2,652 research outputs found
Canonical equivariant extensions using classical Hodge theory
Lin and Sjamaar have used symplectic Hodge theory to obtain canonical
equivariant extensions for Hamiltonian actions on closed symplectic manifolds
that have the strong Lefschetz property. Here we obtain canonical equivariant
extensions much more generally by means of classical Hodge theory.Comment: 10 page
Principal fibrations from noncommutative spheres
We construct noncommutative principal fibrations S_\theta^7 \to S_\theta^4
which are deformations of the classical SU(2) Hopf fibration over the four
sphere. We realize the noncommutative vector bundles associated to the
irreducible representations of SU(2) as modules of coequivariant maps and
construct corresponding projections. The index of Dirac operators with
coefficients in the associated bundles is computed with the Connes-Moscovici
local index formula. The algebra inclusion A(S_\theta^4) \into A(S_\theta^7)
is an example of a not trivial quantum principal bundle.Comment: 23 pages. Latex. v3: Additional minor corrections, version published
in CM
Hodge polynomials of some moduli spaces of Coherent Systems
When , we study the coherent systems that come from a BGN extension in
which the quotient bundle is strictly semistable. In this case we describe a
stratification of the moduli space of coherent systems. We also describe the
strata as complements of determinantal varieties and we prove that these are
irreducible and smooth. These descriptions allow us to compute the Hodge
polynomials of this moduli space in some cases. In particular, we give explicit
computations for the cases in which and is even,
obtaining from them the usual Poincar\'e polynomials.Comment: Formerly entitled: "A stratification of some moduli spaces of
coherent systems on algebraic curves and their Hodge--Poincar\'e
polynomials". The paper has been substantially shorten. Theorem 8.20 has been
revised and corrected. Final version accepted for publication in
International Journal of Mathematics. arXiv admin note: text overlap with
arXiv:math/0407523 by other author
Two-dimensional categorified Hall algebras
In the present paper, we introduce two-dimensional categorified Hall algebras
of smooth curves and smooth surfaces. A categorified Hall algebra is an
associative monoidal structure on the stable -category
of complexes of sheaves with
bounded coherent cohomology on a derived moduli stack .
In the surface case, is a suitable derived enhancement
of the moduli stack of coherent sheaves on the surface. This
construction categorifies the K-theoretical and cohomological Hall algebras of
coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case,
we define three categorified Hall algebras associated with suitable derived
enhancements of the moduli stack of Higgs sheaves on a curve , the moduli
stack of vector bundles with flat connections on , and the moduli stack of
finite-dimensional local systems on , respectively. In the Higgs sheaves
case we obtain a categorification of the K-theoretical and cohomological Hall
algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in
the other two cases our construction yields, by passing to , new
K-theoretical Hall algebras, and by passing to ,
new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and
the non-abelian Hodge correspondences can be lifted to the level of our
categorified Hall algebras of a curve.Comment: 54 page
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