Seminar on "Moduli Spaces and Related Topics"
Organizers:
| Xiang He | xianghe"at"mail.tsinghua.edu.cn |
| Chenglong Yu | yuchenglong"at"mail.tsinghua.edu.cn |
| Dingxin Zhang | dingxin"at"mail.tsinghua.edu.cn |
| Zhiwei Zheng | zhengzw11"at"163.com |
| Jie Zhou | jzhou2018"at"mail.tsinghua.edu.cn |
Venue&Time:
All seminars will be held in C654, Shuangqing Complex Building on Wednesday at 14:00-15:00, unless marked in red.Spring 2024
| Date | Speaker | Title | Abstract |
| May 22 | Siqi He (MCM) |
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| May 15 | Yi Wang (Purdue University) |
"Counting" pseudo-holomorphic disks via loop space | 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. |
| Apr 24 | Xucheng Zhang (YMSC) |
A geometric approach to identifying the stability condition | 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. |
| Apr 18 10:00-11:00 | Zhangchi Chen (MCM) |
Universal holomorphic maps, conflict between fully hypercyclicity and slow growth II | This talk contains more details about the previous one. | .
| Apr 17 | Zhangchi Chen (MCM) |
Universal holomorphic maps, conflict between fully hypercyclicity and slow growth I | 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. |
| Apr 10 | Xiaoyu Su (BUPT) |
A Noether-Lefschetz type theorem for spectral varieties with applications | 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. |
| Mar 27 | Yichen Qin (Humboldt Universität zu Berlin) |
Hodge properties of confluent hypergeometric connections. | 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. |
| Mar 20 | Haoyu Wu (Fudan University) |
Curves on K3 surfaces and Mukai’s program | 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. |
| Mar 13 | Yiming Zhong (BICMR) |
On the moduli space of certain plane sextic curves | 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. |
| Mar 6 11:00-12:00 | Yin Tian (BNU) |
Higher dimensional Heegaard Floer homology and Hecke algebras | 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. |
Fall 2023
| Date | Speaker | Title | Abstract |
| Dec 27 | Fei Si
(BICMR) |
Cohomology of moduli spaces of cubic 4-folds | In this talk, we will introduce Kirwan’s package on computing the equivariant cohomology of GIT quotient and a blowup formula for intersection cohomology. As an application, the cohohomology of Kirwan’s resolution of GIT moduli space of cubic 4-folds is computed, where the resolution space has only isolated quotient singularities. On the other hand, the Kirwan’s resolution provides an explicit resolution of birational period map between GIT moduli space and Baily-Borel’s compactification of locally symmetric variety from the cubic 4-folds. Using this geometric input, we can obtain the intersection cohomology of Baily-Borel’s compactification. |
| Dec 20 | Boris Shapiro
(Stockholm Univ) |
On Hurwitz-Severi numbers | For a point p in CP^2 and a triple (g, d, l) of non-negative integers we define a Hurwitz–Severi number H_g,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. |
| Dec 13 | Yang Zhou
(SCMS) |
Mixed-Spin-P fields for GIT quotients | 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. |
| Dec 6 | Bin Wang (CUHK) |
Springer correspondence and mirror symmetry for Sp/SO Hitchin Systems | 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. |
| Nov 27 10:00am Shuangqing B626 | Yinbang Lin (Tongji Univ) |
Gromov-Witten/Pandharipande-Thomas correspondence via conifold transitions | 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. |
| Nov 22 | Ruiran Sun (McGill Univ) |
Rigidity problems on moduli spaces of polarized manifolds | 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. |
| Nov 8 |
Chuanhao Wei (Westlake Univ) |
Kodaira-type and Bott-type vanishings via Hodge theory | 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. |
| Nov 1 |
Zili Zhang (Tongji Univ) |
Simpson's correspondence and the P=W conjecture | 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. |
| Oct 27 13:30pm Shuangqing A513 | Dali Shen
(BIMSA) |
Tropicalizations of Riemann surfaces and their moduli | 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. |
| Oct 18 | Yongqiang Liu (USTC) |
L^2 type invariants of hyperplane arrangement complement | 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. |
| Oct 11 | Penghui Li (YMSC) |
Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on C^2 | 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. |
Spring 2023
| Date | Speaker | Title | Abstract |
| June 21 15:30am, Jinchunyuan West Building 3rd floor | Renjie Lyu (AMSS) |
Degeneration of Hodge structures and cubic hypersurfaces | 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. |
| June 14 | Jia Choon Lee (BICMR) |
Moduli spaces of modules over even Clifford algebra and Prym varieties | 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. |
| June 6 10:00am, Jinchunyuan West Building 3rd floor | Yao Yuan (Capital Normal University) |
Sheaves on non-reduced curves in a projective surface | 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. |
| May 31 | Zheng Zhang (ShanghaiTech University) |
Cubic threefolds with an involution and their intermediate Jacobians | 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. |
| May 17 | Emanuel Scheidegger (BICMR) |
On attractor points on the moduli space of Calabi-Yau threefolds | 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. |
| May 24 | Songyan Xie (CAS) |
Universal meromorphic functions with slow growth | 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. |
| May 10 | GADEPs focused conference II (BIMSA) |
Periods of Calabi-Yau varieties and Gromov-Witten invariants | The seminar "Geometry, Arithmetic and Differential Equations of Periods" (GADEPs), started in the pandemic year 2020 and its aim is to gather people in different areas of mathematics around the notion of periods which are certain multiple integrals. This is the second GADEPs conference focused on periods and Hodge theory of Calabi-Yau varieties with applications toward computing Gromov-Witten invariants. The study of higher genus Gromov-Witten invariants is one of the core problems in both symplectic and enumerative algebraic geometry. For Calabi-Yau threefolds, mirror symmetry invented by physicists has made remarkable conjectures on the differential structures of higher genus Gromov-Witten invariants known as BCOV. It uses the Hodge theory and periods of the mirror Calabi-Yau threefold. Recently, there has been many progress both in the mathematical framework of BCOV using enhanced Calabi-Yau varieties and also in proving these conjectures rigorously for quintic. The main aim of the workshop, is to gather experts ranging from enumerative algebraic geometry and Hodge theory to symplectic geometry. |
| May 3 | Holiday |
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| Apr 26 | Yi Xie (BICMR) |
Stable (parabolic) holomorphic vector bundles over complex curves and instanton Floer homology | 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. |
| Apr 19 | Shizhang Li (CAS) |
On the cohomology of BG | 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). |
| Apr 12 | Xiang He (YMSC, Tsinghua) |
Tropicalization of curves and applications | 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. |
| Apr 5 | Holiday |
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| Mar 29 | Zhiwei Zheng (YMSC, Tsinghua) |
Moduli spaces of hyperKahler manifolds and cubics | 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. |