Venue

Lecture Hall (Building 1-Section B-3rd Floor)

USTC Shanghai Institute for Advanced Studies

Schedule(August 3-7)

TBA

Titles & Abstracts

*Vestislav Apostolov (Université du Québec à Montréal)

Title: Non-K\"ahler Calabi-Yau geometries on 3-folds

Abstract: Calabi-Yau 3-folds, i.e. complex 3-dimensional K\”ahler manifolds with holonomy group contained in SU(3), play a central role in complex, symplectic and algebraic geometry, mathematical physics, and more.  It is natural to seek extensions beyond the setting of K\"ahler geometry.

In this talk,  I will discuss compact non-K\”ahler Hermitian 3-folds with pluriclosed fundamental $2$-form,  whose Bismut-Ricci form is identically zero. These geometries appeared in the 1980’s in mathematical physics as string backgrounds extending the K\”ahler Calabi-Yau condition.

It turns out that compact non-K\”ahler Hermitian manifolds with $\partial \bar \partial$-closed fundamental $2$-form and zero Bismut-Ricci form have a canonical symmetry reduction to complex co-dimension one.  In  complex dimension 3, the transversal geometry is actually K\''ahler and governed by a single 6th-order scalar PDE for the K\”ahler potential of the underlying K\”ahler metric. In the quasi-regular case, i.e. when the $3$-fold fibers over a compact complex orbifold surface, the PDE has an infinite dimensional momentum map interpretation, similar to the much studied K\"ahler metrics of constant scalar curvature.  We use this to obtain obstructions for the existence of solutions in terms of the automorphism group, paralleling results by Futaki and Calabi-Lichnerowicz–Matsushima in the cscK case. As an application, we characterize the standard Samelson’s examples, i.e. the Hermitian 3-folds which are covered by either $SU(2) \times {\mathbb R }^3$ or $\SU(2) \times \SU(2)$ equipped with a bi-invariant metric and a left invariant complex structure, as the only regular examples having Bott-Chern number $h_{BC}^{1,1} =2$.  We also obtain explicit solutions of the PDE on orthotoric K\"ahler orbifold surfaces, which yield infinitely many non-K\”ahler $3$-folds with $\partial \bar \partial$-closed fundamental $2$-form and zero Bismut-Ricci form on $S^3 × S^3$ and $S^1 × S^2 × S^3$, which are not locally isometric to a Samelson geometry. These appear to be the first such examples.

This talk is based on joint works with Barbaro, Lee and Streets,  and Lahdili and Lee, available on arXiv:2408.09648 and arXiv:2601.04937.

Hanwool Bae (Chonnam Nattional University)

Title: Rabinowitz Fukaya category as a Tate $A_\infty$ category and Calabi--Yau structures

Abstract: The notion of Tate vector space enables one to formulate self-duality for infinite-dimensional vector spaces. It was shown by Cieliebak-Hingston-Oancea that the Rabinowitz Floer homology of a Liouville manifold admits a 2-dimensional TQFT structure and, as a consequence, satisfies a Poincar\'{e} duality in the sense of Tate vector spaces.

In this talk, I will introduce the notion of an $A_\infty$-category enriched in Tate vector spaces. or more generally in ultrasolid modules, which I will call a Tate $A_\infty$-category. I will then explain that the Rabinowitz Fukaya category of a Liouville manifold carries the structure of Tate $A_\infty$-category, and admits a categorical Poincar\'{e} duality, namely, a strong proper Calabi--Yau structure. This is based on joint work with Yuan Gao.

Huai-Liang Chang (Wuhan University)

Title: Mixed-Spin-P field theory of \(X_{3,3}\subset\mathbb{P}^2\times\mathbb{P}^2\)

Abstract: This survey introduces Mixed-Spin-P (MSP) field theory, the most effective approach to organizing phase transitions and all genus mirror predictions among all Gromov-Witten-type theories on compact Calabi-Yau threefolds. MSP theory mixes Calabi-Yau and Landau-Ginzburg phases, and supplies comprehensive algebraic geometric explanations for BCOV theory, holomorphic anomaly equations, Yamaguchi-Yau functional equations, high genus mirror symmetry, and analyticity of all-genus curve-counting potentials Fg’s. The theory naturally provides proofs of various conjectures for diverse phases of CY, including FJRW invariants or, more generally, hybrid Landau-Ginzburg models. I will review the story of MSP theory and sketch its recent developments, including quintic-FJRW phases and also the Calabi-Yau hypersurface \(X_{3,3}\subset\mathbb{P}^2\times\mathbb{P}^2\).

