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On the Hilbert scheme of points of an almost complex fourfold

Claire Voisin — 2000

Annales de l'institut Fourier

If S is a complex surface, one has for each k the Hilbert scheme Hilb k ( S ) , which is a desingularization of the symmetric product S ( k ) . Here we construct more generally a differentiable variety Hilb k ( X ) endowed with a stable almost complex structure, for every almost complex fourfold X . Hilb k ( X ) is a desingularization of the symmetric product X ( k ) .

A geometric application of Nori’s connectivity theorem

Claire Voisin — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study (rational) sweeping out of general hypersurfaces by varieties having small moduli spaces. As a consequence, we show that general K -trivial hypersurfaces are not rationally swept out by abelian varieties of dimension at least two. As a corollary, we show that Clemens’ conjecture on the finiteness of rational curves of given degree in a general quintic threefold, and Lang’s conjecture saying that such varieties should be rationally swept-out by abelian varieties, contradict.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin — 2013

Annales scientifiques de l'École Normale Supérieure

We prove that Bloch’s conjecture is true for surfaces with p g = 0 obtained as 0 -sets X σ of a section σ of a very ample vector bundle on a variety X with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as 0 on holomorphic 2 -forms of  X σ , then it acts as 0 on  0 -cycles of degree 0 of  X σ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general X σ implies the...

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

Claire Voisin — 2002

Journal of the European Mathematical Society

We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: For a smooth projective curve C of genus g in characteristic 0, the condition Cliff C > l is equivalent to the fact that K g - l ' - 2 , 1 ( C , K C ) = 0 , ∀ l ' ≤ l . We propose a new approach, which allows up to prove this result for generic curves C of genus g ( C ) and gonality gon(C) in the range g ( C ) 3 + 1 ≤ gon(C) ≤ g ( C ) 2 + 1 .

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