Mini-Workshop in Algebraic Geometry
Title：Rationality of Multi-variable Poincare Series
Speaker：Xi Chen (University
Abstract：Zariski 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.
Title：Moduli of symmetric cubic fourfolds and nodal sextic curves
Speaker：Chenglong Yu (University of Pennsylvania)
Abstract：Period 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.
Title：The L2 representation of intersection cohomology
Speaker：Junchao Shentu (University of Science&Technology of China)
Abstract：The 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.
Title：Cone spherical metrics, 1-forms and
stable vector bundles on Riemann surfaces
Speaker：Jijian Song (Center for Applied Mathematics, Tianjin
Abstract：A 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.
Title：Lyubeznik numbers of
irreducible projective varieties
Speaker：Botong Wang (University of Wisconsin-Madison)
Abstract：Lyubeznik 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.
Organizers：Mao Sheng (University of Science and
Technology of China)
Sponsors：School of Mathematical Sciences, USTC