Introduction to the Gross-Siebert program
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May 7
Introduction to the Gross-Siebert program (Ruddat)
We introduce the most versatile mirror construction known to date. The strength of this program is that it incorporates various points of view simultaneously.
Every Batyrev-Borisov mirror pair fits in, so there are numerous examples. The construction centers at a maximal degeneration of the geometry which is in line with mathematical physics ("large complex structure limit"). Furthermore, the Strominger-Yau-Zaslow conjecture is naturally in line with this framework. It gives an intrinsic construction of mirror pairs as dual torus fibrations over a real affine manifold. We explain these concepts and give examples.
Log geometry and counts of stable log maps (Garrel)
We introduce some of the aspects of log geometry that are crucial to the Gross-Siebert program. To aid familiarity, the focus will be on examples. We will end with a theorem by F. Kato classifying the local structure of log smooth curves.
Toric degenerations, affine manifolds (Overholser)
The toric degeneration is at the heart of the Gross-Siebert program, providing the link between algebraic and affine geometry. I will give its definition, some examples, and show how such a degeneration leads to two related affine structures.
May 8
Scattering and reconstruction of Calabi-Yau manifolds from degeneration data (Ruddat)
Starting with combinatorial degeneration data, Gross and Siebert found a canonical formal smoothing of the associated degenerate Calabi-Yau space. This provides a core step in their mirror construction. We explain the basics of this procedure which consists of gluing infinitesimal finite order smoothings by an algorithm that is governed by tropical geometry and so called scattering diagrams.
Tropical geometry and curve counting (Overholser)
Tropical geometry has become a popular tool of enumerative geometry. I will give some background that helps to explain its relevance to this area, a selection of known results, and an introduction to the tropical structures that will be needed in later talks.
Curve counting and a theorem by Nishinou-Siebert (Garrel)
Mikhalkin in 2005 proved a groundbreaking result expressing counts of rational curves in the complex projective plane as tropical curve counts in the real plane. A flurry of activity aiming at proving similar results for more general geometries has since ensued. For the Gross-Siebert program, the most important such result is a theorem by Nishinou-Siebert, which we introduce in this lecture.
May 9
Scattering, tropical geometry, and mirror symmetry for P2 (Overholser)
Gross used tropical geometry to reinterpret Barannikov's mirror symmetry for P2, discovering that the mirror map becomes very straightforward in this language. A Landau-Ginzburg potential can be defined using tropical analogues of holomorphic disks; period integrals glue these disks into tropical curves whose related invariants govern the quantum cohomology of P2. This construction features wall-crossing, scattering diagrams, and broken lines.
Periods, tropical cycles and canonical coordinates (Ruddat)
In joint work with Siebert, I compute certain period integrals for these families and show that the mirror map is trivial. In other words, the canonical coordinate of Gross-Siebert is a canonical coordinate in the sense of Hodge theory as defined by Morrison. As a consequence, the formal families lift to analytic families. We introduce tropical 1-cycles that we turn in to ordinary n-cycles whose periods have a simple log pole. We show that such cycles generate the dual of the tangent space to the CY moduli space.
The tropical vertex (Garrel)
This talk introduces work by Gross-Pandharipande-Siebert. We will start by introducing some relative Gromov-Witten invariants of some toric surfaces. Surprisingly, these invariants are calculated by ordered product factorizations in the tropical vertex group, which we describe.
May 10
Directions from tropical mirror symmetry for P2 (Overholser)
I will present a recent generalization of Gross’s mirror symmetry for P2 that yields new tropical interpretations of certain descendent Gromov-Witten invariants. If time permits, I will also give a short introduction to the work of Gross, Hacking, and Keel on log Calabi-Yau surfaces. Both depend on generalizations of the scattering structures of my previous talk.
Relative BPS state counts (Garrel)
The tropical vertex paper explains how to extract BPS state counts from relative Gromov-Witten invariants. I will describe how in the smooth divisor case these invariants are related to local BPS state counts via generalized Donaldson-Thomas invariants of loop quivers. This is joint work with T. Wong and G. Zaimi.
Conifold transitions and a proof of Morrison's conjecture (Ruddat)
Morrison conjectured that mirror symmetry dualizes conifold transitions of Calabi-Yau threefolds. Since mirror symmetry is a phenomenon at a maximally unipotent boundary point of the Calabi-Yau moduli space, in order to prove the conjecture, one needs a theory combining conifold transitions with maximal degenerations. I will report on joint work with Siebert where we produce such a theory by giving a comprehensive account on conifold transitions in the Gross-Siebert program. We exhibit tropical homology groups that control the obstructions à la Friedman-Tian and Smith-Thomas-Yau and that are naturally identified via discrete Legendre transform alias mirror symmetry. This proves Morrison's conjecture.