Titles & Abstracts

• ::: Hiroshi Iritani ::: Equivariant quantum cohomology and its applications
  The Seidel (or shift) operators endow equivariant quantum cohomology (D-module) with the structure of a difference module with respect to the equivariant parameters. I will explain an application of this structure to the quantum cohomology of projective bundles. This is based on joint works with Fumihiko Sanda and Yuki Koto.

• ::: Atsushi Kanazawa ::: Mirror symmetry for generalized K3 surfaces slides
  Hitchin’s invention of generalized Calabi-Yau structures is a key to unify the Calabi-Yau geometry (complex geometry of Calabi-Yau manifolds) and symplectic geometry. Such structures have been extensively studied in 2-dimensions by Huybrechts. Based upon his fundamental work, we will provide a formulation of mirror symmetry for generalized K3 surfaces by introducing lattice polarization on the Mukai lattice. As a byproduct, we will obtain the notion of rigid Kahler structures, and solve the problem of mirror symmetry for singular K3 surfaces in a natural way.

• ::: Chin-Lung Wang ::: A blowup formula in QH
  We study analytic continuations of quantum cohomology $QH(Y)$ under a blowup $\phi: Y \to X$ of complex projective manifolds along the extremal ray quantum variable $q^{\ell}$. Under the decomposition $H(Y) = \phi^* H(X) \oplus K$ where $K = \ker \phi_*$, we show that (i) the restriction of quantum product along the $\phi^*H(X)$ direction, denoted by $QH(Y)_X$, is meromorphic in $x := 1/q^\ell$, (ii) $K$ deforms uniquely to a quantum ideal $\widetilde K$ in $QH(Y)_X$. Moreover, the quotient ring $QH(Y)_X/\widetilde K$ is regular over $x$, and its restriction to the infinity divisor $x = 0$ leads to a natural isomorphism with $QH(X)$. This is a joint work with Y.-P. Lee and H.-W. Lin.

• ::: Tatsuki Kuwagaki ::: Fukaya categories of exact symplectic manifolds over the Novikov ring via sheaf theory
  Studying Fukaya categories of exact symplectic manifolds through sheaf theory is now an active topic. In particular, Ganatra—Pardon—Shende theorem states that Nadler—Shende’s microlocal category of a Weinstein manifold is equivalent to its wrapped Fukaya category. In a joint work with Yuichi Ike, we are studying the Novikov ring version of Nadler—Shende's construction and its consequences. In this talk, I’d like to explain this work and necessary backgrounds.

• ::: Emanuel Scheidegger ::: On genus one fibered Calabi-Yau threefolds with 5-sections and 6-sections slides
  Genus one fibered Calabi-Yau threefolds play an important role from various points of view in string theory and algebraic geometry. In this talk we will present a class of such threefolds with 5-sections that arise as homologically projective dual pairs. Their topological string partition functions show an interesting modular behaviour. We will extend this analysis to examples of such threefolds with 6-sections. The latter have much in common with the Calabi-Yau threefold studied by Hosono and Takagi.

• ::: Tsung-Ju Lee ::: On the solutions to GKZ D-modules with fractional parameters
  Given an integral matrix A and a parameter \beta, one can construct a D-module, called a GKZ D-module, on a certain affine space. GKZ D-modules appear in mirror symmetry as they govern the periods of Calabi--Yau complete intersections in toric varieties. In which case, the parameter \beta is an integral vector. Motivated by the recent work of Hosono, Lian, Takagi, and Yau in the study of mirror symmetry for singular Calabi--Yau varieties, we are interested in GKZ D-modules with a fractional parameter \beta. In this talk, I will give a cohomological description of the solution space to such a GKZ D-module by computing the direct image of an exponentially twisted de Rham complex when \beta is fractional and semi-nonresonant. This is joint work with Dingxin Zhang.

• ::: Fumihiko Sanda ::: Mirror symmetry of Fano manifolds via toric degenerations
  Let X be a Fano manifold and L be a monotone Lagrangian in X. Then (a chart of) a Landau-Ginzburg mirror of X is a Laurent polynomial f which is computed by counting holomorphic disks bounded by L. Suppose that X admits a toric degeneration to a normal toric Fano variety X′. In this talk, I will explain that the Newton polytope of f is equal to the fan polytope of X.

• ::: Atsushi Takahashi ::: On generalized root systems of type D
  Motivated by the geometry of Milnor fiber and vanishing cycles, Kyoji Saito introduced the notion of generalized root systems. A generalized root system consists of a lattice,a set of roots and an element of the Weyl group (called the Coxeter element). Due to the choice of Coxeter element, the notion is finer than the usual one. Indeed, there are several "finite generalized root systems of type D". The question was how to construct these geometrically and whether there exist "natural" Frobenius manifolds for them. To a Laurent polynomial, we associate a finite generalized root system of type D and a "natural" Frobenius manifold whose Frobenius potential is a rational function, which support the Dubrovin's conjecture on algebraic Frobenius manifolds. This is a joint work in progress with Akishi Ikeda, Takumi Otani and Yuuki Shiraishi.

• ::: Charles Doran ::: The Mirror Clemens-Schmid Sequence
  I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a "mirror P=W" conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface and Calabi-Yau threefold settings. This is joint work with Alan Thompson (arXiv:2109.04849).

