Moduli Spaces and Related Topics

报告人 Speaker:Siqi He (何思奇),中科院晨兴数学中心
组织者 Organizer:Xiang He, Chenglong Yu, Dingxin Zhang, Jie Zhou
时间 Time:Wednesday, 14:00-15:00, May 15, 2024
地点 Venue:C654, Shuangqing Complex Building A

(All seminars will be held in C654, Shuangqing Complex Building on Wednesday at 14:00-15:00, unless marked in red.

 

Upcoming Talk:


Title: Hitchin morphism for projective varieties

Speaker: Siqi He (何思奇),中科院晨兴数学中心

Time: 2024.05.22 (Wed.) 2:00-3:00 pm
Place: Shuangqing Complex Building A, C654

Abstract: The Hitchin morphism is a map from the moduli space of Higgs bundles to the Hitchin base, which is generally not surjective when the dimension of the variety is greater than one. Chen-Ngo introduced the concept of the spectral base, which is a closed subscheme of the Hitchin base. They conjectured that the Hitchin morphism is surjective to the spectral base and also proved that the surjectivity is equivalent to the existence of finite Cohen-Macaulayfications of the spectral varieties. For rank two Higgs bundles, we will discuss an explicit construction of the Cohen-Macaulayfication of the spectral variety. In addition, we will discuss several applications using the spectral base to the topology of projective variety. This talk is based on some collaborative work with J. Liu and N. Mok.



 

Past Talks:

Title: "Counting" pseudo-holomorphic disks via loop space
Speaker: Yi Wang (王怡), Purdue University, Postdoc
Time: 2024.05.15 (Wed.) 2:00-3:00 pm
Place: Shuangqing Complex Building A, C654


Abstract:

Let M be a symplectic manifold and L be a Lagrangian submanifold. Celebrated work of Fukaya-Oh-Ohta-Ono constructs a filtered cyclic A-infinity algebra structure on the cochain (cohomology) group of L by studying moduli spaces of pseudo-holomorphic disks to (M,L) with boundary mark points. In this talk, I will explain how this A-infinity algebra lifts to the free loop space of L in a way compatible with the string topology of loop spaces, as well as some applications of this formalism.




Speaker: 张旭成 (YMSC)

Title: A geometric approach to identifying the stability condition

Time: 2024.4.24 (Wed) 14:00--15:00

Place: Shuangqing Complex Building A, C654


Abstract:

For any reductive group we find a geometric interpretation of the stability condition for principal bundles over a curve: it is the unique maximal open locus that admits a schematic moduli space. Some applications and further progress will be discussed. This is a joint work with Dario Weissmann.




Title: Universal holomorphic maps, conflict between fully hypercyclicity and slow growth

Speaker: Zhangchi Chen (陈张弛,中科院晨兴数学中心)

Time: first talk 2024.4.17 (Wed) 2:00-3:00 pm, second talk 2024.4.18 (Thurs) 10:00-11:00 am (Remark: the first talk will be general, and the second talk will contain more details)

Place: Shuangqing Complex Building A, C654


Abstract: 

In the space O(C,C) of entire functions, equipped with the open-compact topology, an element is called universal if its translation orbit is dense. It is hypercyclic w.r.t some translation operator if its orbit under this operator is dense. It is fully hypercyclic if it is simultaneously hypercyclic to all non-trivial translations in all directions.

Universal entire functions are transcendental, hence their Nevanlinna characteristic functions grows faster than O(log r). Dinh-Sibony asked what the slowest Nevanlinna growth of universal entire curves is. In a joint work with Dinh Tuan Huynh and Song-Yan Xie, we solved their question completely, by constructing universal entire curves in CP^n whose Nevanlinna growth is slower than any given transcendental entire function.

Bin Guo and Song-Yan Xie discovered the conflict between fully hypercyclicity and slow growth. They proved that if the growth is too slow then the hypercyclic directions in [0,2pi) has Hausdorff dimension 0.

Replace C by the unit disc D, and translations by Aut(D), one can also talk about universal holomorphic discs. Transcendental functions defined on D with bounded Nevanlinna characteristic functions are called of bounded type, which is the analogous property of having slow growth. In a joint work with Bin Guo and Song-Yan Xie, we constructed universal discs in CP^n of bounded type. We also discovered a weak-conflict between fully hypercyclicity and slow growth. If the disc is of bounded type, then the hypercyclic directions in [0,2pi) has Lebesgue measure 0.




