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November 3

Registration

November 4

Chair: Qiongling Li 李琼玲 (Morning), Ling He 何玲 (Afternoon)

07:20-08:20 Breakfast
08:20-08:30 Opening Ceremony
08:30-09:20 Asymptotic behaviour of the Hitchin metric of the moduli space of Higgs bundles
Takuro Mochizuki (Kyoto University)
The moduli space of stable Higgs bundles of degree \(0\) is equipped with the hyperkähler metric, called the Hitchin metric. On the locus where the Hitchin fibration is smooth, there is the hyperkähler metric called the semi-flat metric, associated with the algebraic integrable systems with the Hitchin section. As predicted by Gaiotto, Moore and Neitzke, the difference between the metrics along the curve \((E,t\theta)\) decays in an exponential way. In this talk, we shall discuss how to improve the exponential rate in the rank \(2\) case.
09:20-09:40 Tea Break
09:40-10:30 Metric asymptotics on the irregular Hitchin moduli space
陈杲 (中国科学技术大学)
In 1987, Hitchin constructed a complete hyperkähler metric on the moduli space of Higgs bundles, which can be generalized to accommodate singularities. In this talk, we consider Higgs bundles with irregular singularities over the Riemann sphere. We construct a generic ray in the moduli space and study the asymptotic behavior of the Hitchin metric along this ray. Using the techniques developed by Biquard-Boalch, Fredrickson-Mazzeo-Swoboda-Weiss, and Mochizuki, we show that the Hitchin metric is exponentially close to a simpler semi-flat metric. In dimension four, we obtain an explicit asymptotic formula for the semi-flat metric, which is of type ALG or ALG*. This is a joint work with Nianzi Li.
10:30-10:50 Tea Break
10:50-11:40 Rank one symmetric differentials over projective variety
何思奇 (中国科学院)
Rank one symmetric differentials, a concept introduced by Taubes, play a significant role in gauge theory and differential geometry. In this talk, we’ll dive into the world of rank one symmetric differentials over projective varieties. We’ll explore how rank one symmetric differentials are connected to Higgs bundle and a recent proposal by Chen-Ngo. Furthermore, we will explain how rank one symmetric differentials could play a role in the Simpson integral conjecture. We will discuss a new proof using Finsler metric rigidity to prove characteristic rigidity and integrality for arithematic varieties with rank bigger than one. This talk is based on collaborated work with J.Liu and another work with J.Liu and N.Mok.
12:00-13:00 Lunch
14:00-14:50 LG/CY correspondence between tt*-geometries
范辉军 (北京大学)
tt*-geometry structure was found by physicists in the 1980’s, and defined and developed later in mathematics at the beginning of 90’s. It is an integrable structure mixed with the holomorphic and anti-holomorphic parts, and has close connections with Higgs bundles, Frobenius manifolds and other interesting structures. It is believed that it can be applied to more important occasions. The tt* geometrical structures of Calabi-Yau manifolds have been built long ago in the name of “special geometry”. In this talk, I will explain my construction of tt*-geometry for Landau-Ginzburg model via geometrical analysis method long time ago and formulate very recent results building the explicit LG/CY isomorphism between tt* geometrical structures for projective CY hypersurfaces. The latter work appears in arxiv: 2210.16747.
14:50-15:10 Tea Break
15:10-16:00 Kodaira-type vanishings via Nonabelian Hodge Theory
魏传豪 (西湖大学)
In the past decade, T. Mochizuki has completed the spectacular theory of mixed Twistor D-modules. In this talk, I will first briefly introduce this result. Then, I will show that Kodaira-type vanishing still holds under the setting of mixed Twistor D-modules, which generalizes Saito vanishing under the setting of mixed Hodge Modules. I will also introduce a relative version and a version of Kawamata-Viehweg vanishing with Q-divisors, under this general setting. Both of them seem to be new even in the mixed Hodge Modules setting.
16:00-16:20 Tea Break
16:20-17:10 Around irregular nonabelian Hodge theory
黄鹏飞 (Heidelberg University)
Nonabelian Hodge theory studies a correspondence between Higgs bundles and local systems, via the intermediate objects: integrable connections. In this talk, we will first give a quick review of the classical theory. Then we will talk about integrable connections with irregular singularities and show a nonabelian Hodge correspondence for them. This correspondence is applicable to general complex reductive groups as the structure group. Finally we will provide a construction of moduli spaces of various filtered local systems. Based on some recent joint work with Hao Sun.
18:00-20:00 Banquet

November 5

Chair: Hao Sun 孙浩 (Morning), Bin Xu 许斌 (Afternoon)

