202 research outputs found

    Some new results on modified diagonals

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    O'Grady studied recently mm-th modified diagonals for a smooth projective variety, generalizing the Gross-Schoen modified small diagonal. These cycles Γm(X,a)\Gamma^m(X,a) depend on a choice of reference point a∈Xa\in X (or more generally a degree 11 zero-cycle). We prove that for any X,aX,a, the cycle Γm(X,a)\Gamma^m(X,a) vanishes for large mm. We also prove the following conjecture of O'Grady: if XX is a double cover of YY and Γm(Y,a)\Gamma^m(Y,a) vanishes (where aa belongs to the branch locus), then Γ2m−1(X,a)\Gamma^{2m-1}(X,a) vanishes, and we provide a generalization to higher degree finite covers. We finally prove the vanishing Γn+1(X,oX)=0\Gamma^{n+1}(X,o_X)=0 when X=S[m]X=S^{[m]}, SS a K3K3 surface, and n=2mn=2m, which was conjectured by O'Grady and proved by him for m=2,3m=2,3.Comment: Final version, to appear in Geometry and Topolog

    Remarks on curve classes on rationally connected varieties

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    We study for rationally connected varieties XX the group of degree 2 integral homology classes on XX modulo those which are algebraic. We show that the Tate conjecture for divisor classes on surfaces defined over finite fields implies that this group is trivial for any rationally connected variety.Comment: A few typos correcte

    Hodge structures on cohomology algebras and geometry

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    We study restrictions on cohomology algebras of Kaehler compact manifolds, not depending on the h^{p,q} numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the Lefschetz property but not having the cohomology algebra of a compact Kaehler manifold. We also prove the stability of these restrictions under products.Comment: Final version, to appear in Math. Annalen 200

    Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal

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    Given a smooth projective 3-fold Y, with H3,0(Y)=0H^{3,0}(Y)=0, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Markoushevich. We then conclude that the Hodge conjecture holds for degree 4 integral Hodge classes on fibrations into cubic threefolds over curves, with restriction on singular fibers

    A geometric application of Nori's connectivity theorem

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    We prove among other things that a general Calabi-Yau hypersurface in projective space is not rationally swept out by abelian varieties of dimension greater than or equal to 2

    Green's generic syzygy conjecture for curves of even genus lying on a K3 surface

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    We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: {\it For a smooth projective curve CC of genus gg in characteristic 0, the condition CliffC>l{\rm Cliff} C>l is equivalent to the fact that Kg−l′−2,1(C,KC)=0,∀l′≤lK_{g-l'-2,1}(C,K_C)=0, \forall l'\leq l.} We propose a new approach, which allows up to prove this result for generic curves CC of genus g(C)g(C) and gonality gon(C){\rm gon(C)} in the range \frac{g(C)}{3}+1\leq {\rm gon(C)}\leq\frac{g(C)}{2}+1.$

    Unirational threefolds with no universal codimension 2 cycle

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    We prove that the general quartic double solid with k≤7k\leq 7 nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal CH0{\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

    Green's canonical syzygy conjecture for generic curves of odd genus

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    We prove the Green conjecture for generic curves of odd genus. That is we prove the vanishing Kk,1(X,KX)=0K_{k,1}(X,K_X)=0 for XX generic of genus 2k+12k+1. The curves we consider are smooth curves XX on a K3 surface whose Picard group has rank 2. This completes our previous work, where the Green conjecture for generic curves of genus gg with fixed gonality dd was proved in the range d≥g/3d\geq g/3, with the possible exception of the generic curves of odd genus.Comment: Final version to appear in Compositio Mathematic
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