# Workshop on Moduli spaces of curves and mirror symmetry

March 19-23, 2018

## Week schedule

### Monday

9:30-10:30 Sara Angela Filippini

Title: Stability data, irregular connections and tropical curves

Abstract: I will outline the construction of isomonodromic families of irregular meromorphic connections on P^1 with values in the derivations of a class of infinite-dimensional Poisson algebras, and describe two of their scaling limits. In the "conformal limit" we recover a version of the connections introduced by Bridgeland and Toledano-Laredo, while in the "large complex structure limit" the connections relate to tropical curves in the plane and, through work of Gross, Pandharipande and Siebert, to tropical/GW invariants. This is joint work with M. Garcia-Fernandez and J. Stoppa.

11:00-12:00 Andreas Gross

Title: Tropical Jacobians, Theta Divisors, and the Poincaré Formula

Abstract: Poincaré’s formula relates the homology classes of the loci of effective divisors on the Jacobian of a curve with powers of the homology class of the theta divisor. In the tropical world, there are analogous versions of Jacobians, theta divisors, and homology, so it is possible to state an analogous formula using these tropical objects. In my talk, I will present joint work with Farbod Shokrieh in which we prove that this tropical Poincaré formula does indeed hold.

14:00-15:00 Walter Gubler

Title: Lagerberg's superforms and non-archimedean Arakelov theory

Abstract: We explain how Lagerberg's superforms were used by Chambert-Loir and Ducros to define forms and currents on Berkovich spaces. Then we will explain tropical intersection theory can be used to enrich the forms with tropical cycles to get applications in non-archimedean Arakelov theory.

This is joint work with Klaus Künnemann plus some recent progress.

15:30-16:30 Lionel Lang

Title: Monodromy groups of lattice polygons (joint with J.Forsgård, B.Shapiro)

Abstract: Any lattice polygon $\Delta$ together with a distinguish interior lattice point (a,b) allow to define a fibration of $(\C^*)^2$ over $\C$ given by $(z,w) \to f(z,w)/(z^aw^b)$ where f is a Laurent polynomial of degree $\Delta$. The monodromy action of this fibration distinguishes a subgroup $G_{\Delta, (a,b)}$ of the group of diffeomorphisms of a generic fiber. In some cases, we can explicitly compute this subgroup by looking at the coamoeba of the fiber. Moreover, this description does not depend on the lattice point (a,b). Unfortunately, this only works when the coamoeba is modeled on a minimal bipartite graph, and it is not know in general if such coamoebas exist. Fortunately, minimal bipartite graphs always exist and allows us to construct a combinatorial version $G_{\Delta}$ of the above monodromy group that depend only on $\Delta$.

In this talk, we will describe the interaction between the latter objects via concrete examples. We will also address the following questions: are the groups $G_{\Delta, (a,b)}$ independent of (a,b) in general? If yes, do they coincide with $G_{\Delta}$?

### Tuesday

9:30-10:30 Sam Payne

11:00-11:40 Andrea Petracci, Part 1

Title: Mirror Symmetry and Fano varieties

Abstract: Fano varieties are the basic building blocks of algebraic varieties. It is known that in each dimension there are only finitely many deformation families of smooth Fano varieties, which have been classified up to dimension 3. Already in dimension 4 the number of smooth Fano variety is completely unknown. Recent work of Coates, Corti, Galkin, Golyshev, and Kasprzyk aims to use Mirror Symmetry to give some insight on this problem. In these two talks I will give an overview to this research area by introducing the tools involved, such as Fano polytopes, mutations, Laurent polynomials, Gromov-Witten invariants, toric deformation/degenerations. I will also present some new results, some of which are due to ongoing collaboration with Alessio Corti and Paul Hacking.

11:50-12:30 Andrea Petracci, Part 2

14:00-15:00 Karin Schaller

Title: Stringy invariants and combinatorial identities

Abstract: We give a combinatorial interpretation of the stringy Libgober-Wood identity in terms of generalized stringy Hodge numbers and intersection products of stringy Chern classes for arbitrary projective Q-Gorenstein toric varieties. Simultaneously we introduce stringy E-functions and stringy Chern classes in general for projective varieties with at worst log-terminal singularities.

As a first application we derive a novel combinatorial identity relating arbitrary-dimensional reflexive polytopes to the number 24. Followed by another generalization extending the well-known formula for reflexive polygons including the number 12 to LDP-polygons and toric log del Pezzo surfaces, respectively. Our third application is motivated by computations of stringy invariants of non-degenerated affine hypersurfaces in 3-dimensional algebraic tori whose minimal models are K3-surfaces, giving rise to a combinatorial identity for the Euler number 24. Using combinatorial interpretations of the stringy E-function and the stringy Libgober-Wood identity, we show with purely combinatorial methods that this identity holds for any 3-dimensional lattice polytope containing exactly one interior lattice point. This talk is based on joint work with Victor Batyrev.

15:30-16:30 Arthur Renaudineau

Title: Bounding Betti numbers of real hypersurfaces near the tropical limit

Abstract: In general, bounding Betti numbers of real algebraic varieties is a very difficult question. In this talk I will explain our proof of a conjecture of Itenberg which tackle this question for real algebraic projective hypersurfaces near the (non-singular) tropical limit. Those hypersurfaces are the one coming from Viro's combinatorial patchworking method.

To prove the bounds conjectured by Itenberg we develop a real analogue of tropical homology and use a spectral sequence to compare it to the usual tropical homology of Itenberg, Katzarkov, Mikhalkin, Zharkov.

This is joint work in progress with Kristin Shaw.

