08w5106 Hodge Theory

Arriving Sunday, April 6 and departing Friday, April 11, 2008

Organizers: Patrick Brosnan (University of British Columbia), Mark Green (University of California, Los Angeles), Ludmil Katzarkov (University of Miami), Gregory Pearlstein (Michigan State University).

Confirmed Participants

Press Release: Recent Progress on Hodge Theory

Information for Participants

Preliminary Programme

Final Report (PDF file)

Objectives

Title:

Objectives:

We propose the following two topics as the main ones for the conference:

(1) Recent results related to the Hodge conjecture and algebraic cycles. (2) Recent results concerning the asymptotics of variations of mixed Hodge structure.

In 1920, Lefschetz proved that every integral Hodge class of type (1,1) on a smooth projective surface could be represented via the fundamental class of an algebraic cycle. In his 1950 ICM address, Hodge conjectured that every integral (p,p) class on as smooth projective variety was the fundamental class of a codimension p algebraic cycle. This conjecture was later amended by the work of Atiyah and Hirzebruch who showed that one must use rational rather than integral coefficients. However, other than that, it has remained open ever since.

Efforts to prove the Hodge conjecture by extending Lefschetz approach to the (1,1)-theorem using normal functions informed the initial theory of the variations of Hodge structures and their degenerations by Griffiths school in the 1970\'s. However, it was eventually discovered that progress on the Hodge conjecture via this route is blocked by the failure of Jacobi inversion for for higher codimension cycles.

Recently work of Phillip Griffiths, Mark Green, Richard Thomas, Herb Clemens and others has shown that obstacle can be avoided by replacing Lefschetz pencils with families of hyperplane sections of sufficiently high degree.

Working from a completely different view point, Kazuya Kato, Sampei Usui and others have completed Griffiths dream of constructing good, Hodge theoretic partial compactifications of the arithmetic quotients of Griffiths period domains by marrying the original work of Griffiths school on degenerations of Hodge structure with the new theory of log geometry.

One of the primary goals of this conference is to bring together researchers working on applications of Hodge theory to algebraic geometry, such as Griffiths and Green, with researchers working on topics internal to Hodge theory, such as Kato and Usui\'s work on compactifications of period domains. In particular, while there are conferences fairly regularly on the subject of algebraic cycles and also on the applications of Hodge theory to such subjects as Gromov-Witten theory (e.g., the conference on Calabi-Yau varieties and Mirror Symmetry at Banff in 2003), conferences on subjects internal to Hodge theory have been rather rare. The last one in North America was the conference on Hodge theory and Log geometry at Japanese American Mathematical Institute at Johns Hopkins University in Baltimore. However, as illustrated above, there is a great need for conferences in this field because the theory is experiencing renewed progress on many fronts.

This conference will also have a large component emphasizing the training of graduate students and postdocs working in the area. We will reserve several slots for younger mathematicians getting started in the area. We also feel that it is important to include women and minorities.