Wojciech J. Gajda

Abstract:

The Chow ring of an algebraic variety X is the Q-algebra generated by the classes of algebraic cycles on X, divided out by rational equivalence. It is graded (by the codimension of the cycles) and endowed with the intersection product. In the case where X is an abelian variety, the addition on X defines a second ring structure given by the Pontryagin product and we may also define a Fourier transform, an involution exchanging the two ring structures on the Chow ring. When X=J=J(C) is the Jacobian of a (proper and smooth) curve C, the theta divisor defines a tautological class in the Chow ring and Beauville defined the tautological ring of J to be the smallest subring of the Chow ring which is closed under all natural operations. He has given a detailed description of this ring. If C carries a non trivial automorphism σ this gives rise to other obvious classes in the Chow ring. In his thesis, Thomas Richez studied the algebra generated by these extra classes, constructing a tautological ring of (J,σ) extending Beauville's ring. In my talk I will review the work of Beauville and explain the main results obtained by my student Thomas Richez. I will also indicate the principal ideas of the proofs.