Geometry Seminar

Let F(x_1,x_2) denote a homogeneous binary form of order $d$. Assume that d factors as d= rm. The Hilbert covariant H_{r,d}(F) is a binary form (whose coefficients are polynomials in the coefficients of F) with the following property: H_{r,d}(F) vanishes identically, exactly when F is a perfect m-th power of an order r form. It was constructed by Hilbert in 1885; and in particular, H_{1,d}(F) is the Hessian of F. In geometric terms, we have an SL_2-equivariant imbedding of P^r into P^d, and the coefficients of H_{r,d} give a set of defining equations for the image variety. These covariants are part of a beautiful classical landscape called the 'invariant theory of binary forms', whose origins lie in the mid-19th century. In this talk, I will exhibit two entirely different approaches to the construction of H, and outline a proof of the fact that they lead to the same object. I will also mention some results and problems about the ideal generated by the coefficients of H. All of this is joint work with Abdelmalek Abdesselam from the University of Virginia.

Except a few special cases (e.g. abelian varieties and K3 surfaces), the images of period maps for families of algebraic varieties satisfy non-trivial Griffiths' transversality relations. It is of interest to understand these images of period maps, especially for Calabi-Yau threefolds. In this talk, I will discuss the case when the images of period maps can be described algebraically. Specifically, I will show that if a horizontal subvariety Z of a period domain D is semi-algebraic and is stabilized by a large discrete group, then Z is automatically Hermitian symmetric with a totally geodesic embedding into the period domain D. Additionally, I will discuss the classification of the semi-algebraic cases for variations of Hodge structures of Calabi-Yau type. This is joint work with R. Friedman.