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Mini-Workshop in Algebraic Geometry

Mini-Workshop in Algebraic Geometry

 

Time201979

Room:管理科研楼1308

 

Schedule

09:00-10:00

TitleRationality of Multi-variable Poincare Series

SpeakerXi Chen  (University of Alberta)

AbstractZariski conjectured that the Poincare series of a divisor on a smooth projective surface is rational. This was proved by Cutkosky and Srinivas in 1993. We are considering a generalization of this statement to multi-variable Poincare series. This is a joint work with J. Elizondo.

Tea Break

10:30-11:30

TitleModuli of symmetric cubic fourfolds and nodal sextic curves

SpeakerChenglong Yu  (University of Pennsylvania)

AbstractPeriod map is a powerful tool to study geometric objects related to K3 surfaces and cubic 4-folds. In this talk, we focus on moduli of cubic 4-folds and sextic curves with specified symmetries and singularities. We identify the geometric (GIT) compactifications with the Hodge theoretic (Looijenga, mostly Baily-Borel) compactifications of locally symmetric varieties. As a corollary, the algebra of GIT invariants is identified with the algebra of automorphic forms on the corresponding period domains. One of the key inputs is the functorial property of semi-toric compactifications of locally symmetric varieties. Our work generalizes results of  Matsumoto-Sasaki-Yoshida, Allcock-Carlson-Toledo, Looijenga-Swierstra and Laza-Pearlstein-Zhang. This is joint work with Zhiwei Zheng.

Lunch


13:30-14:30

TitleThe L2 representation of intersection cohomology

SpeakerJunchao Shentu  (University of Science&Technology of China)

AbstractThe intersection cohomology is introduced by Goresky MacPherson as a cohomology theory on singular spaces (e.g. algebraic varieties) that satisfies the Poincare duality. After series of works by Goresky-MacPherson, Steenbrink, Beilinson- Bernstein- Deligne-Gabber, M. Saito and Cataldo-Migliorini, the whole (absolute and relative) Hodge-Lefschetz theorems are established. This makes intersection cohomology the most natural cohomology theory which is pure in the sense of Hodge theory. In this talk I will explain how to use differential forms to represent the intersection complex on an algebraic variety, at least when the variety admits only equi-singularities. This is an ongoing project, joint with Chen Zhao, willing to fill the analytic part (the de Rham Theorem) of the Hodge theory of the intersection cohomology.

14:40-15:40

TitleCone spherical metrics, 1-forms and stable vector bundles on Riemann surfaces

SpeakerJijian Song  (Center for Applied Mathematics, Tianjin University)

AbstractA cone spherical metric on a compact Riemann surface X is a conformal metric of constant curvature +1 with finitely many conical singularities. The singularities of the metric can be described by a real divisor D. An open question called Picard-Poincar′e problem is whether there exists a cone spherical metric for properly given (X; D) such that the singularities of the metric are described by the divisor D. In this talk, I will report an existence result of meromorphic 1-forms with real periods on Riemann surface and an angle constraint for reducible metrics on the Riemann sphere. For the irreducible metrics, by using projective structures, we prove that if D is effective, then they always can be obtained from rank 2 stable vector bundles with line subbundles. At last, I will talk about how to construct a special class of cone spherical metrics by Strebel differentials. This is a joint work with Yiran Cheng, Bo Li, Lingguang Li and Bin Xu.

Tea Break

16:00-17:00

TitleLyubeznik numbers of irreducible projective varieties

SpeakerBotong Wang  (University of Wisconsin-Madison)

AbstractLyubeznik numbers are invariants of singularities that are defined algebraically, but has topological interpretations. In positive characteristics, it is a theorem of Wenliang Zhang that the Lyubeznik numbers of the cone of a projective variety do not depend on the choice of the projective embedding. Recently, Thomas Reichelt, Morihiko Saito and Uli Walther related the problem with the failure of Hard Lefschetz theorem for singular varieties. And they constructed examples of reducible complex projective varieties whose Lyubeznik numbers depend on the choice of projective embeddings. I will discuss their works and a generalization to irreducible projective varieties.

 


OrganizersMao Sheng  (University of Science and Technology of China)

SponsorsSchool of Mathematical Sciences, USTC



  科大主页 | 国家数学与交叉科学中心(合肥) | 中科院吴文俊数学重点实验室 |
中科院数学与系统科学研究院 | 北京国际数学研究中心 | 安徽省数学会