Joint seminar with BICMR (PKU) and CAS
(Contact: Chenyang Xu (BICMR), Yuguang Zhang (MSC), Eduard Looijenga (MSC))
NB: Because of the opening of the Conference Center at Sanya this week, the talk of Dec. 19 has been postponed to Dec 26. Place (BICMR) and time (13:30-15:05) are unchanged.
Aim of the seminar
A log Calabi-Yau variety is a nonsingular quasi-projective variety U (of dimension n, say) which admits a smooth completion Ū by means of a normal crossing divisor D with the property that -D is anticanonical. The focus will here be on the case when D contains a point where it has the maximal number (n) of local branches (this is in fact a property of U). There are plenty of examples of these, some of which are quite classical. During the past few years there has been some spectacular progress on mirror symmetry for such incomplete varieties. One of the surprising outcomes is that they come with a natural collection of functions, called by their discoverers-for good reasons- Theta functions. It is part of a vast program of which the principal actors are Mark Gross, Paul Hacking, Sean Keel and Bernd Siebert. The goal of the seminar is to become familiar with part of this project.
Meeting place
The seminar will meet on Thursdays during the time slot 13:30-15:05 and the location will alternate between the Mathematical Sciences Center at Tsinghua (Lecture Hall on 3rd floor) and the Beijing International Center for Mathematical Research at Peking University (Quanzai 29). The first meeting is on Sept. 26. Note that we do not meet on Oct. 3 (falls in a holiday week) and Oct. 17 (falls in the week of the East Asia Algebraic Geometry conference).
Preliminary program
This is to a large extent modeled after the program of a workshop recently held at MIT, which was designed by Mark Gross.
Sept 26 (MSC), lecture by Eduard Looijenga (MSC).
Introduction. Mostly be based on Keel's talk: Mirror Symmetry Made Easy, it will also make a brief mention of the SYZ conjecture.
[Sept 30-Oct 4 holiday week]
Oct 10 (BICMR), lecture by Jiang Qingyuan (MSC).
The key point of the Gross-Siebert program is the hope that starting with an integral affine manifold B with singularities and some additional data one may produce a toric degeneration whose (dual) intersection complex is B ([GS08], [CDM], §10, 11 and [TGMS], Chapter 6). We stick to the two-dimensional case, where the argument follows [KS] fairly closely. The Mumford degeneration will be covered in detail: see [GS08], §1 and [TGMS], §6.2.1. This includes the case when B is a lattice polytope.
[Oct 14-18 5th Algebraic Geometry conference in East Asia]
Oct 24 (BICMR) Jiang Qingyuan will need about 15 minutes to finish his talk of last time. After that:
45 min. lecture by Tatsuki Hayama (MSC).
The case for B general without singularities can now be dealt with ([TGMS], §6.2.2). This requires developing an understanding of various local systems living on B. This also makes connection to the construction of the Tate elliptic curve in the case when B = R/dZ is a circle ([TGMS], §6.2.2 as well as [Clay09], §8.4). If time permits, there can be some discussion as to what goes wrong in the case of singularities ([GS08], §2.3 and [TGMS], §6.2.3).
45 min. lecture by Xu Ze (BICMR).
Continuation with Givental's construction for Fano toric mirror symmetry with more details. [GHK23] is actually a generalization of that construction. §1.3 in [GHK23] is a good reference.
Oct 31 (MSC), lecture by Zhang Yuguang(MSC).
The case for B with singularities, but without scattering. Do the examples of [GS08], §3. Explain how modifying gluing maps via automorphisms attached to walls fixes the problems caused by monodromy. These corrections can also be explained in terms of Maslov index zero disks in the Fano context; this is a good point at which to make contact with Auroux’ work [A07].
Nov 7 (BICMR), lecture by Lu Wenxuan (MSC).
(The scattering process.) Start with the examples of [GS08], §4.1, 4.2. Discuss the Kontsevich-Soibelman lemma in dimension two: see [GPS], Theorem 1.4 or [TGMS] §6.3.1. Finish the outline of the algorithm for producing smoothings in dimension two, following the outline in [CDM], §10 and all details covered in [TGMS], Chapter 6.
Nov 14 (MSC), lecture by Zhou Jian (Tsinghua).
The enumerative interpretation of the Kontsevich-Soibelman lemma as given in [GPS]. The survey [GP09], as well as [CDM] §11 or [TGMS], §6.3.2 can be consulted. This gives evidence that the algorithm for producing smoothings morally is correcting the complex structure using Maslov index zero disks on the mirror side, fitting with the philosophy of [A07].
Nov 21 (BICMR), lecture by Xu Chenyang (BICMR).
