Abstracts

The Hodge structure on the (co)homology of a smooth compact complex algebraic variety is a collection of linear algebraic data that provides many obstructions and insights about its topology. In the seventies, Deligne showed that the theory can be extended to both non-smooth and non-compact varieties by expanding the notion of Hodge structure to that of a mixed Hodge structure, enabling these tools to be used in a much broader setting. In this course, we will give an introduction to Deligne’s theory of mixed Hodge complexes of sheaves in the setting of smooth non-compact varieties, as well as talk about recent developments using these techniques.

Given a normal crossings degeneration f:(X, ωX ) → ∆ of complex Kähler manifolds, in recent joint work with T. Pelka, we have shown how to associate a smooth locally trivial fibration fA : A → ∆log over the real blow-up of the disc ∆. It is moreover endowed with a closed 2-form ωA giving it the structure of a symplectic fibration. The restriction of ωA to every fibre of fA at positive radius (that is, over a point of ∆\{0}) is the modification by a potential of the restriction of ωX to the same fibre. The construction is so that:

(1)  We can produce symplectic representatives of the monodromy with very special dynamics, and based on this and on a spectral sequence due to McLean, prove the family version of Zariski’s multiplicity conjecture.
(2)  If f is a maximal Calabi-Yau degeneration we can produce Lagrangian torus fibrations over the complement of a codimension 2 set over the (expanded) essential skeleton of the degeneration, satisfying many of the properties conjectured by Kontsevich and Soibelman.

In this lecture series, I will highlight the main aspects of the construction, and at the end of it, depending on time, we will discuss some of the applications (1) and/or (2).

When X is a finite dimensional representation on which a linear algebraic group G acts with finitely many orbits, the equivariant Riemann-Hilbert correspondence implies that the simple G-equivariant DX-modules are classified by the irreducible G-equivariant local systems on the orbits of the G-action. If in addition all stabilizers are connected, then all such local systems are trivial, and therefore there is a one to one correspondence between orbits O and simple equivariant DX-modules. The simple DX-module corresponding to O is denoted L(O,X) and called the intersection homology D-module associated to O. Articulated by MacPherson and Vilonen, it is an important problem to construct L(O,X) explicitly, as well as to realize explicitly the category of G-equivariant DX-modules. I will introduce the basic theory and discuss further applications, focusing on the case when X is a space of matrices and G is the natural symmetry group acting via row and column operations.

Take a rational normal curve of even degree 2n, and consider its secant varieties, especially the one swept out by the secant (n − 1)-planes, which is a hypersurface. In these lectures, my plan is to describe how to compute several interesting invariants of this projective hypersurface (and of the cone over it in affine space): singular and intersection cohomology, nearby and vanishing cycles, and (maybe) the Bernstein-Sato polynomial. One goal is to demonstrate in a special case how one can do such computations. Most of what I am going to say is based on the Ph.D. thesis of my student Dan Brogan.