Euler characteristics with weight (−q)χ, where q is a parameter. (Note the difference with the traditional weighing by qχ as in [MNOP06]. The change of sign of q fits much better with all correspondences.) Informally, q is referred to as the boxcounting parameter. The main feature of the correspondence is that the 3-dimensional boxcounting parameter q becomes in dimension 5 the equivariant parameter for the C×-action Received 15 April 2014, accepted in final form 9 October 2015. 2010 Mathematics Subject Classification 14N35.

Topological Strings from Quantum Mechanics

We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov–Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local $${{\mathbb{P}}^2}$$ , local $${{\mathbb{P}}^1 \times {\mathbb{P}}^1}$$ and local $${{\mathbb{F}}_1}$$ . In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi–Yau manifolds, in which the genus expansion emerges as a ’t Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background-independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry.

Intersections on tropical moduli spaces

This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a ``boundary'' divisor, and we prove general tropical versions of the WDVV, respectively, topological recursion equations (under some assumptions). As a direct application, we prove that, for the toric varieties $\PP ^1$, $\PP ^2$, $\PP ^1 \times \PP ^1$ and with $\Psi $-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\PP ^2$ in \cite {MR08}). Our approach uses tropical intersection theory and unifies and simplifies some parts of the existing tropical enumerative geometry (for rational curves).

Tropical Aspects in Geometry, Topology and Physics

The workshop Tropical Aspects in Geometry, Topology and Physics was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject. The development of tropical geometry is based on deep links between problems in real and complex enumerative geometry, symplectic geometry, quantum fields theory, mirror symmetry, dynamical systems and other research areas. On the other hand, new interesting phenomena discovered in the framework of tropical geometry (like refined tropical enumerative invariants) pose the problem of a conceptual understanding of these phenomena in the “classical” geometry and mathematical physics.

Gromov compactness in non-archimedean analytic geometry

Abstract Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

Littlewood–Richardson coefficients for reflection groups

In this paper we explicitly compute all Littlewood–Richardson coefficients for semisimple and Kac–Moody groups G, that is, the structure constants (also known as the Schubert structure constants) of the cohomology algebra H⁎(G/P,C), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood–Richardson coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of G. However, if some off-diagonal entries of the Cartan matrix are 0 or −1, the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies aijaji≥4 for all i, j, then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood–Richardson coefficients. We extend this and other results to the structure coefficients of the T-equivariant cohomology of flag varieties G/P and Bott–Samelson varieties Γi(G).

Degrees of Projections of Rank Loci

We provide formulas for the degrees of the projections of the locus of square matrices with given rank from linear spaces spanned by a choice of matrix entries. The motivation for these computations stems from applications to “matrix rigidity”; we also view them as an excellent source of examples to test methods in intersection theory, particularly computations of Segre classes. Our results are generally expressed in terms of intersection numbers in Grassmannians, which can be computed explicitly in many cases. We observe that, surprisingly (to us), these degrees appear to match the numbers of Kekulé structures of certain benzenoid hydrocarbons, and arise in many other contexts with no apparent direct connection to the enumerative geometry of rank conditions.

Enumerative Geometry, Tau-Functions and Heisenberg–Virasoro Algebra

In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich-Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the tau-functions in terms of matrix integrals, Virasoro constraints and Kac-Schwarz operators. All constructed operators belong to the algebra (or group) of symmetries of the KP hierarchy.

Modifications of Hodge bundles and enumerative geometry: the stable hyperelliptic locus

We study the stable hyperelliptic locus, i.e. the closure, in the Deligne–Mumford moduli of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated with a family of stable curves, we construct a bundle map (‘degree-2 Brill–Noether’) from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton–MacPherson residual intersection theory, the class of the structure sheaf and various other sheaves supported on the stable hyperelliptic locus can be computed by the Porteous formula and similar tools.

Real solutions to systems of polynomial equations and parameter continuation

Abstract Given a parameterized family of polynomial equations, a fundamental question is to determine upper and lower bounds on the number of real solutions a member of this family can have and, if possible, compute where the bounds are sharp. A computational approach to this problem was developed by Dietmaier in 1998 who used a local linearization procedure to move in the parameter space to change the number of real solutions. He used this approach to show that there exists a Stewart-Gough platform that attains the maximum of forty real assembly modes. Due to the necessary ill-conditioning near the discriminant locus, we propose replacing the local linearization near the discriminant locus with a homotopy-based method derived from the method of gradient descent arising in optimization. This new hybrid approach is then used to develop a new result in real enumerative geometry.

