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

Location: MATH 731





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 $f$.

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.

February 21st: Daxin Xu

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

March 7th: Joe Knight

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.

April 4th: Heng Du

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.

April 18th: Donu Arapura