202 research outputs found

### Some new results on modified diagonals

O'Grady studied recently $m$-th modified diagonals for a smooth projective
variety, generalizing the Gross-Schoen modified small diagonal. These cycles
$\Gamma^m(X,a)$ depend on a choice of reference point $a\in X$ (or more
generally a degree $1$ zero-cycle). We prove that for any $X,a$, the cycle
$\Gamma^m(X,a)$ vanishes for large $m$. We also prove the following conjecture
of O'Grady: if $X$ is a double cover of $Y$ and $\Gamma^m(Y,a)$ vanishes (where
$a$ belongs to the branch locus), then $\Gamma^{2m-1}(X,a)$ vanishes, and we
provide a generalization to higher degree finite covers.
We finally prove the vanishing $\Gamma^{n+1}(X,o_X)=0$ when $X=S^{[m]}$, $S$
a $K3$ surface, and $n=2m$, which was conjectured by O'Grady and proved by him
for $m=2,3$.Comment: Final version, to appear in Geometry and Topolog

### Remarks on curve classes on rationally connected varieties

We study for rationally connected varieties $X$ the group of degree 2
integral homology classes on $X$ 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

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

Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, 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

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

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

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

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

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

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