Title: A Noether-Lefschetz type theorem for spectral varieties with applications

Speaker: Xiaoyu Su (苏晓羽,北京邮电大学)

Time: 2024.4.10 (Wed) 2-3 pm

Place: Shuangqing Complex Building A, C654


Abstract: 

In this talk, we will discuss spectral variety and related moduli spaces of Higgs pairs on surfaces. We will first introduce the geometry of moduli of Higgs pairs and the spectral varieties. Indeed, we will talk about the Noether-Lefschetz type problems and the Picard schemes of the spectral varieties. If time permits, the Picard group of generic (very general) spectral varieties and its geometric applications will also be discussed. This is a joint work with Bin Wang.




Title: Hodge properties of confluent hypergeometric connections

Speaker: Yichen Qin (Humboldt Universität zu Berlin)

Time: Wednesday, 14:00-15:00, March 27, 2024

Place: C654, Shuangqing Complex Building A


Abstract:

Sabbah and Yu computed the irregular Hodge numbers associated with hypergeometric connections. In this talk, we introduce a new approach for hypergeometric connections whose defining parameters are rational numbers. Our method relies on a geometric interpretation of hypergeometric connections, which enables us to describe the irregular Hodge filtrations explicitly and derive several arithmetic applications on hypergeometric sums. This research is conducted in collaboration with Daxin Xu.



Title: Curves on K3 surfaces and Mukai’s program

Speaker: Haoyu Wu (Fudan University)

Time: Wednesday, 14:00-15:00, March 20, 2024

Place: C654, Shuangqing Complex Building A


Abstract:

The Mukai’s program seeks to recover a K3 surface X from any curve C on it by exhibiting it as a Fourier-Mukai partner to a Brill--Noether locus of vector bundles on the curve. In this talk, I will give an introduction to work of Feyzbakhsh for Picard number one K3 and primitive curve C. We extend the results to the case of non-primitive curves by introducing the tools of destabilizing regions. As an application, we show that there are hyper-K\"{a}hler varieties as Brill-Noether loci of curves in every dimension. This is a joint work with Yiran Chen and Zhiyuan Li.




Title: On the moduli space of certain plane sextic curves

Speaker: Yiming Zhong (BICMR)

Time: Wednesday, 14:00-15:00, March 13, 2024

Place: C654, Shuangqing Complex Building A


Abstract:

We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne–Mostow theory and periods of K3 surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of complex hyperbolic balls. We show that the two ball-quotient constructions can be unified in a geometric way. This is a joint work with Zhiwei Zheng.




Title: Higher dimensional Heegaard Floer homology and Hecke algebras

Speaker: Yin Tian (BNU)

Time:Wednesday at 11:00, March 6, 2024

Place: C654, Shuangqing Complex Building


Abstract:

Higher dimensional Heegaard Floer homology (HDHF) is a higher dimensional analogue of Heegaard Floer homology in dimension three. It's partly used to study contact topology in higher dimensions. In a special case, it's related to symplectic Khovanov homology. In this talk, we discuss HDHF of cotangent fibers of the cotangent bundle of an oriented surface and show that it is isomorphic to various Hecke algebras. This is a joint work with Ko Honda and Tianyu Yuan.




Title: On Hurwitz-Severi numbers

Speaker: Boris Shapiro, Stockholm University

Time: Dec 20, 2023, Wednesday at 10:00am

Abstract: For a point p in CP2 and a triple (g, d, l) of non-negative integers we define a Hurwitz–Severi number Hg,d,l as the number of generic irreducible plane curves of genus g and degree d+l having an l-fold node at p and at most ordinary nodes as singularities at the other points, such that the projection of the curve from p has a prescribed set of local and remote tangents and lines passing through nodes. Under certain conditions we express the above Hurwitz-Severi numbers via appropriate Hurwitz numbers. Several questions will be posed.




