1,023 research outputs found
BPS invariants of semi-stable sheaves on rational surfaces
BPS invariants are computed, capturing topological invariants of moduli
spaces of semi-stable sheaves on rational surfaces. For a suitable stability
condition, it is proposed that the generating function of BPS invariants of a
Hirzebruch surface takes the form of a product formula. BPS invariants for
other stability conditions and other rational surfaces are obtained using
Harder-Narasimhan filtrations and the blow-up formula. Explicit expressions are
given for rank <4 sheaves on a Hirzebruch surface or the projective plane. The
applied techniques can be applied iteratively to compute invariants for higher
rank.Comment: 26 pages, version submitted to journa
Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula
We consider the problem of explicitly computing dimensions of spaces of
automorphic or modular forms in level one, for a split classical group
over such that has discrete series.
Our main contribution is an algorithm calculating orbital integrals for the
characteristic function of at torsion elements of
. We apply it to compute the geometric side in
Arthur's specialisation of his invariant trace formula involving stable
discrete series pseudo-coefficients for . Therefore we
explicitly compute the Euler-Poincar\'e characteristic of the level one
discrete automorphic spectrum of with respect to a
finite-dimensional representation of . For such a group
, Arthur's endoscopic classification of the discrete spectrum
allows to analyse precisely this Euler-Poincar\'e characteristic. For example
one can deduce the number of everywhere unramified automorphic representations
of such that is isomorphic to a given
discrete series representation of . Dimension formulae
for the spaces of vector-valued Siegel modular forms are easily derived.Comment: 89 pages, 28 tables, comments welcome. Much more data available at
http://www.math.ens.fr/~taibi/dimtrace
A complete theory of low-energy phase diagrams for two-dimensional turbulence steady states and equilibria
For the 2D Euler equations and related models of geophysical flows, minima of
energy--Casimir variational problems are stable steady states of the equations
(Arnol'd theorems). The same variational problems also describe sets of
statistical equilibria of the equations. In this paper, we make use of
Lyapunov--Schmidt reduction in order to study the bifurcation diagrams for
these variational problems, in the limit of small energy or, equivalently, of
small departure from quadratic Casimir functionals. We show a generic
occurrence of phase transitions, either continuous or discontinuous. We derive
the type of phase transitions for any domain geometry and any model analogous
to the 2D Euler equations. The bifurcations depend crucially on a_4, the
quartic coefficient in the Taylor expansion of the Casimir functional around
its minima. Note that a_4 can be related to the fourth moment of the vorticity
in the statistical mechanics framework. A tricritical point (bifurcation from a
continuous to a discontinuous phase transition) often occurs when a_4 changes
sign. The bifurcations depend also on possible constraints on the variational
problems (circulation, energy). These results show that the analytical results
obtained with quadratic Casimir functionals by several authors are non-generic
(not robust to a small change in the parameters)
An inverse mapping theorem for blow-Nash maps on singular spaces
A semialgebraic map between two real algebraic sets is called
blow-Nash if it can be made Nash (i.e. semialgebraic and real analytic) by
composing with finitely many blowings-up with non-singular centers. We prove
that if a blow-Nash self-homeomorphism satisfies a lower
bound of the Jacobian determinant condition then is also blow-Nash and
satisfies the same condition. The proof relies on motivic integration arguments
and on the virtual Poincar\'e polynomial of McCrory-Parusi\'nski and Fichou. In
particular, we need to generalize Denef-Loeser change of variables key lemma to
maps that are generically one-to-one and not merely birational
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