Click here for a detailed program with all the necessary references and more.
The seminar will have approximatively 13 talks divided in three different but related topics in Hodge theory. The first topic we are going to cover consists in studying the properties of the Hodge structure on the cohomology of a smooth algebraic variety over the field of complex numbers. With this aim in mind we will start by recalling some results about Kähler manifolds and the main theorems of the classical Hodge theory. Then we proceed introducing the formalism of filtered objects of an abalian category and we study some of their formal properties. At this point we start dealing with Hodge structures and study some classical examples: projective spaces, smooth projective curves, abelian varieties, K3 surfaces, Blow-ups. We then proceed with the main result of Deligne's Theorie de Hodge II which proves the exsistence of a Hodge structure on the cohomology of a smooth algebraic variety over the complex numbers, without the properness assuption. This is done combining the classical results of Hodge, some very detailed analysis of spectral sequences (coming from the formalism of filtered objects) and a combination of the results on compactification (due to Nagata) and desingularization (due to Hironaka) for schemes of finite type over the complex numbers. We plan to give some more applications of this theory following some examples of Deligne and some other explicit computations of Arapura. This part should take approximatively 6 talks.
The second part of the seminar will be devoted to understanding some deformations theory of Hodge structures. Here the seminar will definitively assume a more analytical taste: we will first review some deformation theory of complex manifolds (following Kodaira's approach) and then pass to the study of deformations of Hodge structures. One of the key resuslts we will prove is the upper semi-continuity of the Hodge numbers for a smooth family, and see how they relate to some cohomological vanishing theorem and to the Kodaira embedding theorem.
Finally the third part of the seminar will be devoted to a more advanced study of some specific topics that apply Hodge theory. For this sections we already have plenty of ideas that we are not really able to chose in advance: we will simply decide, together with the partecipants, acoording to their taste and availability of time. This part should take more or less 3 talks.
For a self contined and more detailed desription of the content of each single talk you can consuld the following program. It contains, in addition, some detailed suggestion for the speakers and some organizational remark.
We meet essentially every tuesday at 10 a.m. (it means 10 sharp), beginning from October 21st, in the tea room, which is formally known as WSC-O-3.46. Talks are supposed to be 2 full hours long and should not last more than that. After every talk we usually enjoy a brief discussion where we ask questions and clarify some points that stimulate our curiosity.
Right here you can find a preliminar list of the talks with their respective speakers and date.
For any further information about the seminar you can contact me at lorenzo.mantovani'youknowwhattoputhere(and remove the apostrophes)'uni-due.de