SCIENTIFIC PROGRAMS AND ACTIVITIES

An introduction to Hodge theory

We give an introduction to the basic concepts of Hodge theory, including the notion of a pure Hodge structure and the Hodge filtration. We then discuss the uses of this theory in the study of cohomology, including the Hodge decomposition and the Lefschetz decomposition.

Introduction to nonabelian Hodge theory: Higgs bundles and local sytems I

Hodge theory can be extended to cohomology with coefficients in nonabelian groups. For GLr, this results in a correspondence between flat vector bundles (which, by the Riemann-Hilbert correspondence, are the same as local systems), and so-called Higgs bundles. Over smooth projective varieties, the latter are not only holomorphic, but in fact algebraic, objects. We will discuss this correspondence and how it is useful (among other things) for constructing variations of Hodge structure.

The Kuga–Satake construction

There is a well known construction of Kuga and Satake which embeds the transcendental Hodge structure of any algebraic K3 surface into the second cohomology group of an abelian variety. I will give an overview of this construction and show how it can be turned into an explicit geometric correspondence in some situations.

Algebraic and arithmetic properties of period maps

The three talks will cover Mumford-Tate groups and boundary components, as well as limits of normal
functions and generalized Abel-Jacobi maps.

Classical period domains

I will discuss the classification of Hermitian symmetric domains, the connection between HSD’s and VHS, and some examples of moduli spaces uniformized by HSD’s.

Period domains and their differential geometry revisited

Griffiths’ period domains classify polarized Hodge structures; they have a pure Lie-theoretic description as reductive domains. This can be used to translate differential geometric properties on associated bundles into properties for associated Lie-algebras. In particular, this gives an explanation for the curvature properties for the natural invariant metric on such domains. These results were all known in the seventies of the last century and due to Griffiths and Schmid. They obtained them making heavy use of detailed Lie-theory. The proposed approach avoids this. Mixed Hodge structures can also be described by period domains, but these are no longer reductive. The transformation group acting transitively on such a domain is no longer semi-simple and the natural metric in general is no longer invariant. This complicates the curvature calculations. Nevertheless, in special cases which are of interest in geometric applications one can deduce some properties analogous to what happens in the pure case. The description of mixed period domain was also known for some time and is due to Usui and Kaplan. The curvature calculations were started by Pearlstein. In a joint work in progress we are extending this result and give some applications.
It is my intention to explain this in 2 lectures focusing mainly on the pure case.

Introduction to nonabelian Hodge theory: Higgs bundles and local systems II

We will discuss interesting geometric aspects of the moduli spaces for the objects introduced in Part I
in Garcia–Raboso’s talk.

Degenerations of Hodge structures

This talk concerns the behaviour of geometric variations of Hodge structures near singular fibers in a family as studied by Schmid-Steenbrink. We define the canonical extension of a vector bundle with connection from the punctured disk to the disk and then extend variations of Hodge structures by extending the Gauss-Manin connection. The limiting object to be filled in is a mixed Hodge structure. We define nearby and vanishing cycles sheaves and state various properties of these.

[1] Picard numbers of quintic surfaces

The Picard number is a non-trivial invariant of an algebraic surface which captures much of its inner structure. It is a fundamental problem which Picard numbers occur within a given class of surfaces. For the prototype example of quintics in P3, I will show that all numbers 1 and 45 indeed occur as Picard numbers. The main technique consists in arithmetic deformations.

In a 1943 paper, Benjamino Segre claimed that a smooth complex quartic surface contains at most 64 lines. However, his arguments turn out to be incomplete, and at some places wrong. I will present joint work with S. Rams which uses elliptic fibrations to give a complete proof of the corresponding statement over any field of characteristic other than 2 and 3.

Variations of Hodge structure and the period map

We begin by defining the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection.