146 research outputs found
Connectivity and a Problem of Formal Geometry
Let be a product of weighted
projective spaces, and let be the diagonal of . We prove
an algebraization result for formal-rational functions on certain closed
subvarieties of along the intersection .Comment: 9 pages, to appear in the Proceedings volume "Experimental and
Theoretical Methods in Algebra, Geometry and Topology", series Springer
Proceedings in Mathematics & Statistic
Multicomponent Skyrmion lattices and their excitations
We study quantum Hall ferromagnets with a finite density topologically
charged spin textures in the presence of internal degrees of freedom such as
spin, valley, or layer indices, so that the system is parametrised by a
-component complex spinor field. In the absence of anisotropies, we find
formation of a hexagonal Skyrmion lattice which completely breaks the
underlying SU(d) symmetry. The ground state charge density modulation, which
inevitably exists in these lattices, vanishes exponentially in . We compute
analytically the complete low-lying excitation spectrum, which separates into
gapless acoustic magnetic modes and a magnetophonon. We discuss the
role of effective mass anisotropy for SU(3)-valley Skyrmions relevant for
experiments with AlAs quantum wells. Here, we find a transition, which breaks a
six-fold rotational symmetry of a triangular lattice, followed by a formation
of a square lattice at large values of anisotropy strength.Comment: 4.5 pages, 3 figure
Holomorphic symmetric differentials and a birational characterization of Abelian Varieties
A generically generated vector bundle on a smooth projective variety yields a
rational map to a Grassmannian, called Kodaira map. We answer a previous
question, raised by the asymptotic behaviour of such maps, giving rise to a
birational characterization of abelian varieties.
In particular we prove that, under the conjectures of the Minimal Model
Program, a smooth projective variety is birational to an abelian variety if and
only if it has Kodaira dimension 0 and some symmetric power of its cotangent
sheaf is generically generated by its global sections.Comment: UPDATED: more details added on main proo
Linear Toric Fibrations
These notes are based on three lectures given at the 2013 CIME/CIRM summer
school. The purpose of this series of lectures is to introduce the notion of a
toric fibration and to give its geometrical and combinatorial
characterizations. Polarized toric varieties which are birationally equivalent
to projective toric bundles are associated to a class of polytopes called
Cayley polytopes. Their geometry and combinatorics have a fruitful interplay
leading to fundamental insight in both directions. These notes will illustrate
geometrical phenomena, in algebraic geometry and neighboring fields, which are
characterized by a Cayley structure. Examples are projective duality of toric
varieties and polyhedral adjunction theory
Borcherds symmetries in M-theory
It is well known but rather mysterious that root spaces of the Lie
groups appear in the second integral cohomology of regular, complex, compact,
del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms)
of toroidal compactifications of M theory. Their Borel subgroups are actually
subgroups of supergroups of finite dimension over the Grassmann algebra of
differential forms on spacetime that have been shown to preserve the
self-duality equation obeyed by all bosonic form-fields of the theory. We show
here that the corresponding duality superalgebras are nothing but Borcherds
superalgebras truncated by the above choice of Grassmann coefficients. The full
Borcherds' root lattices are the second integral cohomology of the del Pezzo
surfaces. Our choice of simple roots uses the anti-canonical form and its known
orthogonal complement. Another result is the determination of del Pezzo
surfaces associated to other string and field theory models. Dimensional
reduction on corresponds to blow-up of points in general position
with respect to each other. All theories of the Magic triangle that reduce to
the sigma model in three dimensions correspond to singular del Pezzo
surfaces with (normal) singularity at a point. The case of type I and
heterotic theories if one drops their gauge sector corresponds to non-normal
(singular along a curve) del Pezzo's. We comment on previous encounters with
Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real
fermionic simple roots when they would naively aris
On the canonical map of surfaces with q>=6
We carry out an analysis of the canonical system of a minimal complex surface
of general type with irregularity q>0. Using this analysis we are able to
sharpen in the case q>0 the well known Castelnuovo inequality K^2>=3p_g+q-7.
Then we turn to the study of surfaces with p_g=2q-3 and no fibration onto a
curve of genus >1. We prove that for q>=6 the canonical map is birational.
Combining this result with the analysis of the canonical system, we also prove
the inequality: K^2>=7\chi+2. This improves an earlier result of the first and
second author [M.Mendes Lopes and R.Pardini, On surfaces with p_g=2q-3, Adv. in
Geom. 10 (3) (2010), 549-555].Comment: Dedicated to Fabrizio Catanese on the occasion of his 60th birthday.
To appear in the special issue of Science of China Ser.A: Mathematics
dedicated to him. V2:some typos have been correcte
Formal properties in small codimension
In this note we extend connectedness results to formal properties of inverse images under proper maps of Schubert varieties and of the diagonal in products of projective rational homogeneous spaces
Cohomological characterizations of projective spaces and hyperquadrics
We confirm Beauville's conjecture that claims that if the p-th exterior power
of the tangent bundle of a smooth projective variety contains the p-th power of
an ample line bundle, then the variety is either the projective space or the
p-dimensional quadric hypersurface.Comment: Added Lemma 2.8 and slightly changed proof of Lemma 6.2 to make them
apply for torsion-free sheaves and not only to vector bundle
Fibrations on four-folds with trivial canonical bundles
Four-folds with trivial canonical bundles are divided into six classes
according to their holonomy group. We consider examples that are fibred by
abelian surfaces over the projective plane. We construct such fibrations in
five of the six classes, and prove that there is no such fibration in the sixth
class. We classify all such fibrations whose generic fibre is the Jacobian of a
genus two curve.Comment: 28 page
Deformation of canonical morphisms and the moduli of surfaces of general type
In this article we study the deformation of finite maps and show how to use
this deformation theory to construct varieties with given invariants in a
projective space. Among other things, we prove a criterion that determines when
a finite map can be deformed to a one--to--one map. We use this criterion to
construct new simple canonical surfaces with different and . Our
general results enable us to describe some new components of the moduli of
surfaces of general type. We also find infinitely many moduli spaces having one component whose general point corresponds to a
canonically embedded surface and another component whose general point
corresponds to a surface whose canonical map is a degree 2 morphism.Comment: 32 pages. Final version with some simplifications and clarifications
in the exposition. To appear in Invent. Math. (the final publication is
available at springerlink.com
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