Search results for "Mathematics::Symplectic Geometry"
showing 10 items of 184 documents
Singular levels and topological invariants of Morse Bott integrable systems on surfaces
2016
Abstract We classify up to homeomorphisms closed curves and eights of saddle points on orientable closed surfaces. This classification is applied to Morse Bott foliations and Morse Bott integrable systems allowing us to define a complete invariant. We state also a realization Theorem based in two transformations and one generator (the foliation of the sphere with two centers).
Foliations and Line Bundles
2014
In this chapter we start the global study of foliations on complex surfaces. The most basic global invariants which may be associated with such a foliation are its normal and tangent bundles, and here we shall prove several formulae and study several examples concerning the calculation of these bundles. We shall mainly follow the presentation given in [5]; the book [20] may also be of valuable help.
The structure of Fedosov supermanifolds
2009
Abstract Given a supermanifold ( M , A ) which carries a supersymplectic form ω , we study the Fedosov structures that can be defined on it, through a set of tensor fields associated to any symplectic connection ∇ . We give explicit recursive expressions for the resulting curvature and study the particular case of a base manifold M with constant holomorphic sectional curvature.
Supermanifolds, Symplectic Geometry and Curvature
2016
We present a survey of some results and questions related to the notion of scalar curvature in the setting of symplectic supermanifolds.
Compactifying Torus Fibrations Over Integral Affine Manifolds with Singularities
2021
This is an announcement of the following construction: given an integral affine manifold B with singularities, we build a topological space X which is a torus fibration over B. The main new feature of the fibration X → B is that it has the discriminant in codimension 2.
Numerical Kodaira Dimension
2014
In this chapter we study, following [30] , the first properties of the Zariski decomposition of the cotangent bundle of a nonrational foliation. In particular, we shall give a detailed description of the negative part of that Zariski decomposition, and we shall obtain a detailed classification of foliations whose Zariski decomposition is reduced to its negative part (i.e. foliations of numerical Kodaira dimension 0). We shall also discuss the “singular” point of view adopted in [30].
A special Calabi–Yau degeneration with trivial monodromy
2021
A well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi–Yau threefolds.
Orbit spaces of Small Tori
2003
Consider an algebraic torus of small dimension acting on an open subset of ℂn, or more generally on a quasiaffine variety such that a separated orbit space exists. We discuss under which conditions this orbit space is quasiprojective. One of our counterexamples provides a toric variety with enough effective invariant Cartier divisors that is not embeddable into a smooth toric variety.
Infinite Dimensional Holomorphy
2019
We give an introduction to vector-valued holomorphic functions in Banach spaces, defined through Frechet differentiability. Every function defined on a Reinhardt domain of a finite-dimensional Banach space is analytic, i.e. can be represented by a monomial series expansion, where the family of coefficients is given through a Cauchy integral formula. Every separate holomorphic (holomorphic on each variable) function is holomorphic. This is Hartogs’ theorem, which is proved using Leja’s polynomial lemma. For infinite-dimensional spaces, homogeneous polynomials are defined as the diagonal of multilinear mappings. A function is holomorphic if and only if it is Gâteaux holomorphic and continuous…
On the existence of attractors
2009
On every compact 3-manifold, we build a non-empty open set $\cU$ of $\Diff^1(M)$ such that, for every $r\geq 1$, every $C^r$-generic diffeomorphism $f\in\cU\cap \Diff^r(M)$ has no topological attractors. On higher dimensional manifolds, one may require that $f$ has neither topological attractors nor topological repellers. Our examples have finitely many quasi attractors. For flows, we may require that these quasi attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics.