NATIONAL SCIENCE FOUNDATION
TOKYO REGIONAL OFFICE
The National Science Foundation's (NSF) Tokyo Office periodically receives and disseminates reports on research developments in Japan that are related to the Foundation's mission. NSF-sponsored researchers currently working in Japan prepare many of these reports. These reports present information for use by NSF program managers and policy makers; they are not statements of NSF policy.
Special Scientific Report #01-03 (September 5, 2001)
Algebraic cycles on degenerating families of abelian varieties.
The following report was prepared by B. Brent Gordon, Program Manager in the National Science Foundation's Division of Mathematical Science. Dr. Gordon traveled to Japan in July and August 2001 under a Japan Society for the Promotion of Science (JSPS) Short-Term Invitational Fellowship. Professor M. Hanamura at Kyushu University served as his host.
Dr. Gordon assessment on the state of mathematics in Japan appears as Tokyo Report Memorandum #01-11. He may be reached at bgordon@nsf.gov.
For the first two weeks of my three week stay in Japan I collaborated with my host, Professor M. Hanamura, at Kyushu University in Fukuoka, and Professor J.P. Murre, visiting from Leiden University, on mathematical research concerning algebraic cycles in degenerating families of abelian varieties.
The goal of my collaboration with Hanamura and Murre is to prove the Corti-Hanamura “Motivic Decomposition Conjecture” as well as Murre’s conjecture on the existence of Chow-Künneth decompositions for the case of degenerating families of abelian varieties. To explain the conjectures, let X denote a smooth projective variety of dimension n over the complex numbers. Then with composition of correspondences as the multiplication the middle Chow group CHn(X x X, Q) is a ring, and Murre conjectures the existence of mutually orthogonal idempotents p0 , ..., p2n whose sum is the identity, i.e., the class of the diagonal, such that modulo homological equivalence pj is the jth Künneth component of the diagonal. This can be considered a generalization of the well-known conjecture going back at least to Grothendieck that the Künneth components of the diagonal (modulo homolological equivalence) should be algebraic. Now suppose S denotes a projective variety of dimension d, not necessarily smooth, and p : X ® S is a surjective projective morphism. Then the “Topological Decomposition Theorem” of Beilinson, Bernstein and Deligne describes the cohomology of X in terms of the intersection cohomology of S with local coefficient systems over the strata of S. In this context the Corti-Hanamura conjecture asserts that there exist mutually orthogonal idempotent elements in CHn(X xS X, Q) whose sum is the class of the diagonal and which, modulo homological equivalence, induce the same decomposition of the cohomology of X as given by the Topological Decomposition Theorem. We sometimes refer to this as a “relative Chow-Künneth decomposition” in the sense that X is a “relative” scheme over S.
Prior to our meeting this summer Hanamura, Murre and I had already written and submitted a paper where we verified the Motivic Decomposition Conjecture under the conditions that S has isolated singularities, the generic fiber of p is an abelian variety, and all of the components of the special fibers of p are toric varieties. In the two weeks we had together this summer the first thing we did was to (nearly) finish writing a paper in which we prove that in the situation of the previous paper plus some additional hypotheses it becomes possible to deduce an “absolute” Chow-Künneth decomposition from a given relative Chow-Künneth decomposition; and moreover, show that there are interesting examples where all the hypotheses are satisfied. More precisely, with our current technology in addition to the previous hypotheses we must assume: that p factors through a resolution of singularities T ® S; that T has a Chow-Künneth decomposition; that the divisors of T over the singularities of S are configurations of toric varieties; that the intersection cohomology of S with coefficients in a nontrivial, irreducible local coefficient system vanishes except possibly in the middle degree d; and that the monodromy invariants in the cohomology of a generic fiber of p are algebraic cycles. Some interesting examples where all these hypotheses are satisfied are the smooth toroidal compactifications of fiber products of families of abelian varieties parameterized by Hilbert modular varieties (where an elliptic modular curve is a special case of a Hilbert modular variety).
In our last few days together we considered how we might generalize or eliminate some of the hypotheses from the first theorem, that gives the relative decomposition. The most critical issue is the assumption that the special fibers of p be configurations of toric varieties, since this is an artificial assumption essential only for our present methods. The general case is that the special fibers may be (toroidally) compactified semi-abelian varieties, or equivalently, toric variety bundles over abelian varieties. At first we considered 1-motives as a substitute for semi-abelian varieties, but that did not appear to be a profitable line of investigation. Then Hanamura proved an analogue of the projective bundle formula for the Chow groups of toric variety bundles, namely that the Chow group of a toric variety bundle is the tensor product of the Chow group of the base with the Chow group of the toric variety. It is not yet clear how this result will be helpful, but it is a nice proposition, and I am optimistic.