arXiv subjects

Mark Gross

Publications and source records attributed to Mark Gross.

At least 37 records · Page 2Linked to original sources

arXiv2016

Towards Mirror Symmetry for Varieties of General Type

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf. This exchange can be anticipated from an analysis of Hochschild homology of the relevant categories arising in homological mirror symmetry in this case; we also conjecture that a similar, but different, exchange of dimensions arises from Hochschild cohomology, relating the cohomology of sheaves of polyvector fields on the hypersurface to the cohomology of the critical locus.

math.AG↗

arXiv2015

The greedy basis equals the theta basis

We prove the equality of two canonical bases of a rank 2 cluster algebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich.

math.QA↗

arXiv2015

Mirror symmetry for log Calabi-Yau surfaces I

We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.

math.AG↗

arXiv2014

Moduli of surfaces with an anti-canonical cycle

We prove a global Torelli theorem for pairs (Y,D), where Y is a smooth projective rational surface and D is an effective anti-canonical divisor which is a cycle of rational curves. This Torelli theorem was conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.

math.AG↗

arXiv2014

Birational geometry of cluster algebras

We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer's example of an upper cluster algebra which is not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.

math.AG↗

arXiv2014

Local mirror symmetry in the tropics

We discuss how the Gross-Siebert reconstruction theorem applies to the local mirror symmetry of Chiang, Klemm, Yau and Zaslow. The reconstruction theorem associates to certain combinatorial data a degeneration of (log) Calabi-Yau varieties. While in this case most of the subtleties of the construction are absent, an important normalization condition already introduces rich geometry. This condition guarantees the parameters of the construction are canonical coordinates in the sense of mirror symmetry. The normalization condition is also related to a count of holomorphic disks and cylinders, as conjectured in our work and partially proved in various works of Chan, Cho, Lau, Leung and Tseng. We sketch a possible alternative proof of these counts via logarithmic Gromov-Witten theory. There is also a surprisingly simple interpretation via rooted trees marked by monomials, which points to an underlying rich algebraic structure both in the relevant period integrals and the counting of holomorphic disks.

math.AG↗

arXiv2013

Mirror Symmetry and the Strominger-Yau-Zaslow conjecture

This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. We begin with the original differential geometric conjecture and its refinements, and explain how it led to the algebro-geometric program developed by myself and Siebert. After explaining the overall philosophy, I explain how recent results fit into this program.

math.AG↗

arXiv2013

Gromov-Hausdorff collapsing of Calabi-Yau manifolds

This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the base manifold. Secondly, if the fibered Calabi-Yau manifolds are Lagrangian fibrations of holomorphic symplectic manifolds, the metrics on the regular parts of the limits are special Kähler metrics. By combining these two results, we extend arXiv:math/0008018 to any fibered projective K3 surface without any assumption on the type of singular fibers.

math.DG↗

arXiv2012

Logarithmic Gromov-Witten invariants

The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li, and completes a program first proposed by the second named author in 2002. One considers target spaces X carrying a log structure. Domains of stable log curves are log smooth curves. Algebraicity of the stack of such stable log maps is proven, subject only to the hypothesis that the log structure on X is fine, saturated, and Zariski. A notion of basic stable log map is introduced; all stable log maps are pull-backs of basic stable log maps via base-change. With certain additional hypotheses, the stack of basic stable log maps is proven to be proper. Under these hypotheses and the additional hypothesis that X is log smooth, one obtains a theory of log Gromov-Witten invariants.

math.AG↗

arXiv2012

From real affine geometry to complex geometry

We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This paper is the key step of the program we initiated in math.AG/0309070.

math.AG↗

arXiv2012

Collapsing of abelian fibred Calabi-Yau manifolds

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit. In this setting we prove an analog of a result of Gross-Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.

math.DG↗

arXiv2012

Theta functions and mirror symmetry

This is a survey covering aspects of varied work of the authors with Mohammed Abouzaid, Paul Hacking, and Sean Keel. While theta functions are traditionally canonical sections of ample line bundles on abelian varieties, we motivate, using mirror symmetry, the idea that theta functions exist in much greater generality. This suggestion originates with the work of the late Andrei Tyurin. We outline how to construct theta functions on the degenerations of varieties constructed in previous work of the authors, and then explain applications of this construction to homological mirror symmetry and constructions of broad classes of mirror varieties.

math.AG↗

arXiv2009

Mirror Symmetry via Logarithmic Degeneration Data II

This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B, and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry, and gives an explicit description of the monodromy of a smoothing.

math.AG↗

arXiv2009

Mirror symmetry for P^2 and tropical geometry

This paper explores the relationship between mirror symmetry for P^2, at the level of big quantum cohomology, and tropical geometry. The mirror of P^2 is typically taken to be ((C^*)^2,W), where W is a Landau-Ginzburg potential of the form x+y+1/xy. The complex moduli space of the mirror is the universal unfolding of W, and oscillatory integrals produce a Frobenius manifold structure on this universal unfolding. We show that W can be deformed by counting Maslov index two tropical disks, and the natural paramaters appearing in this deformation are then the flat coordinates on the moduli space. Furthermore, the oscillatory integrals are shown to read off directly tropical curve counts from the potential. Thus we show in fact that mirror symmetry for P^2 is equivalent in a strong sense to tropical curve counting formulas, including tropical formulas for gravitational descendent invariants.

math.AG↗

arXiv2009

The tropical vertex

Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory.

math.AG↗

arXiv2009

Calabi-Yau Three-folds and Moduli of Abelian Surfaces II

This is the sequel to arXiv:math/0001089. In this paper, we complete the promised description of moduli of abelian surfaces of low degree, covering the cases of degree (1,12), (1,14), (1,16), (1,18) and (1,20). In each case, we describe birational models for the moduli space of abelian surfaces with polarization of the given degree and either a level structure, or what we call a partial level structure. In every case, the moduli space is rational or unirational. In additional, several new Calabi-Yau threefolds fibred in abelian surfaces are described.

math.AG↗

arXiv2009

An invitation to toric degenerations

This is an expository paper which explores the ideas of the authors' paper "From Affine Geometry to Complex Geometry", arXiv:0709.2290. We explain the basic ideas of the latter paper by going through a large number of concrete, increasingly complicated examples.

math.AG↗

arXiv2008

The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations

This survey article begins with a discussion of the original form of the Strominger-Yau-Zaslow conjecture, surveys the state of knowledge concering this conjecture, and explains how thinking about this conjecture naturally leads to the program initiated by the author and Bernd Siebert to study mirror symmetry via degenerations of Calabi-Yau manifolds and log structures.

math.AG↗