scholarly journals Complex hyperkähler structures defined by Donaldson–Thomas invariants

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Tom Bridgeland ◽  
Ian A. B. Strachan
Keyword(s):  
Complex Manifold ◽  
Geometric Structure ◽  
Tangent Bundle ◽  
Total Space ◽  
Stability Conditions ◽  
Link Type ◽  
Hyperkähler Metrics

AbstractThe notion of a Joyce structure was introduced in Bridgeland (Geometry from Donaldson–Thomas invariants, preprint arXiv:1912.06504) to describe the geometric structure on the space of stability conditions of a $$\hbox {CY}_3$$ CY 3 category naturally encoded by the Donaldson-Thomas invariants. In this paper we show that a Joyce structure on a complex manifold defines a complex hyperkähler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkähler metrics in geometric terms.

scholarly journals Negative Vector Bundles and Complex Finsler Structures

1975 ◽  
Vol 57 ◽  
pp. 153-166 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Complex Manifold ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Finsler Structure

A complex Finsler structure F on a complex manifold M is a function on the tangent bundle T(M) with the following properties. (We denote a point of T(M) symbolically by (z, ζ), where z represents the base coordinate and ζ the fibre coordinate.)

The Tangent Bundle of an Almost Complex Manifold

2001 ◽  
Vol 44 (1) ◽  
pp. 70-79 ◽  
Author(s):  
László Lempert ◽  
Róbert Szőke
Keyword(s):  
Complex Manifold ◽  
Tangent Bundle ◽  
Deformation Theory ◽  
Complex Structure ◽  
Almost Complex Structure ◽  
Holomorphic Maps ◽  
Almost Complex Manifolds ◽  
Almost Complex Manifold ◽  
Almost Complex ◽  
Natural Way

AbstractMotivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complexmanifold with an almost complex structure. We describe various properties of this structure.

scholarly journals The projective curvature of the tangent bundle with natural diagonal metric

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 401-410 ◽  
Author(s):  
Cornelia-Livia Bejan ◽  
Simona-Luiza Druţă-Romaniuc
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Total Space ◽  
Projectively Flat ◽  
Projective Curvature ◽  
Necessary And Sufficient

Our study is mainly devoted to a natural diagonal metric G on the total space TMof the tangent bundle of a Riemannian manifold (M, 1). We provide the necessary and sufficient conditions under which (TM,G) is a space form, or equivalently (TM,G) is projectively Euclidean. Moreover, we classify the natural diagonal metrics G for which (TM,G) is horizontally projectively flat (resp. vertically projectively flat).

scholarly journals Parabolic conformally symplectic structures I; definition and distinguished connections

2018 ◽  
Vol 30 (3) ◽  
pp. 733-751 ◽  
Author(s):  
Andreas Čap ◽  
Tomáš Salač
Keyword(s):  
Lie Algebra ◽  
Geometric Structure ◽  
Tangent Bundle ◽  
Smooth Manifold ◽  
Symplectic Structure ◽  
Linear Connection ◽  
Underlying Structure ◽  
Normalization Condition ◽  
Lie Algebra Cohomology ◽  
First Order

AbstractWe introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type {C_{n}} and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle {\mathrm{TM}}. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

scholarly journals STABILITY CONDITIONS ON GENERIC COMPLEX TORI

2012 ◽  
Vol 23 (05) ◽  
pp. 1250035 ◽  
Author(s):  
S. MEINHARDT
Keyword(s):  
Complex Manifold ◽  
Stability Conditions ◽  
Connected Component ◽  
Complex Torus ◽  
Complex Tori ◽  
Simply Connected

In this paper we describe a simply connected component of the complex manifold Stab(X) of stability conditions on a generic complex torus X. A generic complex torus is a complex torus X with Hp,p(X) ∩ H2p(X, ℤ) = 0 for all 0 < p < dim X.

scholarly journals DIRAC STRUCTURES FROM LIE INTEGRABILITY

2012 ◽  
Vol 09 (04) ◽  
pp. 1220005 ◽  
Author(s):  
MIRCEA CRASMAREANU
Keyword(s):  
Tangent Bundle ◽  
Riemannian Geometry ◽  
Fiber Bundle ◽  
Symplectic Structure ◽  
Total Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
3 Dimensional ◽  
Ehresmann Connection

We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.

scholarly journals Lagrange geometry on tangent manifolds

2003 ◽  
Vol 2003 (51) ◽  
pp. 3241-3266 ◽  
Author(s):  
Izu Vaisman
Keyword(s):  
Tangent Bundle ◽  
Tensor Field ◽  
Finsler Geometry ◽  
Interesting Case ◽  
Lagrangian Function ◽  
Total Space ◽  
Lagrangian Functions ◽  
Tangent Manifold

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a nondegenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper, we study a generalization which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples and find the cohomological obstructions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, while giving an index-free presentation of these connections.

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