Automorphisms of surfaces of general type with q>=2 acting trivially in cohomology

A compact complex manifold X is said to be rationally cohomologically rigidified if its automorphism group Aut(X) acts faithfully on the cohomology ring H*(X,Q). In this note, we prove that, surfaces of general type with irregularity q>2 are rationally cohomologically rigidified, and so are minimal surfaces S with q=2 unless K^2=8X. This answers a question of Fabrizio Catanese in part. As examples we give a complete classification of surfaces isogenous to a product with q=2 that are not rationally cohomologically rigidified. These surfaces turn out however to be rigidified.Comment: 18 pages; a remark and a closely relevant reference are adde

Niveau and coniveau filtrations on cohomology groups and Chow groups

The Bloch-Beilinson-Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology -- i.e. the cohomology -- of $X$. In this paper, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of $X$ has an incidence on the Chow groups of $X$. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in low-dimensional cases or when a variety is known to have small Chow groups. For instance, we give a new example of a fourfold of general type with trivial Chow group of zero-cycles and we prove Murre's conjectures for threefolds dominated by a product of curves, for threefolds rationally dominated by the product of three curves, for rationally connected fourfolds and for complete intersections of low degree. The BBM conjectures are closely related to Kimura-O'Sullivan's notion of finite-dimensionality. Assuming the standard conjectures on algebraic cycles the former is known to imply the latter. We show that the missing ingredient for finite-dimensionality to imply the BBM conjectures is the coincidence of a certain niveau filtration with the coniveau filtration on Chow groups.Comment: Final versio

Weak approximation over function fields

We prove that rationally connected varieties over the function field of a complex curve satisfy weak approximation for places of good reduction.Comment: 22 page

Unirational threefolds with no universal codimension 2 cycle

We prove that the general quartic double solid with $k\leq 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal ${\rm CH}_0$ group. The same holds if we replace in this statement "Chow theoretic" by "cohomological". In particular, it is not stably rational. We also prove that the general quartic double solid with seven nodes does not admit a universal codimension 2 cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibres of its Jacobian by a family of 1-cycles. This implies that its third unramified cohomology group is not universally trivial.Comment: Final version to appear in Invent. Mat