Qianyu Chen (University of Michigan) 

Title: Filtrations on Local Cohomology Modules

Abstract: Introduced by Grothendieck, local cohomology modules are classical invariants in algebraic geometry. They carry several natural filtrations arising from different perspectives. On the algebraic side, one has the Ext filtration and the pole-order filtration, while on the Hodge-theoretic side, Saito’s theory of mixed Hodge modules equips local cohomology modules with a Hodge filtration. In this talk, I will discuss the relationships among these filtrations.

Siqi He (AMSS)

Title: Deformation theory of Z2 harmonic spinors

Abstract: Z2 harmonic spinors, introduced by Taubes, describe the boundary of the two spinor Seiberg-Witten moduli space, playing a key role in the study of analytic aspects of a new gauge theoretical equation. Understanding their boundary is essential for potential new gauge theory invariants for 3 and 4-manifolds. Based on joint work with G. Parker and T. Walpuski, we will discuss some recent progress on the deformation problem for Z2 harmonic spinors on 4-manifolds.

Shuho Kanda (The University of Tokyo)

Title: Left-Invariant Complex Structures on Solvable LieGroups: A Non-Stein Example

Abstract: Let G be a simply connected Lie group, and let J be a left-invariant complex structure on G. What can be said about the resulting complex manifold (G,J)? In 2010, Hasegawa conjectured that if G is nilpotent, then (G,J) is biholomorphic to C^n, and, more generally, that (G,J) is Stein whenever G is unimodular and solvable. Recently, this conjecture was resolved affirmatively in the nilpotent case and negatively in the solvable case. In this talk, I will discuss the background of the problem and describe the construction of a counterexample in the unimodular  solvable setting.

Tatsuki Kuwagaki (Kyoto University)

Title: Sheaves and Fukaya category over the Novikov ring

Abstract: Over the past decade, the relationship between the Fukaya categories of exact Lagrangian submanifolds and microlocal sheaf theory has been understood, and has been used as an important technique for studying symplectic topology and mirror symmetry. In this talk, I would like to review recent progress in our understanding of how such correspondences can be generalized to non-exact Lagrangian submanifolds, as well as their applications to homological mirror symmetry. 

Jungsoo Kang (Seoul National University)

Title: Floer homology for prequantization bundles

Abstract: A prequantization bundle is a circle bundle that admits a contact/symplectic structure. I will first present a quantum version of the Gysin sequence for prequantization bundles, which relates the Rabinowitz Floer homology of the total space to the quantum homology of the base. We then consider a Lagrangian submanifold in the base and its Legendrian lift in the total space, whose singular homologies are related via the transfer map. I will present a quantum version of this relation. This is based on joint work with Hanwool Bae, Joonghyun Bae, and Sungho Kim.

Naichung Conan Leung (Chinese University of Hong Kong)

Title: 3d mirror symmetry and mirror symmetry

Abstract: We will explain relationship between 3d mirror symmetry for Rozansky-Witten theory and 2d mirror symmetry for string theory.

Mingyang Li (Stony Brook University)

Title: Gravitational instantons and harmonic maps

Abstract: It is known from general relativity that axisymmetric stationary black holes can be reduced to axisymmetric harmonic maps into the hyperbolic plane H^2, while in the Riemannian setting, 4d Ricci-flat metrics with torus symmetry can also be locally reduced to such harmonic maps satisfying a tameness condition. We study such harmonic maps and application includes a construction of infinitely many new complete, asymptotically flat, Ricci-flat 4-manifolds with arbitrarily large second Betti number b_2. Joint work with Song Sun.

Yang Li (Cambridge University)

Title: On the metric SYZ conjecture

Abstract: For a polarised degeneration family of Calabi-Yau manifolds near the large complex structure limit, the metric SYZ conjecture asks for a special Lagrangian torus fibration on the generic region of the Calabi-Yau manifolds. I will talk about some recent progress.

Natsuo Miyatake (Toyoku University)

Title: Equilibrium Metrics and Complete Harmonic Metrics

Abstract: An equilibrium metric on a positive line bundle $L$ over a compact Riemann surface $X$ is a special singular Hermitian metric whose curvature current induces an equilibrium measure. It is defined from data consisting of a continuous Hermitian metric $e^{-\phi}h_\ast$ on $L$ and a nonpolar compact subset $K\subseteq X$, using the envelope of subharmonic weight functions bounded above by $\phi$ on $K$. The approximation of equilibrium metrics by sequences of singular metrics has been studied in a variety of contexts, including complex geometry, potential theory, dynamical systems, and probability theory.

