Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
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48 results for “generalized complex structure”
Higher complex structuresmath.DG
New geometric structure on surfaces generalizing complex structures.
problem Defining and analyzing new geometric structures on surfaces.
method Using the punctual Hilbert scheme of the plane to define higher complex structures.
result Moduli space of higher complex structures is a generalization of Teichmüller space and conjecturally isomorphic to Hitchin's component.
The study characterizes real flag manifolds with invariant generalized almost complex structures.
problem Characterizing real flag manifolds with invariant generalized almost complex structures.
method Characterization through invariant -transformations and classification of structures.
result No -maximal real flag manifolds admit integrable invariant generalized almost complex structures.
The paper explores new structures in generalized geometry and their relationships.
problem Exploring new structures in generalized geometry.
method Discussing the relation between VB-Courant algebroids and E-Courant algebroids, introducing generalized complex structures, and studying their properties.
result Generalized complex structures on E-Courant algebroids unify different types of structures.
Regular Jacobi structures on line bundles lead to generalized contact bundles.
problem Understanding the relationship between Jacobi structures and generalized contact bundles.
method Investigating weakly regular Jacobi structures and their transverse complex structures.
result Conditions for a pair of a regular Jacobi structure and a transverse complex structure to form a generalized contact structure.
New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.
problem Stable generalized complex structures in higher dimensions with self-crossing singularities.
method Extending stable generalized complex structures to include anticanonical sections with normal self-crossings.
result Construction of large families of stable generalized complex manifolds in four dimensions.
No direct generalized complex structure can be induced from 's nearly Kähler structure.
problem Existence of generalized complex structures on .
method Defined integrability in terms of the Dorfman bracket and studied 's nearly Kähler structure.
result No generalized complex structure can be induced from 's nearly Kähler structure.
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 …
Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
Stable generalized complex structures on certain surfaces are constant.
problem Existence of stable generalized complex structures on ruled surfaces.
method Analysis of sphere bundles over surfaces of genus ≥2.
result Stable generalized complex structures on these surfaces are of constant type.
The paper classifies invariant generalized complex structures on specific flag manifolds.
problem Classifying invariant generalized complex structures on partial flag manifolds.
method Proved that invariant generalized almost complex structures are constant in each component of the isotropy representation.
result All invariant generalized complex structures on partial flag manifolds with at most four isotropy summands are classified.
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…
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…
The paper studies invariant generalized complex structures on flag manifolds.
problem Classifying invariant generalized complex structures on flag manifolds.
method Analyzing invariant -dimensional generalized almost complex structures restricted to each root space, and studying the Nijenhuis operator for a triple of roots.
result Classification of integrable and -integrable generalized complex structures.
Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.
problem Characterizing types of generalized hypercomplex structures.
method Analysis of -family of generalized complex structures and study of twistor spaces.
result Existence of generalized hypercomplex structures on -dimensional tori with non-maximal types.
New method constructs stable generalized complex structures.
problem Creating stable generalized complex structures.
method Developed Gompf-Thurston symplectic techniques adapted to Lie algebroids, used to construct stable structures from log-symplectic structures.
result Introduced boundary Lefschetz fibrations to obtain stable structures from genus one Lefschetz fibrations.
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-…
Study neighbourhoods of submanifolds in generalized complex geometry.
problem Understanding the structure and deformations of submanifolds in generalized complex geometry.
method Analytical tools including Hodge decompositions and Nash-Moser algorithm.
result Explicit conditions for B-field equivalence of holomorphic Poisson structures.
New subalgebra concept helps classify invariant complex structures.
problem Classifying invariant generalized complex structures on Lie groups.
method Introducing subregular subalgebras and using them to construct invariant structures.
result Computed all invariant generalized complex structures for real forms of G2.
Generalized complex structures on certain torus bundles are explored.
problem Exploring generalized complex structures on specific torus bundles.
method Analyzing principal torus bundles over complex manifolds with even dimensional fibers and characteristic class of type (1,1).
result Generalized complex structures on these bundles are equivalent to products of complex and symplectic structures in tubular neighborhoods of fibers.
New geometric structures on surfaces generalize complex and real Lie algebra properties.
problem Generalizing geometric structures associated with Lie algebras.
method Define and analyze generalizations of punctual Hilbert schemes for complex and real Lie algebras.
result Construct geometric structures homeomorphic to Hitchin components.
Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometrymath.DG
The paper extends symplectic techniques to generalized complex geometry.
problem Creating stable generalized complex structures on high-dimensional manifolds.
method Introducing generalized Luttinger surgery and generalized Gluck twist.
result Produced stable generalized complex structures with non-homotopy-equivalent components.
New aspects of the ddc-lemmamath.DG
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…
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…
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…
Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfacesmath.DG
Study flows on complex surfaces to find weak hyperKähler structures.
problem Classifying nondegenerate generalized Kähler surfaces.
method Generalized Kähler-Ricci flow on complex surfaces with nondegenerate Poisson structure.
result Long time existence and convergence to weak hyperKähler structure.
Study invariant structures on flag manifolds using transformations and pure spinors.
problem Understanding invariant generalized complex and Kähler structures on flag manifolds.
method Description of moduli spaces using invariant structures, Weyl group action, and pure spinors.
result Alternative description and cell decomposition of moduli spaces.
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…
The paper extends a theorem for complex structures on Lie groups to Courant algebroids.
problem Characterizing integrable generalized complex structures on transitive Courant algebroids.
method Analyzing skew-symmetric fields of endomorphisms and their closure under the Dorfman bracket.
result Local form of integrable generalized complex structures is determined under certain conditions.
On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M,j), we define six families of distinguished generalized complex or paracomplex structures on M. Each one of them in…
Characterizes structures on generalized tangent bundles and CRF-structures.
problem Understanding structures on generalized tangent bundles and CRF-structures.
method Equivalent characterizations and spinor formalism for CRF-structures.
result Characterization of generalized complex manifolds as products and infinitesimal deformations of CRF-structures.
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…
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.
Abstracts Morimoto's theorem to generalized F-structures.
problem Characterizing complex structures on non-global product manifolds.
method Abstracts Morimoto's construction to generalized F-structures.
result Characterizes invariant generalized complex structures.
The twistor construction is applied for obtaining examples of generalized complex structures (in the sense of N. Hitchin) that are not induced by a complex or a symplectic structure.
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 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 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.
The basic -lemmamath.DG
Paper develops theory of transverse generalized complex structures and proves a key lemma.
problem Proving equivalent conditions to the basic -lemma.
method Describing transverse symplectic structure and relating the lemma to the Lefschetz map.
result Justified approach and proved equivalent conditions to the basic -lemma.
Constructs complex structures on bundles, leading to new non-Kähler manifolds.
problem Creating integrable complex structures on bundles.
method Construction of integrable complex structures on total spaces of principal bundles.
result Yields new non-Kähler compact complex manifolds and SKT metrics.
Classifies and computes cohomologies of complex structures on Lie groups.
problem Classifying and computing cohomologies of complex structures on Lie groups.
method Complete classification and computation of invariant cohomologies for left invariant structures.
result Computed invariant cohomologies for various generalized complex and Kähler structures.
Algebraic dimension is zero for generic complex structures on hypercomplex nilmanifolds.
problem Understanding the algebraic dimension of complex subvarieties in hypercomplex nilmanifolds.
method Analyzing the algebraic dimension of complex subvarieties of hypercomplex nilmanifolds using properties of hypercomplex structures and Lie algebras.
result For generic complex structures, the algebraic dimension of complex subvarieties in hypercomplex nilmanifolds is zero.
In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which a…
Study integrability of generalized almost complex structures on S^6.
problem Integrability of generalized almost complex structures on the 6-dimensional sphere.
method Local coordinate criteria for integrability with respect to brackets and Courant integrability for strong structures.
result No nontrivial spherical combinations of the canonical structures are integrable with respect to the Levi-Civita connection.
Study integrability of specific geometric structures on odd Courant algebroids.
problem Characterize integrability of B_n-generalized structures on odd exact Courant algebroids.
method Characterize integrability in terms of existence of adapted generalized connections.
result Describe affine spaces of adapted generalized connections for integrable structures.
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 to an arbitrary generalized complex manifold . The theory is invariant under the diffeomor…
We shall introduce the notion of logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a logarithmic symplectic structure has unobstruc…
We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of Conn's linearization theorem.