arXiv research

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.

Introduces semi-abelian generalized complex structures.

The paper extends symplectic techniques to generalized complex geometry.

Let M be a hyperkähler manifold. The S^2-family of complex structures compatible with the hyperkähler metric can be assembled into a single complex structure on Z=MxS^2; the resulting complex manifold is known as the twistor space of M. We describe the analogous construction for generalized complex structures in the se…

We show that all 6-dimensional nilmanifolds admit generalized complex structures. This includes the five classes of nilmanifold which admit no known complex or symplectic structure. Furthermore, we classify all 6-dimensional nilmanifolds according to which of the four types of left-invariant generalized complex structu…

The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin (we call these ``generalized Lagrangian submanifolds'' in our paper), we introduce and study three other classes of submanifolds. …

We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a …

We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini's model is $A$ model in the case that the generalized complex structuredepends on only a symplectic structure. Our new model is $B$ model in the case that the generalized complex structure depends…

The study characterizes real flag manifolds with invariant generalized almost complex structures.

Four-manifold theory is employed to study the existence of (twisted) generalized complex structures. It is shown that there exist (twisted) generalized complex structures that have more than one type change loci. In an example-driven fashion, (twisted) generalized complex structures are constructed on a myriad of four-…

We introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson-Dirac submanifolds. Our characterization recovers a result of Vaisman. An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of a…

We study massless deformations of generalized calibrated cycles, which describe, in the language of generalized complex geometry, supersymmetric D-branes in N=1 supersymmetric compactifications with fluxes. We find that the deformations are classified by the first cohomology group of a Lie algebroid canonically associa…

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…

New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.

This paper proves a generalization bound for complex-valued neural networks scaling with spectral complexity.

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…

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 construct a natural generalized complex structure on the total space of any bundle endowed with a Chern connection and whose typical fibre is a homogeneous symplectic manifold. This extends known constructions of generalized complex structures on Lie groups and leads to natural examples of holomorphic maps between g…

Generalizes complex manifolds to manifolds with corners and generalized corners.

Proposes CXNs for neural network computations on cell complexes.

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…

We study generalized complex manifolds from the point of view of symplectic and Poisson geometry. We start by showing that every generalized complex manifold admits a canonical Poisson structure. We use this fact, together with Weinstein's classical result on the local normal form of Poisson manifolds, to prove a local…

We produce examples of generalized complex structures on manifolds by generalizing results from symplectic and complex geometry. We produce generalized complex structures on symplectic fibrations over a generalized complex base. We study in some detail different invariant generalized complex structures on compact Lie g…

Stable generalized complex structures on certain surfaces are constant.

Generalized complex structures on certain torus bundles are explored.

Generalizes Leighton's theorem to cube complexes.

We construct a three-dimensional topological sigma model which is induced from a generalized complex structure on a target generalized complex manifold. This model is constructed from maps from a three-dimensional manifold $X$ to an arbitrary generalized complex manifold $M$. The theory is invariant under the diffeomor…

In this paper we develop a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and nabla a flat connection on V. We define the notion of a nabla-semi-flat generalized com…

Proposes Neural Complexity (NC) for predicting and explaining generalization in deep neural networks.

The paper defines new modular forms from almost complex manifolds and derives anomaly cancellation formulas.

We present a novel notion of complexity that interpolates between and generalizes some classic existing complexity notions in learning theory: for estimators like empirical risk minimization (ERM) with arbitrary bounded losses, it is upper bounded in terms of data-independent Rademacher complexity; for generalized Baye…

The paper connects fibrations to generalized complex structures in semi-toric geometry.

In this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids and construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifo…

Calculates cobordism ring of stably almost complex C_p-manifolds.

We introduce blow-up and blow-down operations for generalized complex 4-manifolds. Combining these with a surgery analogous to the logarithmic transform, we then construct generalized complex structures on nCP2 # m \bar{CP2} for n odd, a family of 4-manifolds which admit neither complex nor symplectic structures unless…

We introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold containing a symplectic 2-torus with trivial normal bundle and whose output is a 4-manifold endowed with a generalized complex structure exhibiting type change along a 2-torus. Performing this surgery on a K3 surface, we obt…

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…

No direct generalized complex structure can be induced from $\mathbb S^6$'s nearly Kähler structure.

Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.

Classifies and computes cohomologies of complex structures on Lie groups.

Let $G$ be a complex semi-simple Lie group and form its maximal flag manifold $\mathbb{F}=G/P=U/T$ where $P$ is a minimal parabolic subgroup, $U$ a compact real form and $T=U\cap P$ a maximal torus of $U$. The aim of this paper is to study invariant generalized complex structures on $\mathbb{F}$. We describe the invari…

A Jacobi structure $J$ on a line bundle $L\to M$ is weakly regular if the sharp map $J^\sharp : J^1 L \to DL$ has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair …

Complexity measures for neural nets with general activations using path-based norms.

This paper provides a general result on controlling local Rademacher complexities, which captures in an elegant form to relate the complexities with constraint on the expected norm to the corresponding ones with constraint on the empirical norm. This result is convenient to apply in real applications and could yield re…

Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.

The paper extends a theorem for complex structures on Lie groups to Courant algebroids.

Improved generalization bounds for CNNs using Rademacher complexity.

We look at generalized complex structures from the point of view of Poisson and Dirac geometry and we remark that the puzzling equations underlying the notion of generalized complex structure have miraculously simple meaning when passing to Lie algebroids/groupoids.