07:20-08:20 Breakfast
08:30-09:20 Constructing abelian schemes of \(\mathrm{GL}_2\)-type over \(4\)-punctured complex projective lines via \(p\)-adic Hodge theory and Langlands correspondence
左康 (武汉大学)
This is a joint work with Jinbang Yang. We construct infinitely many non-isotrivial abelian schemes of \(\mathrm{GL}_2\)-type over \(4\)-punctured complex projective lines with bad reduction of type-\((1/2)_\infty\) via \(p\)-adic Hodge theory and Langlands correspondence. Recently Lin-Sheng-Wang proved conjecture on the torsionness of zeros of Kodaira-Spencer maps of those type abelian schemes Based on their theorem we show the set of those type schemes is exactly parameterized by torsion sections of the universal family of elliptic curves. We note that, recently Lam-Litt gave a totally different construction of those abelian schemes by applying Katz's middle convolution functor. It could be highly interesting to compare these two very different constructions.
09:20-09:40 Photo & Tea Break
09:40-10:30 A higher-dimensional Chevalley restriction theorem for orthogonal groups
许金兴 (中国科学技术大学)
The classical Chevalley restriction theorem asserts that for a semisimple complex Lie group \(G\), the ring of \(G\)-invariant polynomials on the Lie algebra \(\mathfrak{g}\) is isomorphic through restriction to the ring of Weyl group invariant polynomials on the Cartan subalgebra. In studying the Hitchin morphism of principal \(G\)-Higgs bundles over higher dimensional varieties, Chen and Ngo conjectured a multi-variable generalization of the Chevalley restriction theorem, and they proved the \(\mathrm{GL}\) and \(\mathrm{Sp}\) cases. We then prove the orthogonal group case and our treatment can apply to the \(\mathrm{GL}\) and \(\mathrm{Sp}\) cases in a uniform way. In this talk I will explain the main ideas of the proof and the backgrounds from Hitchin morphism and algebraic invariant theory. This is a joint work with Lei Song and Xiaopeng Xia.
10:30-10:50 Tea Break
10:50-11:40 Springer Correspondence and mirror symmetry for \(\mathrm{Sp}/\mathrm{SO}\) Hitchin Systems
王彬 (香港中文大学)
Starting from special nilpotent orbits in \(\mathrm{Sp}_{2n}/\mathrm{SO}_{2n+1}\) which are related by Springer correspondence, we construct various Hitchin systems on curves with marked points. We resolve associated planar singularities and apply it to analyze the corresponding affine Spaltenstein fibers. As a result, we obtain (Strominger-Yau-Zaslow) mirror symmetry for those Hitchin systems. This is a joint work with Xiaoyu Su, Xueqing Wen and Yaoxiong Wen.
12:00-13:00 Lunch
14:00--14:50 \(p\)-adic Simpson correspondence via exponential twisting
盛茂 (中国科学技术大学)
In this talk, I shall explain the work of G. Faltings on a \(p\)-adic Simpson correspondence, using exponential twisting. This reinterpretation of the correspondence makes clearer analogy between the \(p\)-adic Simpson correspondence of Faltings and the char \(p\) Simpson correspondence due to Ogus-Vologodsky. This is a joint work with Zhaofeng Yu.
14:50-15:10 Tea Break
15:10-16:00 Kollar's package with coefficients
申屠钧超 (中国科学技术大学)
Kollar proved in 1986 a package of theorems on the pushforward of dualizing sheaves (torsion freeness, injectivity theorem, vanishing theorem, decomposition theorem). They have been generalized into two directions: 1) Kollar's package for dualizing sheaves twisted by a multiplier ideal sheaf; 2) Kollar's package for Hodge theoretic objects such as Hodge modules. In this talk I will survey the resent progress on this topic and explain how to unify the above results by using L2 analytic method, and show how to establish the Kollar's package in the context of nonabelian Hodge theory. This is a joint work with C. Zhao.
16:00-16:20 Tea Break
16:20-17:10 Non-abelian Hodge correspondence and the P=W conjecture
张子立 (同济大学)
For a complex projective curve \(C\) and a reductive group \(G\), the character variety \(M_B\) and the moduli of Higgs bundles \(M_{Dol}\) are canonically homeomorphic via the non-abelian Hodge correspondence and hence the cohomology groups of them are naturally identified. The geometric structures of the moduli spaces induce various filtrations in the cohomology groups. De Cataldo-Hausel-Migliorini conjectured in 2012 that the Perverse filtration (P) of \(M_{Dol}\) is identical to the Hodge-theoretic Weight filtration (W) of \(M_B\); the P=W conjecture. We will introduce the background recent progress of the P=W conjecture.
18:00-19:00 Dinner