In this talk, I will introduce two functions, called entropy and free energy, associated with a semipositive singular Hermitian metric on the canonical bundle of a Riemann surface. These quantities are defined using an extension of the notion of complete harmonic metrics on cyclic Higgs bundles, whose existence and uniqueness were established by Li--Mochizuki, to complete Hermitian metrics associated with a semipositive singular metric on the canonical bundle that is not necessarily induced by a holomorphic $r$-differential. I will give several estimates for these functions and discuss my ongoing research aimed at quantitatively establishing the principles of entropy increase and free energy decrease in various situations where an equilibrium metric is approximated by a sequence of singular metrics.

Ngoc Cuong Nguyen (KAIST)

Title: Fine properties of functions in complex Sobolev Spaces

Abstract: We study comprehensively local properties of functions in complex Sobolev spaces on a bounded open subset of C^n. The main tool is the corresponding functional capacity for the space which is inspired by the global one due to Vigny [2007]. An inequality between this capacity and the Bedford-Taylor capacity for plurisubharmonic functions is proved, which is sharp as far as the exponents are concerned. Moreover, it is shown that the functional capacity is a Choquet capacity. 

Zhiyu Tian (Peking University)

Title: Integral Hodge conjecture for one-cycles

Abstract: Hodge conjecture is well-known to be false with integral coefficients. However, there is a huge interest in understanding the integral version (and its l-adic analogue, the integral Tate conjecture) for some special classes of algebraic varieties, in view of the possible arithmetic applications. In this talk, I will give a survey of these questions and present some recent results for one-cycles on quotients of projective varieties. This is based on joint work with Yang Cao at Shandong University.

*Valentino Tosatti (New York University)

Title: Holomorphic tubular neighborhoods

Abstract: A classical result in differential topology shows that every submanifold of an ambient manifold admits a tubularneighborhood diffeomorphic to a neighborhood of the zero section of its normal bundle. The analogous result in the category of complex manifolds does not hold in general, already for plane algebraic curves of degree at least 2, and deciding in which cases it holds can be a very difficult question. In 1975, Ogus studied the case of certain elliptic curves embedded in the blowup of the plane at 9 points, which have topologically trivial (but not holomorphically torsion) normal bundle. Arnol'd showed in 1976 that almost all of these (in the measure-theoretic sense) admit a holomorphic tubular neighborhood, but not a single one was known which does not admit such a neighborhood.  I will discuss recent work with Simion Filip where we construct many such examples, answering Ogus's question.

Weiwei Wu (Zhejiang University)

Title: On the global monodromy of a Lefschetz fibration

Abstract: In this talk, we explain a proof that the fixed-point Floer cohomology of the global monodromy of a Lefschetz fibration, whose total space is a Weinstein manifold, is isomorphic to the Hochschild cohomology with coefficients in the global monodromy bimodule.

Moreover, this isomorphism is given by the closed-open string map with a twisting. This concludes, partially, a question posed by Paul Seidel. Previously, Bai-Seidel showed that the twisted closed-open string map is injective.

This is an ongoing joint work with Y Gao, L Han and S Zhang.

Qizheng Yin (Peking University)

Title: Lagrangian fibrations and hyper-Kähler varieties: an analogy

Abstract: We discuss several open questions concerning the topology and algebraic geometry of (holomorphic) Lagrangian fibrations and hyper-Kähler varieties, and draw an analogy between the relative and absolute settings. We also explain how this analogy serves as a guiding principle for solving some of these questions. Based on series of joint work with Younghan Bae, Davesh Maulik, Junliang Shen, and Ruxuan Zhang.

Zhengyi Zhou (MCM, CAS)

Title: On symplectic CPn

Abstract: We show that the existence of a pseudo-holomorphic line passing through two generic points on a symplectic manifold X, phrased using Gromov-Witten invariants, implies that X is homotopy equivalent to CPn with identical first Chern class and small quantum cohomology. We then deduce from this result and its generalizations some rigidity results regarding symplectic hyperplanes in CPn+1 as well as symplectic fillings. The proof is based on Rabinowitz Floer homology.