Title:Mixed-Spin-P fields for GIT quotients

Speaker:Zhou Yang 周杨 (复旦大学上海数学中心)
Time: Dec 13, 2023, Wednesday at 10:00am

Venue:C654, Shuangqing Complex Building

Abstract:

The theory of Mixed-Spin fields was introduced by Chang-Li-Li-Liu for the quintic threefold, aiming at studying its higher genus Gromov-Witten invariants. Chang-Guo-Li has successfully applied it to prove famous conjectures on the higher-genus Gromov-Witten invariants proposed by physicists. In this talk I will explain a generalization of the construction to more spaces. The generalization usually depends on some choices and I will give some concrete examples in the talk.

The key is a stability condition which guarantees the separatedness and properness of certain moduli spaces. It also generalizes the construction of the mathematical Gauged Linear Sigma Model by Fan-Jarvis-Ruan, removing their technique assumption about "good lifitings".

This is a joint work with Huai-Liang Chang, Shuai Guo, Jun Li and Wei-Ping Li.




Title: Springer correspondence and mirror symmetry for Sp/SO Hitchin Systems 

Speaker: Bin Wang 王彬 (Postdoc, 香港中文大学)

Time:Wednesday at 10:00, Dec 6, 2023

Place: C654, Shuangqing Complex Building


Abstract:

Starting from special nilpotent orbits in Sp_{2n}/SO_{2n+1} which are related by Springer correspondence, we construct various Hitchin systems on curves with marked points. We resolve singularities of generic spectral curves. We then apply it to analyze the corresponding affine Spaltenstein fibers,which can be treated as the local version of (parabolic) Hitchin fibers. As a result, we obtain the (Strominger-Yau-Zaslow) mirror symmetry for these Hitchin systems. This is a joint work with X. Su, X. Wen and Y. Wen. 




Title: Gromov--Witten/Pandharipande--Thomas correspondence via conifold transitions 

Speaker: Lin Yinbang 林胤榜(同济大学)

Time: Monday at 10:00 am, Nov 27, 2023

Venue: B626, Shuangqing Complex Building


Abstract:

Given a (projective) conifold transition of smooth projective threefolds from $X$ to $Y$, we show that if the Gromov--Witten/Pandharipande--Thomas descendent correspondence holds for the resolution $Y$, then it also holds for the smoothing $X$ with stationary descendent insertions. As applications, we show the correspondence in new cases. This is joint work with Sz-Sheng Wang.




Title: Rigidity problems on moduli spaces of polarized manifolds

Speaker: Ruiran Sun (McGill University)

Time: Wednesday at 10:00, Nov 22, 2023

Place: C654, Shuangqing Complex Building


Abstract: I will survey the recent progress on the rigidity problems on moduli spaces of polarized manifolds. This talk is based on the joint works with Ariyan Javenpaykar, Steven Lu and Kang Zuo, and with Chenglong Yu and Kang Zuo.




Title:Kodaira-type and Bott-type vanishings via Hodge theory

Speaker: Wei Chuanhao, Westlake University (魏传豪,西湖大学)

Time: Nov 8 ,10:00-11:00 am

Venue: C654, Shuangqing Complex Building


Abstract:

I will first give a brief introduction to T. Mochizuki's Theory of twistor D-modules.  Then, we use it to study Kodaira-type vanishings. In particular, we will generalize Saito vanishing, and give a Kawamata-Viehweg type statement. As an application, we will also prove a Bott-type Vanishing using M. Saito's mixed Hodge module.




Title: Simpson's correspondence and the P=W conjecture

Speaker:Zhang Zili, Tongji University (张子立, 同济大学)

Time: Nov 1 ,10:00-11:00 am

Venue:C654, Shuangqing Complex Building


Abstract:

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 Simpson's 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 some background and recent progress of the P=W conjecture.




Title: Tropicalizations of Riemann surfaces and their moduli

Speaker: Dali Shen, BIMSA

Time: Oct 27 (Fri.) 1:30pm-2:30pm

Place: Shuangqing Complex Building A513


Abstract: The tropical methods have already been used to study the moduli theory of algebraic curves during the past decade. In this talk, I will first discuss about the tropicalization of a smooth pointed Riemann surface via its (hyperbolic) pair of pants decomposition, and then about how to compactify the moduli space of tropicalizations in a geometrically meaningful way.