• ::: Bong Lian ::: Periods of singular cycle covers slides
  We will consider a class of (typically) singular Calabi-Yau varieties given by cyclic branched covers of a fixed semi Fano manifold. The first prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of P^1 branched over 4 points. Two-fold covers of P^2 branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli spaces and their comparisons. I will outline a higher dimensional generalization from the viewpoint of mirror symmetry, and discuss the Riemann-Hilbert problem for periods of these singular varieties. We will introduce a new compactification of the moduli space cyclic covers, using the idea of ‘abelian gauge fixing’ and ‘fractional complete intersections’. This produces a moduli problem that is amenable to tools in toric geometry, particularly those that we have developed jointly in the mid-90’s with S. Hosono and S.-T. Yau in our study of toric Calabi-Yau complete intersections. In dimension 2, this construction gives rise to new and interesting identities of modular forms and mirror maps associated to certain K3 surfaces. We also present an essentially complete mirror theory in dimension 3, and discuss generalization to higher dimensions. The lecture is based on on-going joint work with S. Hosono, T.-J. Lee, H. Takagi, S.-T. Yau.

• Dialogue with Professor Shing-Tung Yau

Topics include the Calabi conjecture, mirror symmetry and possible future research directions.

• ::: Conan Leung ::: Constructing SYZ mirror via Maurer-Cartan equations slides
  In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture which relates the deformation theory of the mirror X with a multi-valued Morse theory on the base of a SYZ fibration, corresponding to the theory of pseudo-holomorphic curves on a Calabi-Yau manifold. In this talk, I will present a joint work with Kwokwai Chan and Ziming Ma in which we prove a reformulation of this correspondence where multi-valued Morse theory on the base is replaced by tropical geometry on the Legendre dual. In the proof, we apply techniques of asymptotic to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part Xsf⊆X allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.

• ::: Adrian Clingher ::: On Two-Elementary K3 Surfaces slides
  This talk will discuss special families of complex algebraic K3 surfaces with generic Picard lattice of two-elementary type. We shall look at geometric features, moduli spaces, as well as possible applications to string dualities. This is joint work with A. Malmendier.

• ::: Sergey Galkin ::: Graph potentials TQFTs and mirror symmetry for character varieties slides
  I will describe the progress on mirror symmetry between SU(2) character varieties and graph potentials, and applications towards TQFTs of Donaldson-(Atiyah?)-Floer type. The talk is based on my joint works with Grigory Mikhalkin (2203.10043), and with Pieter Belmans and Swarnava Mukhopadhyay (2009.05568v3, 2205.07244, 2206.11584, …), which in turn are based on my Tokyo period joint works with Alexander Usnich (IPMU 10-0100), Alexey Bondal (11-0101) and John Alexander Cruz Morales (12-0110).

• ::: Hiromichi Takagi ::: Key varieties for prime Q-Fano 3-folds of codimension 4 slides
  An affine variety K with good properties (many symmetries, good singularities, etc) are called (informally) a key variety for a Q-Fano 3-fold X if a weighted projectivization of K produces X as its weighted complete intersection. In this talk, I will explain how to construct key varieties for prime Q-Fano 3-folds of codimension 4 whose numerical data are presented in the online database http://www.grdb.co.uk/forms/fano3. Especially, I will empharsize that three different classes of key varieties are constructed by three types of degenerations of 9-dimensional Jordan algebras of cubic forms.

• ::: Mark Gross ::: Open FJRW theory notes
  I will describe joint work with Tyler Kelly and Ran Tessler. FJRW (Fan-Jarvis-Ruan-Witten) theory is an enumerative theory of quasi-homogeneous singularities, or alternatively, of Landau-Ginzburg models. It associates to a potential W:C^n -> C given by a quasi-homogeneous polynomial moduli spaces of (orbi-)curves of some genus and marked points along with some extra structure, and these moduli spaces carry virtual fundamental classes as constructed by Fan-Jarvis-Ruan. Here we specialize to the case W=x^r+y^s and construct an analogous enumerative theory for disks. We show that these open invariants provide perturbations of the potential W in such a way that mirror symmetry becomes manifest. Further, these invariants are dependent on certain choices of boundary conditions, but satisfy a beautiful wall-crossing formalism.

• ::: Keiji Oguiso ::: Real forms of smooth projective surfaces of Kodaira dimension zero
  Real form problem asks how many different ways one can describe a given complex projective variety by a system of equations with real coefficients, up to isomorphisms over the real number field. In this talk, based on existing results, we first recall the finiteness of real forms of a minimal smooth projective surface of Kodaira dimension zero. Then, we show that there are a one point blow up of some K3 surface and a one point blow up of some Enriques surface, admitting infinitely many real forms, while smooth projective surfaces birational to an abelian surface and a hyperelliptic surface always have at most finitely many real forms. This answers questions by Professors Mukai and Kondo to us. If time allows, we would like to discuss some expected relations with finite generation of the discrete part of the automorphism group and cone conjecture. This is a joint work in progress with Professors Tien-Cuong Dinh and Xun Yu.

• ::: Shinobu Hosono ::: Global aspects of mirror symmetry of Calabi-Yau manifolds
  It is almost 30 years since the discovery of mirror symmetry of Calabi-Yau manifolds. Starting with quintic hypersurfaces in $\mathbb{P}^4$, we have now many interesting Calabi-Yau manifolds for which we can describe the symmetry explicitly. Grassmannian and Pfaffian Calabi-Yau manifolds, and also Reye congruence and double quintic symmetroid, are among such interesting Calabi-Yau manifolds. In this talk, I will describe mirror symmetry of a Calabi-Yau manifold fibered by abelian surfaces with (1,8) polarization. Describing its mirror family globally, we will find that many interesting aspects of mirror symmetry are encoded at the boundary points of the parameter space of the family. This is based on a work with Hiromichi Takagi, arXiv:2103.08150.