6,951 research outputs found
Orbital degeneracy loci and applications
Degeneracy loci of morphisms between vector bundles have been used in a wide
variety of situations. We introduce a vast generalization of this notion, based
on orbit closures of algebraic groups in their linear representations. A
preferred class of our orbital degeneracy loci is characterized by a certain
crepancy condition on the orbit closure, that allows to get some control on the
canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits,
and also for partially decomposable skew-symmetric three-forms in six
variables. In order to illustrate the efficiency and flexibility of our
methods, we construct in both situations many Calabi--Yau manifolds of
dimension three and four, as well as a few Fano varieties, including some new
Fano fourfolds.Comment: To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5
Signatures of hermitian forms and the Knebusch Trace Formula
Signatures of quadratic forms have been generalized to hermitian forms over
algebras with involution. In the literature this is done via Morita theory,
which causes sign ambiguities in certain cases. In this paper, a hermitian
version of the Knebusch Trace Formula is established and used as a main tool to
resolve these ambiguities.
The last page is an erratum for the published version. We inadvertently (I)
gave an incorrect definition of adjoint involutions; (II) omitted dealing with
the case . As , the
omission does not affect our reasoning or our results. For the sake of
completeness we point out where some small changes should be made in the
published version.Comment: This is the final version before publication. The last page is an
updated erratum for the published versio
Regular cell complexes in total positivity
This paper proves a conjecture of Fomin and Shapiro that their combinatorial
model for any Bruhat interval is a regular CW complex which is homeomorphic to
a ball. The model consists of a stratified space which may be regarded as the
link of an open cell intersected with a larger closed cell, all within the
totally nonnegative part of the unipotent radical of an algebraic group. A
parametrization due to Lusztig turns out to have all the requisite features to
provide the attaching maps. A key ingredient is a new, readily verifiable
criterion for which finite CW complexes are regular involving an interplay of
topology with combinatorics.Comment: accepted to Inventiones Mathematicae; 60 pages; substantially revised
from earlier version
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