Search results for "Mathematics::Symplectic Geometry"
showing 10 items of 184 documents
TANGENTIAL DEFORMATIONS ON FIBRED POISSON MANIFOLDS
2005
In a recent article, Cattaneo, Felder and Tomassini explained how the notion of formality can be used to construct flat Fedosov connections on formal vector bundles on a Poisson manifold $M$ and thus a star product on $M$ through the original Fedosov method for symplectic manifolds. In this paper, we suppose that $M$ is a fibre bundle manifold equipped with a Poisson tensor tangential to the fibers. We show that in this case the construction of Cattaneo-Felder- Tomassini gives tangential (to the fibers) star products.
Diffeomorphisms, Noether charges, and the canonical formalism in two-dimensional dilaton gravity
1995
We carry out a parallel study of the covariant phase space and the conservation laws of local symmetries in two-dimensional dilaton gravity. Our analysis is based on the fact that the Lagrangian can be brought to a form that vanishes on-shell giving rise to a well-defined covariant potential for the symplectic current. We explicitly compute the symplectic structure and its potential and show that the requirement to be finite and independent of the Cauchy surface restricts the asymptotic symmetries.
Quotients of the Dwork Pencil
2012
In this paper we investigate the geometry of the Dwork pencil in any dimension. More specifically, we study the automorphism group G of the generic fiber of the pencil over the complex projective line, and the quotients of it by various subgroups of G. In particular, we compute the Hodge numbers of these quotients via orbifold cohomology.
Groups acting freely on Calabi-Yau threefolds embedded in a product of del Pezzo surfaces
2011
In this paper, we investigate quotients of Calabi-Yau manifolds $Y$ embedded in Fano varieties $X$, which are products of two del Pezzo surfaces — with respect to groups $G$ that act freely on $Y$. In particular, we revisit some known examples and we obtain some new Calabi-Yau varieties with small Hodge numbers. The groups $G$ are subgroups of the automorphism groups of $X$, which is described in terms of the automorphism group of the two del Pezzo surfaces.
Splittings of Toric Ideals
2019
Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient condition for this splitting in terms of the integer matrix that defines $I$. When $I = I_G$ is the toric ideal of a finite simple graph $G$, we give additional splittings of $I_G$ related to subgraphs of $G$. When there exists a splitting $I = I_1+I_2$ of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of $I$ in terms of the (multi-)graded Betti numbers of $I_1$ and $I_2$.
Complex powers and non-compact manifolds
2002
We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, ``extended Weyl algebras,'' whose definition was inspired by Guillemin's paper on the subject. An extended Weyl algebra can be thought of as an algebra of ``abstract pseudodifferential operators.'' Many algebras of pseudodifferential operators are extended Weyl algebras. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between apropriate Sobolev spaces, >...) generalize to…
On deformation of Poisson manifolds of hydrodynamic type
2001
We study a class of deformations of infinite-dimensional Poisson manifolds of hydrodynamic type which are of interest in the theory of Frobenius manifolds. We prove two results. First, we show that the second cohomology group of these manifolds, in the Poisson-Lichnerowicz cohomology, is ``essentially'' trivial. Then, we prove a conjecture of B. Dubrovin about the triviality of homogeneous formal deformations of the above manifolds.
The linearized Calderón problem on complex manifolds
2019
International audience; In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions , the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot by treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holo-morphic functions with approximately prescribed critical points.…
Integrable systems, Frobenius manifolds and cohomological field theories
2022
In this dissertation, we study the underlying geometry of integrable systems, in particular tausymmetric bi-Hamiltonian hierarchies of evolutionary PDEs and differential-difference equations.First, we explore the close connection between the realms of integrable systems and algebraic geometry by giving a new proof of the Witten conjecture, which constructs the string taufunction of the Korteweg-de Vries hierarchy via intersection theory of the moduli spaces of stable curves with marked points. This novel proof is based on the geometry of double ramification cycles, tautological classes whose behavior under pullbacks of the forgetful and gluing maps facilitate the computation of intersection…
Equidistribution and Counting of Common Perpendiculars in Quotient Spaces
2019
In this chapter, we use the results of Chapter 11 to prove equidistribution and counting results in Riemannian manifolds (or good orbifolds) and in metric and simplicial graphs (of groups).