2,778 research outputs found
Higher vector bundles and multi-graded symplectic manifolds
A natural explicit condition is given ensuring that an action of the
multiplicative monoid of non-negative reals on a manifold F comes from
homotheties of a vector bundle structure on F, or, equivalently, from an Euler
vector field. This is used in showing that double (or higher) vector bundles
present in the literature can be equivalently defined as manifolds with a
family of commuting Euler vector fields. Higher vector bundles can be therefore
defined as manifolds admitting certain -grading in the structure sheaf.
Consequently, multi-graded (super)manifolds are canonically associated with
higher vector bundles that is an equivalence of categories. Of particular
interest are symplectic multi-graded manifolds which are proven to be
associated with cotangent bundles. Duality for higher vector bundles is then
explained by means of the cotangent bundles as they contain the collection of
all possible duals. This gives, moreover, higher generalizations of the known
`universal Legendre transformation' T*E->T*E*, identifying the cotangent
bundles of all higher vector bundles in duality. The symplectic multi-graded
manifolds, equipped with certain homological Hamiltonian vector fields, lead to
an alternative to Roytenberg's picture generalization of Lie bialgebroids,
Courant brackets, Drinfeld doubles and can be viewed as geometrical base for
higher BRST and Batalin-Vilkovisky formalisms. This is also a natural framework
for studying n-fold Lie algebroids and related structures.Comment: 27 pages, minor corrections, to appear in J. Geom. Phy
The Baum-Connes Conjecture via Localisation of Categories
We redefine the Baum-Connes assembly map using simplicial approximation in
the equivariant Kasparov category. This new interpretation is ideal for
studying functorial properties and gives analogues of the assembly maps for all
equivariant homology theories, not just for the K-theory of the crossed
product. We extend many of the known techniques for proving the Baum-Connes
conjecture to this more general setting
Twisted Fourier-Mukai functors
Due to a theorem by Orlov every exact fully faithful functor between the
bounded derived categories of coherent sheaves on smooth projective varieties
is of Fourier-Mukai type. We extend this result to the case of bounded derived
categories of twisted coherent sheaves and at the same time we weaken the
hypotheses on the functor. As an application we get a complete description of
the exact functors between the abelian categories of twisted coherent sheaves
on smooth projective varieties.Comment: 16 pages. Minor changes. Final version to appear in Adv. Mat
Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's
It is well known that the Fano scheme of lines on a cubic 4-fold is a
symplectic variety. We generalize this fact by constructing a closed p-form
with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y
of degree n. We provide several definitions of this form - via the Abel-Jacobi
map, via Hochschild homology, and via the linkage class, and compute it
explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show
that the Fano scheme is birational to a certain moduli space of sheaves on a
p-dimensional Calabi--Yau variety X arising naturally in the context of
homological projective duality, and that the constructed form is induced by the
holomorphic volume form on X. This remains true for a general non Pfaffian
hypersurface but the dual Calabi-Yau becomes non commutative.Comment: 34 pages; exposition of Hochschild homology expanded; references
added; introduction re-written; some imrecisions, typos and the orbit diagram
in the last section correcte
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