175 research outputs found
Fully faithful Fourier-Mukai functors and generic vanishing
The aim of this mainly expository note is to point out that, given an
Fourier-Mukai functor, the condition making it fully faithful is an instance of
\emph{generic vanishing}. We test this point of view on some fairly classical
examples, including the strong simplicity criterion of Bondal and Orlov, the
standard flip and the Mukai flop.Comment: In memory of Alexandru T. Lasc
Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality
We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis
inequality for surfaces. The proof is based on the theory of Generic Vanishing
and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative
algebra. Along the way we give a positive answer, in the setting of K\"ahler
manifolds, to a question of Green-Lazarsfeld on the vanishing of higher direct
images of Poincar\'e bundles. We indicate generalizations to arbitrary
Fourier-Mukai transforms.Comment: 12 pages; some improvements according to suggestions from the
referees, to appear in Duke Math.
Wealth distribution and collective knowledge. A Boltzmann approach
We introduce and discuss a nonlinear kinetic equation of Boltzmann type which
describes the influence of knowledge in the evolution of wealth in a system of
agents which interact through the binary trades introduced in Cordier,
Pareschi, Toscani, J. Stat. Phys. 2005. The trades, which include both saving
propensity and the risks of the market, are here modified in the risk and
saving parameters, which now are assumed to depend on the personal degree of
knowledge. The numerical simulations show that the presence of knowledge has
the potential to produce a class of wealthy agents and to account for a larger
proportion of wealth inequality.Comment: 21 pages, 10 figures. arXiv admin note: text overlap with
arXiv:q-bio/0312018 by other author
Cohomological rank functions on abelian varieties
Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this
paper we consider cohomological rank functions of -twisted
(complexes of) coherent sheaves on abelian varieties. They satisfy a natural
transformation formula with respect to the Fourier-Mukai-Poincar\'e transform,
which has several consequences. In many concrete geometric contexts these
functions provide useful invariants. We illustrate this with two different
applications, the first one to GV-subschemes and the second one to
multiplication maps of global sections of ample line bundles on abelian
varieties.Comment: 28 pages, minor changes. Final version to appear on Annales Scient.
EN
Generic vanishing and minimal cohomology classes on abelian varieties
We establish a, and conjecture further, relationship between the existence of
subvarieties representing minimal cohomology classes on principally polarized
abelian varieties, and the generic vanishing of certain sheaf cohomology. The
main ingredient is the Generic Vanishing criterion of math.AG/0608127, based on
the Fourier-Mukai transform.Comment: 12 pages; final version, to appear in Math. Annalen; contains
expository changes and a proof in the case of abelian varieties of the
generic vanishing criterion we use, according to suggestions from the refere
GV-sheaves, Fourier-Mukai transform, and Generic Vanishing
We use homological methods to establish a formal criterion for Generic
Vanishing, in the sense originated by Green and Lazarsfeld and pursued further
by Hacon and the first author, but in the context of an arbitrary Fourier-Mukai
correspondence. For smooth projective varieties we apply this to deduce a
Kodaira-type generic vanishing theorem for adjoint bundles of the form with a nef line bundle, and in fact a more general generic Nadel-type
vanishing theorem for multiplier ideal sheaves. Still in the context of the
Picard variety, the same method generates various other generic vanishing
results, by reduction to standard vanishing theorems. We further use the formal
criterion in order to address examples related to generic vanishing on higher
rank moduli spaces (on curves and on some threefold Calabi-Yau fiber spaces).Comment: 28 pages; many more corrections and improved statements with respect
to previous versions; especially fixed some inaccuracies pointed out by the
referees when working with singular varieties,and extended the general
results to the Cohen-Macaulay case. Final version, to appear in Amer. J.Mat
Regularity on abelian varieties I
We introduce the notion of Mukai regularity (M-regularity) for coherent
sheaves on abelian varieties. The definition is based on the Fourier-Mukai
transform, and in a special case depending on the choice of a polarization it
parallels and strenghtens the usual Castelnuovo-Mumford regularity. Mukai
regularity has a large number of applications, ranging from basic properties of
linear series on abelian varieties and defining equations for their
subvarieties, to higher dimensional type statements and to a study of special
classes of vector bundles. Some of these applications are explained here, while
others make the subject of upcoming papers.Comment: 18 pages; final version, with substantial changes in the order of
presentation in Section 2, and other minor expository changes, as suggested
by the refere
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