November 6

Chair: Jiayu Li 李嘉禹

07:20-08:20 Breakfast
08:30-09:20 The non-Abelian Hodge correspondence on the non-Kähler case
张希 (南京理工大学)
The non-abelian Hodge correspondence was established by Corlette-Donaldson-Hitchin-Simpson, it states that, on a compact Kähler manifold, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this talk, we consider this correspondence on the non-Kähler case. The key to obtain the extended non-abelian Hodge correspondence lies in our recent works on semi-stable Higgs bundle and the existence of Poisson metric on semisimple projectively flat bundle. Finally, we give sThese works are joint with Chao Li, Jiayu Li, Yanci Nie, Changpeng Pan and Chuanjing Zhang.
09:20-09:30 Tea Break
09:30-10:20 Curvature of the base manifold of a Monge-Ampère fibration and its existence
万学远 (重庆理工大学)
In this talk, we will consider the Monge-Ampère fibration, a relative Kähler fibration that satisfies a homogenous Monge-Ampère equation. There are two generalized Weil-Petersson metrics on its base manifold. For one metric, we present a specific curvature formula, demonstrating its non-positive holomorphic bisectional curvature, along with several other curvatures capped negatively. We'll also prove the conditions under which a holomorphic vector bundle over a compact Kähler manifold corresponds to this Monge-Ampère fibration. Furthermore, we link relative Kähler fibrations to Higgs-flatness of associated infinite rank Higgs bundles. We conclude by highlighting classical Monge-Ampère fibration examples. This work is joint with Xu Wang.
10:20-10:40 Tea Break
10:40-11:30 On the existence of harmonic metrics on non-Hermitian Yang-Mills bundles
沈正晗 (南京理工大学)
In this talk, we will study the non-abelian Hodge correspondence to vector bundles with arbitrary Chern class. We will talk about the non-Hermitian Yang-Mills (NHYM for short) bundles over compact Kähler manifolds. We will show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles. This talk is based on a joint-work with Dr. Changpeng Pan and Prof. Xi Zhang.
11:30-11:40 Tea Break
11:40-12:30 Classificatin of constantly curved holomorphic \(2\)-spheres of degree \(6\) in the complex Grassmannian \(G(2,5)\)
徐言 (南开大学)
Up to now the only known example in the literature of constantly curved holomorphic \(2\)-sphere of degree \(6\) in the complex \(G(2, 5)\) has been the first associated curve of the Veronese curve of degree \(4\). By exploring the rich interplay between the Riemann sphere and projectively equivalent Fano \(3\)-folds of index \(2\) and degree \(5\), we prove, up to the ambient unitary equivalence, that the moduli space of generic (to be precisely defined) such \(2\)-spheres is semi-algebraic of dimension \(2\). All these \(2\)-spheres are verified to have non-parallel second fundamental form except for the above known example. This is a joint work with Professor Q.S. Chi and Z.X. Xie.
12:40-13:40 Lunch
18:00-19:00 Dinner

November 7

Chair: Zhi Hu 胡智

07:20-08:20 Breakfast
08:30-09:20 Cyclic Higgs bundles and minimal surfaces in pseudo-hyperbolic spaces
聂鑫 (东南大学)
Higgs bundles are well-known to have associated harmonic maps into symmetric spaces. We will discuss situations where this harmonic map is the Gauss map of a minimal surface in a pseudo-hyperbolic space. The study of such minimal surface in relation with classical Teichmüller theory (essentially \(\mathrm{SL}(2,\mathbb{R})\)-Higgs bundles) was mainly due to Mess, Bonsante and Schlenker; while a generalization to cyclic \(\mathrm{SO}(2,n)\)-Higgs bundles is more recently given by Collier-Tholozan-Toulisse. We will explain a further generalization to cyclic \(\mathrm{SO}(n+1,n)\)-Higgs bundles, which allows us to cover the exceptional Lie group.
09:20-09:40 Tea Break
09:40-10:30 Loop group methods for the non-abelian Hodge correspondence
Lynn Heller (BIMSA)
The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions of the self-duality equation. In my talk I will explain how to construct such solutions for strongly parabolic \(\mathfrak{sl}(2,\mathbb{C})\) Higgs fields on a \(4\)-punctured sphere with parabolic weights \(t\sim 0\) using loop groups methods through an implicit function theorem argument. The methods and computations are based on Deligne’s interpretion of the twistor space via \(\lambda\)-connections. As a first application of this approach, I identify the rescaled limit hyper-Kähler moduli space at \(t = 0\) as the Eguchi-Hanson space, and I show that the hyper-Kähler metric can be expanded to arbitrary order in terms of multiple-polylogarithms. Finally, the geometric properties lead to some identities involving multiple-polylogarithms. This talk is based on joint work with Sebastian Heller and Martin Traizet.
10:30-10:50 Tea Break
10:50-11:40 Some estimates on Higgs bundles over Riemann surfaces
戴嵩 (天津大学)
Given a Riemann surface, the non-Abelian Hodge theory roughly speaking builds a correspondence between Higgs bundles and harmonic maps. In this talk, under the background of the moduli space of the Higgs bundles, we discuss some estimates of the geometric quantities from harmonic maps. We focus on two kinds of estimates: domination and boundedness. We will also show some applications of the estimates. The main results in this talk are joint with Qiongling Li.
12:00-13:00 Lunch
18:00-19:00 Dinner