### Wednesday

9:30-10:30 Grisha Mikhalkin

Title: Maximally writhed real algebraic knots and links

Abstract: The Alexander-Briggs tabulation of knots in R^3 (started almost a century ago, and considered as one of the most traditional ones in classical Knot Theory) is based on the minimal number of crossings for a knot diagram. From the point of view of Real Algebraic Geometry it is more natural to consider knots in RP^3 rather than R^3, and use a different number also serving as a measure of complexity of a knot: the minimal degree of a real algebraic curve representing this knot.

As it was noticed by Oleg Viro about 20 years ago, the writhe of a knot diagram becomes an invariant of a knot in the real algebraic set-up, and corresponds to a Vassiliev invariant of degree 1. In the talk we’ll survey these notions, and consider the knots with the maximal possible writhe for its degree. Surprisingly, it turns out that there is a unique maximally writhed knot in RP^3 for every degree d. Furthermore, this real algebraic knot type has a number of characteristic properties, from the minimal number of diagram crossing points (equal to d(d-3)/2) to the minimal number of transverse intersections with a plane (equal to d-2). Based on a series of joint works with Stepan Orevkov.

11:00-12:00 Hannah Markwig

Title: Tropical mirror symmetry for elliptic curves and beyond

Abstract: Mirror symmetry for elliptic curves relates the generating series of Hurwitz numbers of the elliptic curve (i.e. counts of covers with fixed genus and simple branch points) to Feynman integrals. When passing to the tropical world, the relation works on a "fine level", i.e. summand by summand. We review the known results and then talk about generalizations: we can count covers in a broader context, and we can also count tropical curves in a product of an elliptic curve with the projective line. In both cases, the generating series in question can be related to Feynman integrals summand by summand.

Joint work with Boehm, Bringmann, Buchholz, resp. with Boehm, Goldner.

14:00-15:00 Kristin Shaw

Title: The separating semigroup of a real curve

Abstract: This talk is based on joint work with Mario Kummer where we introduce the separating semigroup of a real algebraic curve of dividing type. The elements of this semigroup record the possible degrees of the covering maps obtained by restricting separating morphisms to the real part of the curve. We also introduce the hyperbolic semigroup which consists of elements of the separating semigroup arising from morphisms which are compositions of a linear projection with an embedding of the curve to some projective space. We completely determine both semigroups in the case of maximal curves. We also prove that any embedding of a real curve to projective space of sufficiently high degree is hyperbolic. Using these semigroups we show that the hyperbolicity locus of an embedded curve is in general not connected. It turns out that this example is a case of the maximally writhed real algebraic knots of Mikhalkin and Orevkov.

15:30-16:30 Philipp Jell

Title: Lefschetz (1,1)-theorem in tropical geometry

Abstract: For a smooth projective complex variety X, the classical Lefschetz (1,1)-theorem characterizes the (1,1)-cohomology classes of X which are Chern classes of line bundles. In joint work with Johannes Rau and Kristin Shaw, we prove a tropical analogue of this. In this talk I will introduce the necessary notions to state the tropical Lefschetz (1,1)-theorem and discuss some applications.

### Thursday

9:30-10:30 Erwan Brugallé

Title: On Betti numbers of tropical varieties

11:00-11:40 Grisha Mikhalkin

Title: An introductory talk on tropical curves and their moduli spaces

11:50-12:30 Grisha Mikhalkin part 2

14:00-15:00 Josephine Yu

Title: Some Topics in Real Tropical Geometry

Abstract: I will discuss two different topics in the intersection of real algebraic geometry and tropical geometry: (1) Berkovich spaces and (2) stable varieties. In a joint work with Claus Scheiderer, we show the real analogue of Payne's result that the analytification is the limit of tropicalizations and identify the real locus in the Berkovich space. In a different joint work with Felipe Rincon and Cynthia Vinzant, we study real stable varieties and show that their tropicalizations are subfans of the type A braid arrangement, generalizing a result of Brändén in the case of homogeneous multi-affine stable polynomials.

15:30-16:30 Johannes Rau

Title: Lower bounds and asymptotics of real double Hurwitz numbers

Abstract: Complex and real double Hurwitz numbers can be computed in terms of certain tropical graph counts. Since the real version of these numbers typically does depend on the explicit choice of branch points, it is of interest to prove lower bounds or asymptotic statements about real Hurwitz numbers which are independent of this choice. We present such results using the tropical approach, in particular, about the equality of the rate of growth of complex vs. real numbers on the logarithmic scale. These results are achieved without the explicit construction of an invariant signed count.

### Friday

9:30-10:30 Ilya Tyomkin

Title: Algebraic-tropical correspondence for rational curves

Abstract: In my talk I'll discuss the problem of enumeration of rational curves with marked points in toric varieties that satisfy toric and multiple cross-ratio constraints. I'll focus on the tropical approach to the problem and will present an algebraic-tropical correspondence theorem that reduces the problem to a combinatorial one. If time permits, I'll briefly explain the proof, which is surprisingly short, involves only the standard theory of schemes, and works in arbitrary characteristic (in small characteristics the algebraic curves should be counted with appropriate multiplicities).

11:00-12:00 Eugenii Shustin

Title: Geometry of the tropical cuspidal edge

Abstract: The family of (uni)cuspidal curves forms a divisor (cuspidal edge) of the Severy variety parameterizing plane curves of given degree and genus.

We study the tropical cuspidal edge that parametrizes plane marked cuspidal tropical curves of given degree and genus: we describe cells, which parametrize curves passing through the maximal possible number of generic points in the plane and consider arrangements of such cells sharing a common codimension one face.