**ARITHMETIC GEOMETRY LEARNING SEMINAR**

**Time: Wednesday, 11:30-1:30pm**

**Location: MATH 731**

**Speakers/Abstracts:**

**January 24 th: Feng Hao**

*Title: ***An algebraic introduction to Gauss-Manin connection
(by Katz and Oda)**In this presentation, I will give
a purely algebraic construction of Gauss-Manin connection arising
from a smooth morphism $f: X \rightarrow Y$ between two smooth
schemes over $k$, which is realized as a connection differential
in the first page of certain spectral sequence. Then using Cech
cohomology calculation, I will show that it is exactly the one
Manin originally defined, as a connecting morphism in a long exact
sequence of hyper-higher direct image sheaves of a
certain short exact sequence De Rham complexes of the
morphism $f$.

Abstract:

**February 7: Pavel Coupek**

*Title: ***Existence of Lefschetz pencils**A Lefschetz
pencil for a variety X is a tool that allows one to fibre X over
the projective line (after a blowup) in a way that the
non-smooth fibres have a unique quadratic singularity. In the
presentation, I will discuss the proof of existence of
Lefschetz pencils for smooth projective varieties in
characteristic 0.

Abstract:

*Title: ***P-adic cohomology
Abstract: **This will be some background and
motivation for Daxin's talk in the Algebraic Geometry
seminar.

*Title:*** Algebraic geometry and analytic geometry**** **A (complex)
analytic space is a ringed space that is locally the zero set
of some analytic functions on C^n. It has a topology
induced from the usual (analytic) topology of C^n, and a
structure sheaf of analytic functions. A complex variety X
gives rise to an analytic space X^{an}, and a coherent
sheaf F on X gives rise to a coherent analytic sheaf F^{an} on
X^{an}. Serre showed that, when X is projective, the
cohomology H(X,F) is isomorphic to H(X^{an},F^{an}).
Moreover, F --> F^{an} induces an equivalence of
categories Coh(X) --> Coh(X^{an}). I will talk about the
proof of these results, and some applications.

Abstract:

*Title: ***Vector
bundles and Koszul complex**The first part of
the talk will be on some basic facts about vector bundles. We
will give several equivalence definitions of vector bundles and
talk about some related topics like projective bundles. The
second part of the talk will be on Koszul complexes. In
particular, we will talk about Koszul complexes associated with
sections of vector bundles. We will give examples when the
Koszul complexes provide locally free resolutions of the zeros
of the sections. And we will probably give some applications of
such kind of results.

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