117 research outputs found
Nowhere minimal CR submanifolds and Levi-flat hypersurfaces
A local uniqueness property of holomorphic functions on real-analytic nowhere
minimal CR submanifolds of higher codimension is investigated. A sufficient
condition called almost minimality is given and studied. A weaker necessary
condition, being contained a possibly singular real-analytic Levi-flat
hypersurface is studied and characterized. This question is completely resolved
for algebraic submanifolds of codimension 2 and a sufficient condition for
noncontainment is given for non algebraic submanifolds. As a consequence, an
example of a submanifold of codimension 2, not biholomorphically equivalent to
an algebraic one, is given. We also investigate the structure of singularities
of Levi-flat hypersurfaces.Comment: 21 pages; conjecture 2.8 was removed in proof; to appear in J. Geom.
Ana
Tameness of holomorphic closure dimension in a semialgebraic set
Given a semianalytic set S in a complex space and a point p in S, there is a
unique smallest complex-analytic germ at p which contains the germ of S, called
the holomorphic closure of S at p. We show that if S is semialgebraic then its
holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic
filtration by the holomorphic closure dimension. As a consequence, every
semialgebraic subset of a complex vector space admits a semialgebraic
stratification into CR manifolds satisfying a strong version of the condition
of the frontier.Comment: Published versio
The {\L}ojasiewicz exponent of a set of weighted homogeneous ideals
We give an expression for the {\L}ojasiewicz exponent of a set of ideals
which are pieces of a weighted homogeneous filtration. We also study the
application of this formula to the computation of the {\L}ojasiewicz exponent
of the gradient of a semi-weighted homogeneous function (\C^n,0)\to (\C,0)
with an isolated singularity at the origin.Comment: 15 page
From error bounds to the complexity of first-order descent methods for convex functions
This paper shows that error bounds can be used as effective tools for
deriving complexity results for first-order descent methods in convex
minimization. In a first stage, this objective led us to revisit the interplay
between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can
show the equivalence between the two concepts for convex functions having a
moderately flat profile near the set of minimizers (as those of functions with
H\"olderian growth). A counterexample shows that the equivalence is no longer
true for extremely flat functions. This fact reveals the relevance of an
approach based on KL inequality. In a second stage, we show how KL inequalities
can in turn be employed to compute new complexity bounds for a wealth of
descent methods for convex problems. Our approach is completely original and
makes use of a one-dimensional worst-case proximal sequence in the spirit of
the famous majorant method of Kantorovich. Our result applies to a very simple
abstract scheme that covers a wide class of descent methods. As a byproduct of
our study, we also provide new results for the globalization of KL inequalities
in the convex framework.
Our main results inaugurate a simple methodology: derive an error bound,
compute the desingularizing function whenever possible, identify essential
constants in the descent method and finally compute the complexity using the
one-dimensional worst case proximal sequence. Our method is illustrated through
projection methods for feasibility problems, and through the famous iterative
shrinkage thresholding algorithm (ISTA), for which we show that the complexity
bound is of the form where the constituents of the bound only depend
on error bound constants obtained for an arbitrary least squares objective with
regularization
Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
Loop quantum gravity is an approach to quantum gravity that starts from the
Hamiltonian formulation in terms of a connection and its canonical conjugate.
Quantization proceeds in the spirit of Dirac: First one defines an algebra of
basic kinematical observables and represents it through operators on a suitable
Hilbert space. In a second step, one implements the constraints. The main
result of the paper concerns the representation theory of the kinematical
algebra: We show that there is only one cyclic representation invariant under
spatial diffeomorphisms.
While this result is particularly important for loop quantum gravity, we are
rather general: The precise definition of the abstract *-algebra of the basic
kinematical observables we give could be used for any theory in which the
configuration variable is a connection with a compact structure group. The
variables are constructed from the holonomy map and from the fluxes of the
momentum conjugate to the connection. The uniqueness result is relevant for any
such theory invariant under spatial diffeomorphisms or being a part of a
diffeomorphism invariant theory.Comment: 38 pages, one figure. v2: Minor changes, final version, as published
in CM
Residue currents associated with weakly holomorphic functions
We construct Coleff-Herrera products and Bochner-Martinelli type residue
currents associated with a tuple of weakly holomorphic functions, and show
that these currents satisfy basic properties from the (strongly) holomorphic
case, as the transformation law, the Poincar\'e-Lelong formula and the
equivalence of the Coleff-Herrera product and the Bochner-Martinelli type
residue current associated with when defines a complete intersection.Comment: 28 pages. Updated with some corrections from the revision process. In
particular, corrected and clarified some things in Section 5 and 6 regarding
products of weakly holomorphic functions and currents, and the definition of
the Bochner-Martinelli type current
Poisson-de Rham homology of hypertoric varieties and nilpotent cones
We prove a conjecture of Etingof and the second author for hypertoric
varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is
isomorphic to the de Rham cohomology of its hypertoric resolution. More
generally, we prove that this conjecture holds for an arbitrary conical variety
admitting a symplectic resolution if and only if it holds in degree zero for
all normal slices to symplectic leaves.
The Poisson-de Rham homology of a Poisson cone inherits a second grading. In
the hypertoric case, we compute the resulting 2-variable Poisson-de
Rham-Poincare polynomial, and prove that it is equal to a specialization of an
enrichment of the Tutte polynomial of a matroid that was introduced by Denham.
We also compute this polynomial for S3-varieties of type A in terms of Kostka
polynomials, modulo a previous conjecture of the first author, and we give a
conjectural answer for nilpotent cones in arbitrary type, which we prove in
rank less than or equal to 2.Comment: 25 page
Background independent quantizations: the scalar field II
We are concerned with the issue of quantization of a scalar field in a
diffeomorphism invariant manner. We apply the method used in Loop Quantum
Gravity. It relies on the specific choice of scalar field variables referred to
as the polymer variables. The quantization, in our formulation, amounts to
introducing the `quantum' polymer *-star algebra and looking for positive
linear functionals, called states. Assumed in our paper homeomorphism
invariance allows to derive the complete class of the states. They are
determined by the homeomorphism invariant states defined on the CW-complex
*-algebra. The corresponding GNS representations of the polymer *-algebra and
their self-adjoint extensions are derived, the equivalence classes are found
and invariant subspaces characterized. In the preceding letter (the part I) we
outlined those results. Here, we present the technical details.Comment: 51 pages, LaTeX, no figures, revised versio
Background independent quantizations: the scalar field I
We are concerned with the issue of quantization of a scalar field in a
diffeomorphism invariant manner. We apply the method used in Loop Quantum
Gravity. It relies on the specific choice of scalar field variables referred to
as the polymer variables. The quantization, in our formulation, amounts to
introducing the `quantum' polymer *-star algebra and looking for positive
linear functionals, called states. The assumed in our paper homeomorphism
invariance allows to determine a complete class of the states. Except one, all
of them are new. In this letter we outline the main steps and conclusions, and
present the results: the GNS representations, characterization of those states
which lead to essentially self adjoint momentum operators (unbounded),
identification of the equivalence classes of the representations as well as of
the irreducible ones. The algebra and topology of the problem, the derivation,
all the technical details and more are contained in the paper-part II.Comment: 13 pages, minor corrections were made in the revised versio
- …