A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define a…
Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible wit…
The paper extends symplectic techniques to generalized complex geometry.
Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex o…
Survey on holomorphic structures on complex manifolds.
The paper connects fibrations to generalized complex structures in semi-toric geometry.
We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry. We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the eq…
Study on complex tori foliations and flat geometries.
A closed 3-form defines an extension of by . This fact leads to the definition of the group of -twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold. The Hamiltonian redu…
We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini's model is model in the case that the generalized complex structuredepends on only a symplectic structure. Our new model is model in the case that the generalized complex structure depends…
Classifies holomorphic parabolic geometries on complex manifolds.
These are the lecture notes from the 26th Winter School "Geometry and Physics", Czech Republic, Srni, January 14 - 21, 2006. These lectures are an introduction into the realm of generalized geometry based on the tangent plus the cotangent bundle. In particular we discuss the relation of this geometry to physics, namely…
Survey of geometry developments, including complex structures on surfaces.
We define the notion of a moment map and reduction in both generalized complex geometry and generalized Kähler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $\C¶^n$, Hirzebruch surfaces, the blow up of $\CP^2$ at arbitrarily many points, and other toric varieties,…
Unified framework for complex, split-complex, and dual numbers.
Book introduces generalized Ricci flow for constructing canonical metrics.
We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists. T-duality relates topologically distinct torus bundles, and prescribes a method for transporting geometrical structu…
Study complex lines in symplectic geometry, generalizing previous results.
Model structures on multicomplexes help study complex geometry.
This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the ch…
We revisit our earlier work on the AKSZ formulation of topological sigma model on generalized complex manifolds, or Hitchin model. We show that the target space geometry geometry implied by the BV master equations is Poisson--quasi--Nijenhuis geometry recently introduced and studied by Stiénon and Xu (in the untwisted …
The paper proves complex geometry results for manifolds of the form X × R², answering a 1994 conjecture.
We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometr…
The study identifies two sources of invariants in 2--nondegenerate CR geometries.
In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mapping…
On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
Survey explores interactions between convex and complex geometry.
Paper generalizes toric concepts to nonrational settings.
The aim of this article is to use generalized complex structures in order to extend the definition of twistor spaces given by Penrose. We will adapt the integrability result of Atiyah, Hitchin and Singer. We will deduce new correspondences betwenn differential geometry and (generalized) complex geometry. In the last se…
The abstract discusses how generalized Calabi-Gray manifolds help solve non-Kähler geometry questions.
We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B-field equivalent to a holomorphic Poisson structure. This is intimately tied with our second main re…
First we introduce a generalization of symmetric spaces to parabolic geometries. We provide construction of such parabolic geometries starting with classical symmetric spaces and we show that all regular parabolic geometries with smooth systems of involutive symmetries can be obtained this way. Further, we investigate …
Generalized complex (GC) geometry interpolates between ordinary symplectic and complex geometry. Stable generalized complex manifolds (first introduced by Cavalcanti, Gualtieri in 2015) carry a Poisson structure which is generically symplectic, but degenerates on a (real) codimension-2 submanifold. Up to gauge equivale…
This is largely a survey paper, dealing with Cartan geometries in the complex analytic category. We first remind some standard facts going back to the seminal works of F. Klein, E. Cartan and C. Ehresmann. Then we present the concept of a branched holomorphic Cartan geometry which was introduced by the authors in [BD].…
Hitchin's generalized complex geometry has been shown to be relevant in compactifications of superstring theory with fluxes and is expected to lead to a deeper understanding of mirror symmetry. Gualtieri's notion of generalized complex submanifold seems to be a natural candidate for the description of branes in this co…
Researchers compute the cohomology of an elliptic tangent bundle.
We present a sigma model field theoretic realization of Hitchin's generalized complex geometry, which recently has been shown to be relevant in compactifications of superstring theory with fluxes. Hitchin sigma model is closely related to the well known Poisson sigma model, of which it has the same field content. The c…
A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of …
In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in to complex values of a generalized cross-ratio by considering as a real section of the complex Plücker quadric, realized as the space of two-spheres in We develop the geometry of the Plücker…
New geometric structures on 3-manifolds discovered and proven for all closed orientable ones.
We revisit the backgrounds of type IIB on manifolds with -structure and discuss two sets of solutions arising from internal geometries that are complex and symplectic respectively. Both can be realized in terms of generalized complex geometry. We identify a map which relates the complex and symplectic supersymme…
The first part of these notes is a self-contained introduction to generalized complex geometry. It is intended as a `user manual' for tools used in the study of supersymmetric backgrounds of supergravity. In the second part we review some past and recent results on the generalized complex structure of supersymmetric ty…
A theorem of Lawson and Simons states that the only stable minimal submanifolds in complex projective spaces are complex submanifolds. We generalize their result to the cases of quaternionic and octonionic projective spaces. Our approach gives a unified viewpoint towards conformal and projective geometries.
We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisso…
We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kähler manifolds.
Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.
PINNs solve differential geometry problems in complex shapes.
Authors discuss complex and non-Archimedean geometry, proving a conjecture.