77 research outputs found
The Toda hierarchy and the KdV hierarchy
The Toda hierarchy of size is well known to be analogous to the KdV
hierarchy at goes to infinity. This paper shows that given a periodic
function, there is a canonical way of defining the initial data for the Toda
lattice equations so that the evolution of this data under the Toda lattice
hierarchy looks asymptotically like the evolution of under the KdV
hierarchy. Further, the conserved quantities of and those of the Toda
hierarchy match.Comment: AMSTe
Simply connected projective manifolds in characteristic have no nontrivial stratified bundles
We show that simply connected projective manifolds in characteristic
have no nontrivial stratified bundles. This gives a positive answer to a
conjecture by D. Gieseker. The proof uses Hrushovski's theorem on periodic
points.Comment: 16 pages. Revised version contains a more general theorem on torsion
points on moduli, together with an illustration in rank 2 due to M. Raynaud.
Reference added. Last version has some typos corrected. Appears in
Invent.math
Semistability vs. nefness for (Higgs) vector bundles
According to Miyaoka, a vector bundle E on a smooth projective curve is
semistable if and only if a certain numerical class in the projectivized bundle
PE is nef. We establish a similar criterion for the semistability of Higgs
bundles: namely, such a bundle is semistable if and only if for every integer s
between 0 and the rank of E, a suitable numerical class in the scheme
parametrizing the rank s locally-free Higgs quotients of E is nef. We also
extend this result to higher-dimensional complex projective varieties by
showing that the nefness of the above mentioned classes is equivalent to the
semistability of the Higgs bundle E together with the vanishing of the
discriminant of E.Comment: Comments: 20 pages, Latex2e, no figures. v2 includes a generalization
to complex projective manifolds of any dimension. To appear in Diff. Geom.
App
A GIT interpretration of the Harder-Narasimhan filtration
An unstable torsion free sheaf on a smooth projective variety gives a GIT
unstable point in certain Quot scheme. To a GIT unstable point, Kempf
associates a "maximally destabilizing" 1-parameter subgroup, and this induces a
filtration of the torsion free sheaf. We show that this filtration coincides
with the Harder-Narasimhan filtration.Comment: 19 pages; Comments of the referees and references added. The
construction for holomorphic pairs (Sections 6 and 7 from previous version)
will appear in a further publication. To appear in Rev. Mat Complutens
Harper operators, Fermi curves, and Picard-Fuchs equations
This paper is a continuation of the work on the spectral problem of Harper
operator using algebraic geometry. We continue to discuss the local monodromy
of algebraic Fermi curves based on Picard-Lefschetz formula. The density of
states over approximating components of Fermi curves satisfies a Picard-Fuchs
equation. By the property of Landen transformation, the density of states has a
Lambert series as the quarter period. A -expansion of the energy level can
be derived from a mirror map as in the B-model.Comment: v2, 13 pages, minor changes have been mad
Geometric invariant theory of syzygies, with applications to moduli spaces
We define syzygy points of projective schemes, and introduce a program of
studying their GIT stability. Then we describe two cases where we have managed
to make some progress in this program, that of polarized K3 surfaces of odd
genus, and of genus six canonical curves. Applications of our results include
effectivity statements for divisor classes on the moduli space of odd genus K3
surfaces, and a new construction in the Hassett-Keel program for the moduli
space of genus six curves.Comment: v1: 23 pages, submitted to the Proceedings of the Abel Symposium
2017, v2: final version, corrects a sign error and resulting divisor class
calculations on the moduli space of K3 surfaces in Section 5, other minor
changes, In: Christophersen J., Ranestad K. (eds) Geometry of Moduli.
Abelsymposium 2017. Abel Symposia, vol 14. Springer, Cha
Elliptic curve configurations on Fano surfaces
The elliptic curves on a surface of general type constitute an obstruction
for the cotangent sheaf to be ample. In this paper, we give the classification
of the configurations of the elliptic curves on the Fano surface of a smooth
cubic threefold. That means that we give the number of such curves, their
intersections and a plane model. This classification is linked to the
classification of the automorphism groups of theses surfaces.Comment: 17 pages, accepted and shortened version, the rest will appear in
"Fano surfaces with 12 or 30 elliptic curves
Asymptotics for Fermi curves: small magnetic potential
We consider complex Fermi curves of electric and magnetic periodic fields.
These are analytic curves in C^2 that arise from the study of the eigenvalue
problem for periodic Schroedinger operators. We characterize a certain class of
these curves in the region of C^2 where at least one of the coordinates has
"large" imaginary part. The new results in this work extend previous results in
the absence of magnetic field to the case of "small" magnetic field. Our
theorems can be used to show that generically these Fermi curves belong to a
class of Riemann surfaces of infinite genus.Comment: 71 pages, 1 figur
Scattering theory for lattice operators in dimension
This paper analyzes the scattering theory for periodic tight-binding
Hamiltonians perturbed by a finite range impurity. The classical energy
gradient flow is used to construct a conjugate (or dilation) operator to the
unperturbed Hamiltonian. For dimension the wave operator is given by
an explicit formula in terms of this dilation operator, the free resolvent and
the perturbation. From this formula the scattering and time delay operators can
be read off. Using the index theorem approach, a Levinson theorem is proved
which also holds in presence of embedded eigenvalues and threshold
singularities.Comment: Minor errors and misprints corrected; new result on absense of
embedded eigenvalues for potential scattering; to appear in RM
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