 

Title: L^2 type invariants of hyperplane arrangement complement

Speaker: Liu Yongqiang, USTC (刘永强,中国科学技术大学)

Time: Wednesday at 10:00, Oct 18, 2023

Place: C654, Shuangqing Complex Building


Abstract:

We first give an brief introduction on the topic of hyperplane arrangement.  Then we give concrete formulas for these L^2 type invariants at degree 1 and study their connections with combinatorics. If time allows, some similar results for smooth complex quasi-projective variety will be discussed.



 

Title: Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on C^2

Speaker: Penghui Li (YMSC)

Time: Wednesday at 10:00, Oct 11, 2023

Place: C654, Shuangqing Complex Building


Abstract:

Using a geometric argument building on our new theory of graded sheaves, we compute the categorical trace and Drinfel'd center of the (graded) finite Hecke category H_W in terms of the category of (graded) unipotent character sheaves, upgrading results of Ben-Zvi-Nadler and Bezrukavninov-Finkelberg-Ostrik. In type A, we relate the categorical trace to the category of 2-periodic coherent sheaves on the Hilbert schemes of points on C^2 (equivariant with respect to the natural C*×C* action), yielding a proof of a conjecture of Gorsky-Negut-Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on Hilbert schemes. As an important computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich on the formality of the Hochschild homology of H_W. This is a joint work with Quoc P. Ho.




Title: Degeneration of Hodge structures and cubic hypersurfaces

Speaker: Renjie Lyu 吕人杰 (中科院AMSS, Postdoc)

Time: June 22 (Thur.) 3:30-5:00 pm

Place: Jinchunyuan West Building, Floor 3

 

Abstract:

The degeneration of Hodge structures is related to how a smooth projective variety degenerate. And it provides a Hodge-theoretic perspective to compactify moduli spaces. In this talk, I will focus on a particular degeneration of cubic hypersurfaces and study the associated limiting mixed Hodge structure. It generalizes some results in Radu Laza’s and Brendan Hassett’s works on cubic fourfolds. This is a joint work with Zhiwei Zheng.



 

Title: Moduli spaces of modules over even Clifford algebra and Prym varieties

Speaker: Jia Choon Lee (BICMR)

Time: Wednesday at 15:30, June 14, 2023

Place: Conference Room 3, Jinchunyuan West Building

 

Abstract:

A conic fibration has an associated sheaf of even Clifford algebra on the base. In this talk, I will discuss the relation between the moduli spaces of modules over the sheaf of even Clifford algebra and the Prym variety associated to the conic fibration. I will begin by motivating the connection between them from the viewpoint of the rationality problem of cubic hypersurfaces. Then I will explain the construction of a rational map from the moduli space of modules over the sheaf of even Clifford algebra to the special subvarieties in Prym varieties. As an application, we get an explicit correspondence between instanton bundles of minimal charge on cubic threefolds and twisted Higgs bundles on curves.

 



Title: Sheaves on non-reduced curves in a projective surface

Speaker: Yao Yuan (Capital Normal University)

Time: Tuesday 10:00am, June 06, 2023

Place: Third floor, Jinchunyuan West Building

 

Abstract:

Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface, and moduli spaces of Higgs bundles as well. We estimate the dimension of the stack M_X(nC, \chi) of pure sheaves supported at the non-reduced curve nC (n ≥ 2) with C an integral curve on X. We prove that the Hilbert-Chow morphism h_{L,\chi} : M_X^H(L, \chi) -> |L| sending each semistable 1-dimensional sheaf to its support have all its fibers of the same dimension for X Fano or with trivial canonical line bundle and |L| contains integral curves.  The strategy is to firstly deal with the case with C smooth and then do induction on the arithmetic genus of C which once can decrease by a blow-up given C singular.  As an application, we generalize the result of Maulik-Shen on the cohomology \chi-independence of M_X^H(L,\chi) to X any del Pezzo surface not necessarily toric.

 



Title: Cubic threefolds with an involution and their intermediate Jacobians

Speaker: Zheng Zhang 张正 (ShanghaiTech University)

Time: Wednesday at 15:30, May 31, 2023

Place: Conference Room 3, Jinchunyuan West Building


Abstract:

We study the moduli space of cubic threefolds admitting an involution via the period map sending such a cubic threefold to the invariant/anti-invariant part of the intermediate Jacobian.  Our main result is global Torelli holds for the period map. Key ingredients of the proof include a description of the invariant/anti-invariant part of the intermediate Jacobian as a Prym variety and a detailed study of certain positive dimensional fibers of the corresponding Prym map. The proof also relies on the results of Donagi-Smith, Ikeda and Naranjo-Ortega on related Prym maps. This is joint work with S. Casalaina-Martin and L. Marquand.

