WORKSHOP ON ALGEBRAIC CYCLES

Ohio State University

Columbus, Ohio

November 7-8, 2009

Titles and Abstracts of Talks

James Lewis - University of Alberta Title: An Archimedean height pairing on the equivalence relation defining Bloch's higher Chow groups Abstract: The existence of a height pairing on the equivalence relation defining Bloch's higher Chow groups is a surprising consequence of some recent joint work by myself and Xi Chen on a nontrivial $K_1$-class on a self-product of a general $K3$ surface. I will explain how this pairing comes about.
Patrick Brosnan- University of British Columbia Title: The zero locus of an admissible normal function Abstract: Let S be an algebraic variety and H a variation of pure Hodge structure on S of odd weight. Then H induces over S a family J(H) of compact complex tori whose fibers are called Griffiths intermediate Jacobians. Admissible normal functions are certain sections of J(H) which are well-behaved at infinity. Since J(H) is an analytic variety, it is obvious that the locus where a normal function is zero is analytic in S. However, it is far from clear that it is algebraic because, in general, the Griffiths intermediate Jacobian is not an algebraic object and, therefore, normal functions cannot be thought of as exactly algebraic objects. I will discuss joint work with Gregory Pearlstein in which we prove that, in fact, the zero locus of an admissible normal function over an algebaic base is algebraic.
S.Mahanta - Johns Hopkins University Title: F_1-geometry and Galois extensions Abstract: The motivation for developing a geometry over F_1 (the field with one element) arises, on the one hand, from Tits' early work on Chevalley group schemes and, on the other, from some considerations in arithmetic geometry. After a brief survey of some of the approaches towards F_1 geometry I shall discuss a notion of finite Galois extensions over F_1 and, if time permits, outline a possible construction of the Witt vectors.
G.V.Ravindra- University of Missouri, St. Louis Title: Curves and bundles on threefolds Abstract: The Noether-Lefschetz theorem says that for a "very general" hypersurface X of degree at least 4 in P^3, any curve C in X occurs as an intersection of X and S where S is another surface in P^3. Motivated by this, Griffiths and Harris asked whether any curve C in X, where X is now a general hypersurface of degree at least 6 in P^4, is an intersection of X with a surface S in P^4. C. Voisin showed that there exists curves which are not of this form. One would still be interested to know whether there is indeed a "generalised Noether-Lefschetz theorem" in this situation. We will explain how arithmetically Cohen-Macaulay (ACM) curves and bundles provide an answer in this direction. 1) In the first part, we shall show that ACM curves provide examples of curves which are not intersections as above and show that Voisin examples can be understood as special cases. 2) We will then sketch a proof of the following characterisation of complete intersection curves on a general smooth projective hypersurface of dimension three and degree at least six: " a curve in such a hypersurface is a complete intersection if and only if it is arithmetically Gorenstein (i.e. it is the zero locus of a non-zero section of a rank two bundle with vanishing intermediate cohomology). " Apart from the obvious motivation for such a theorem, we shall show a) how this theorem can be thought of as a generalisation of the classical Noether-Lefschetz theorem for curves on two dimensional hypersurfaces. b) that this implies the following interesting fact: a "general" homogeneous polynomial in five variables of degree at least six cannot be obtained as the Pfaffian (square root of the determinant) of an even sized "minimal" skew-symmetric matrix with homogeneous polynomial entries. c) Relate it to Horrocks' criteria for a vector bundle on projective space to be split, and finally d) show that it can be viewed as a verification of a (strengthening of a) conjecture of Buchweitz-Greuel-Schreyer in commutative algebra and algebraic geometry. This is based on joint work with N.Mohan Kumar and A.P.Rao in St. Louis and I.Biswas and J.Biswas in India.