Introduction to homological mirror symmetry [K]. Many sources can be used to prepare this material. Explain enough to be able to give the motivation for theta functions, as outlined in §1 of [GS11b].
Nov 28 (MSC), lecture by Xiang Maosong (BICMR).
Talk about the idea of how tropical trees can approximate holomorphic disks contributing to Floer homology between sections. See §8.4 of [Clay09] (NB: if input points coincide, then the moduli space of tropical trees yields not the correct dimension and some perturbation methods are necessary. This doesn’t arise when computing m2, so this is sufficient for our purposes) and §2 of [GS11b].
Dec 5 (BICMR), lecture by Zhang Yugang (MSC).
Explain the notion of a broken line ([GKS23] §2.3 (see also [GS11b], §3.2) and how it gives rise to a construction of theta functions, motivated by the tropical interpretation of Floer homology.
Dec 12 (MSC), lecture by Xu Chenyang (BICMR).
Continuation of [GHK23]. §5 of [GS11b] gives a little bit of exposition of the ideas in this paper. Begin by explaining how a pair (Y, D) of rational surface Y and where D is a cycle of rational curves making up a reduced anticanonical divisor gives rise to an integral affine manifold with singularities. Review scattering diagrams in this context and how they give rise to deformations of the punctured n-vertex. Describe the variant of jagged paths called broken lines and how consistency gives the existence of theta functions. Explain how these are used to extend the deformations of punctured vertex to deformations of the vertex.
Dec 26 (BICMR) (Note the new date!) , lecture by Eduard Looijenga (MSC).
Finish the discussion of [GHK23]. Explain the canonical scattering diagram, making contact with [GPS] and the Kontsevich-Soibelman lemma. This is §3. The basic idea is that using the existence of a toric model and the main result of [GPS], one converts the scattering diagram on the affine manifold associated to (Y,D) to one in R2 which is generated by the Kontsevich-Soibelman process.
References
[Clay03] C. Vafa et al., Mirror symmetry, Clay mathematics monographs vol. 1, 2003.
[Clay09] P. Aspinwall et al., Dirichlet branes and mirror symmetry, Clay mathematics monographs vol. 4, 2009.
[G08] M. Gross, The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations, Proceedings of Symposia
in Pure Mathematics, 2008, arXiv:0802.3407
[GS08] M. Gross and B. Siebert, Invitation to toric degenerations, arXiv:0808.2749
[GP09] M. Gross and R. Pandharipande, Quivers, Curves, and the tropical vertex, arXiv:0909.5153
[CDM] M. Gross, Mirror Symmetry and the Strominger-Yau-Zaslow conjecture, (survey paper for CDM), arXiv:1212.4220
[TGMS] M. Gross, Tropical Geometry and Mirror Symmetry, book, 2011. Can be downloaded here.
[SYZ] A. Strominger, S.-T. Yau, E. Zaslow, Mirror symmetry is T-duality, hep-th/9606040
[PZ] A. Polishchuk and E. Zaslow, Categorical mirror symmetry for the elliptic curve, math/9801119
[K] M. Kontsevich, 1994 ICM address, alg-geom/9411018
[A07] D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor, arXiv:0706.3207
[G99] M. Gross, Topological Mirror Symmetry, math/9909015
[GHK] M. Gross, P. Hacking, S. Keel, Mirror symmetry for log Calabi-Yau surfaces I, arXiv:1106.4977 or rather a new
version of it,
[GHK23] which can be downloaded here.
[GPS] M. Gross, R. Pandharipande, B. Siebert, The tropical vertex, Duke Math. J, 2011, arXiv:0902.0779
[GS03] M. Gross and B. Siebert, Affine manifolds, log structures and mirror symmetry, math/0211094
[GS03b] M. Gross and B. Siebert, Mirror Symmetry via Logarithmic Degeneration Data I, math/0309070
[GS07] M. Gross and B. Siebert, From affine geometry to complex geometry, Ann. of Math. (2011), 1301-1428,
math/0703822
[GS07b] M. Gross and B. Siebert, Mirror Symmetry via Logarithmic Degeneration Data II, arXiv:0709.2290
[GS11b] M. Gross and B. Siebert, Theta functions and mirror symmetry, arXiv:1204.1991
[GS11c] M. Gross and B. Siebert, Logarithmic Gromov-Witten invariants, arXiv:1102.4322
[KS] M. Kontsevich, and Y. Soibelman: Affine structures and non-archimedean analytic spaces, math/0406564
[NS04] T. Nishinou and B. Siebert, Toric degenerations of toric varieties and tropical curves, math/0409060
Recommended is the taped lecture by Sean Keel: Mirror symmetry made easy