 



Title: Universal meromorphic functions with slow growth

Speaker: Songyan Xie (CAS)

Time: Wednesday at 15:30, May 24, 2023

Place: Conference Room 3, Jinchunyuan West Building

 

Abstract:

I will show a solution to a problem asked by Dinh and Sibony in their open problem list, about minimal growth of universal meromorphic functions. This is joint work with Dinh Tuan Huynh and Zhangchi Chen. If time permits, I will also discuss my recent joint work with my Ph.D. student Bin Guo, about the existence of universal holomorphic functions in several variables with slow growth.

 



Title: On attractor points on the moduli space of Calabi-Yau threefolds

Speaker: Emanuel Scheidegger (BICMR)

Time: Wednesday at 15:30, May 17, 2023

Place: Conference Room 3, Jinchunyuan West Building


Abstract:

We briefly review the origin in physics of attractor points on the moduli space of Calabi-Yau threefolds. We turn to their mathematical interpretation as special cases of Hodge loci. This leads to fascinating conjectures on the modularity of the Calabi-Yau threefolds at these points in terms of their periods and L-functions. For hypergeometric one-parameter families of Calabi-Yau threefolds, these conjectures can be verified at least numerically to very high precision.




Title: stable (parabolic) holomorphic vector bundles over complex curves and instanton Floer homology

Speaker: 谢羿(北京大学)

Time: Wednesday at 15:30, Apr 26, 2023

Place: Conference Room 3, Jinchunyuan West Building

 

Abstract:

Stable holomorphic bundles are objects in algebraic geometry which have been studied by many people. Instanton Floer homology is an invariant of 3-manifolds, which has been used to solve many problems in the low dimensional topology. It turns out the two things are closely related: knowledge on the moduli space of stable bundles can help the calculation of Instanton Floer homology.  In this talk, I will explain this relationship and its generalization to stable parabolic bundles. This is joint work with Boyu Zhang.

 


 

Title: On the cohomology of BG

Speaker: 李时璋(中科院晨兴数学中心)

Time: Wednesday at 15:30, Apr 19, 2023

Place: Conference Room 3, Jinchunyuan West Building

 

Abstract:

In this talk, if time permits, we will discuss:

(1) classify order p group schemes over Spec(char p alg closed field) using Dieudonne modules;

(2) a new way of understanding Dieudonne modules in terms of cohomology of BG (due to Mondal);

(3) an attempt of using BG to construct a counterexample that Deligne--Illusie asked for (work of Antieau--Bhatt--Mathew);

(4) why this attempt cannot succeed (joint work in progress with Kubrak--Mondal), and how this attempt can be made successful (due to Petrov).

 


 

Title: Tropicalization of curves and applications

Speaker: Xiang He (YMSC, Tsinghua)

Time: Wednesday at 15:30, Apr 12, 2023

Place: Conference Room 3, Jinchunyuan West Building

 

Abstract:

The tropicalization process assigns to an algbraic variety a polyhedral complex with extra structure that records certain degeneration data. In this talk, I will introduce the tropicalization of (a family of) algebraic curves and explain the connection between the geometry of the algebro-geometric side and the tropical side. I will then discuss the application of this construction to the irreducibility of Severi varieties and the moduli space of curves. This is joint work with Karl Christ and Ilya Tyomkin.

 


 

Title: Moduli spaces of hyperKahler manifolds and cubics

Speaker: Zhiwei Zheng, YMSC, Tsinghua

Time: Mar. 29 (Wed.) 3:30-5:05 pm

Place: Jinchunyuan West Building, Second Floor, Conference Room 3

 

Abstract: 

The study of moduli spaces of hyperKahler manifolds and low dimensional cubic hypersurfaces is an active direction in algebraic geometry. Thanks to kinds of Torelli theorem, many moduli spaces can be realized as locally symmetric varieties of unitary type or orthogonal type. Hodge theory, birational geometry and arithmetic geometry converge in this topic. In this talk I will give a general introduction to the theory and examples, and discuss the future directions.