Asymptotics of unitary multimatrix models: The Schwinger–Dyson lattice and topological recursion

We prove the existence of a 1/N2 expansion in unitary multimatrix models which are Gibbs perturbations of the Haar measure, and express the expansion coefficients recursively in terms of the unique solution of a noncommutative initial value problem. The recursion obtained is closely related to the “topological recursion” which underlies the asymptotics of many random matrix ensembles and appears in diverse enumerative geometry problems. Our approach consists of two main ingredients: an asymptotic study of the Schwinger–Dyson lattice over noncommutative Laurent polynomials, and uniform control on the cumulants of Gibbs measures on product unitary groups. The required cumulant bounds are obtained by concentration of measure arguments and change of variables techniques.

Hilbert schemes and multiple q-zeta values

We present several conjectures on multiple q-zeta values and on the role which they play in certain problems of enumerative geometry.

Structure of node polynomials for curves on surfaces

We provide a structural generalization of a theorem by Kleiman–Piene, concerning the enumerative geometry of nodal curves in a complete linear system on a smooth projective surface S. Provided that r, the number of nodes, is sufficiently small compared to the ampleness of the linear system, we show that, under certain assumptions, the number of r‐nodal curves passing through points in general position on S is given by a Bell polynomial in universally defined integers which we identify, using classical intersection theory, as linear, integral polynomials evaluated in four basic Chern numbers. Furthermore, we provide a decomposition of the as a sum of three terms with distinct geometric interpretations, and discuss the relationship between these polynomials and Kazarian's Thom polynomials for multisingularities of maps.

Quantum curves for the enumeration of ribbon graphs and hypermaps

The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-commutative quantisation whose associated differential operator annihilates the partition function for the spectral curve. In this paper, we determine the quantum curves and partition functions for an infinite sequence of enumerative problems involving generalisations of ribbon graphs known as hypermaps. These results give rise to an explicit conjecture relating hypermap enumeration to the topological recursion and we provide evidence to support this conjecture.

Genus 0 characteristic numbers of the tropical projective plane

AbstractFinding the so-called characteristic numbers of the complex projective plane$ \mathbb{C} {P}^{2} $is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given$d$and$g$one has to find the number of degree$d$genus$g$curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is$3d- 1+ g$so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when$g= 0$. Namely, we show that the tropical problem is well posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of$ \mathbb{C} {P}^{2} $in terms of open Hurwitz numbers.

A new gluing recursive relation for linear Sigma model of ℙ1-orbifold

The study of the moduli space plays an important role in classical enumerative geometry and its interaction with string theory in physics. Given X = [ℙ1/ℤr] and let x′ = ([0]a, [∞]b) the 2-tuple of twisted sectors on X, we construct in this paper two different compactifications of the moduli space M0,2(X, d[ℙ1/ℤr], x′): Nonlinear Sigma Model Mdx′ and Linear Sigma Model Ndx′. Relations between Mdx′ and Ndx′ are studied and a new gluing recursive relation on Ndx′ is derived from Mdx′ due to virtual localization formula.

ON GROMOV–WITTEN INVARIANTS OF DEL PEZZO SURFACES

We compute Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhäuser, Boston, 1995), pp. 81–148; L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ2 and enumerative geometry, J. Differential Geom.48(1) (1998) 61–90], Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on rational surfaces, Manuscripta Math.102 (2000) 53–84]. We solve the problem in two steps: (1) we consider curves on [Formula: see text], the plane blown up at one point, which have given degree, genus, and prescribed multiplicities at fixed generic points on a conic that avoids the blown-up point; then we obtain a Caporaso–Harris type formula counting such curves subject to arbitrary additional tangency conditions with respect to the chosen conic; as a result we count curves of any given divisor class and genus on a surface of type [Formula: see text], the plane blown up at six points on a given conic and at one more point outside the conic; (2) in the next step, we express the Gromov–Witten invariants of [Formula: see text] via enumerative invariants of [Formula: see text], using Vakil's extension of the Abramovich–Bertram formula.

Tautological module and intersection theory on Hilbert schemes of nodal curves

This paper presents the rudiments of Hilbert-Mumford Intersection (HMI) theory: intersection theory on the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension. We introduce an additive group of geometric cycles, called ’tautological module’, generated by diagonal loci, node scrolls, and twists thereof. We determine recursively the intersection action on this group by the discriminant (big diagonal) divisor and all its powers. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry of families of nodal curves.