2008-7-10 Decomposition Theorem and the Topology of Algebraic Maps(II)

北大数院学术报告

Decomposition Theorem and the Topology of Algebraic Maps(II)

报告人：Prof. Li Li (Univ. of Illinois at Urbana-Champaign, USA)

时间：2008-07-10 下午 2:30 - 3:30

地点：北京大学资源大厦1328

Abstract: These talks give an introduction to the intersection cohomology, perverse sheaf and the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. I will first focus on two concrete examples: the resolution of surface singularities and the map from a surface to a curve. Then we will talk on perverse sheaves and the general form of the Decomposition Theorem. Finally the geometrical proof of the theorem by de Cataldo and Migliorini using intersection forms will sketched.

2008-07-09 Li Li: Decomposition Theorem and the Topology of Algebraic Maps(I)

北大数院学术报告

报告人：Prof. Li Li (Univ. of Illinois at Urbana-Champaign, USA)

时间：2008-07-09 上午 9:00 - 11:30

地点：北京大学资源大厦1328

Abstract: These talks give an introduction to the intersection cohomology, perverse sheaf and the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. I will first focus on two concrete examples: the resolution of surface singularities and the map from a surface to a curve. Then we will talk on perverse sheaves and the general form of the Decomposition Theorem. Finally the geometrical proof of the theorem by de Cataldo and Migliorini using intersection forms will sketched.

2008-07-07 Noriko Yui: On the modularity of certain K3 fibered Calabi-Yau threefolds

晨兴-清华学术报告

报告人：Prof.Noriko Yui
Queen’s University

报告内容：On the modularity of certain K3 fibered Calabi-Yau threefolds

时间：7月7日，(周一)
4：30-5：30 pm

地点：晨兴110室

Abstract
We will discuss the modularity of Galois representations associated to Calabi–
Yau threefolds over Q. We will focus on Calabi–Yau threefolds over Q which are
non-rigid, that is, those Calabi–Yau threefolds with the third Betti number B
3
> 2.
When a Calabi–Yau threefold is non-rigid, the dimension B
3
of the Galois repre-
sentation gets rather large, and the modularity question poses a serious challenge.
We will construct explicit examples of non-rigid Calabi–Yau threefolds fibered
over P
1
by non-constant semi-stable K3 surfaces and reaching the Arakelov–Yau
upper bound. For these examples, we prove that the “interesting” part of their
L-series do come from modular forms.
Further, we will consider more general K3-fibered Calabi–Yau threefolds over Q
than the above examples. For these Calabi–Yau threefolds, we decompose Calabi–
Yau threefolds into motives. Then we address the modularity question for some of
the motives. We will compute the motivic zeta-functions and L-series, and establish
the modularity in a number of examples.
We also discuss arithmetic mirror symmetry for these Calabi-Yau threefolds.
This is a joint work in progress with Y. Goto and R. Kloosterman.