The study proves completeness of certain Kähler manifolds and their families.
problem Completeness of projective special Kähler and quaternionic Kähler manifolds.
method Analyzing boundary behavior and cubic prepotential to prove completeness and define families of manifolds.
result Proves completeness of projective special Kähler manifolds with regular boundary behavior and cubic prepotential.
Study on special metrics on complex manifolds, focusing on degeneracy.
problem Understanding degenerate metrics on complex manifolds.
method Investigation of special-Hermitian metrics, including Kähler and locally conformally Kähler metrics.
result Characterization of degenerate metrics on non-Kähler manifolds.
The paper characterizes and classifies projective special Kähler manifolds.
problem Characterizing and classifying projective special Kähler manifolds.
method Intrinsic characterisation using a symmetric tensor with differential and algebraic conditions.
result The tensor vanishes for standard structures, leading to classification of 4D manifolds.
The paper establishes a correspondence between special Kähler manifolds and their deformations.
problem Mapping between affine and projective special Kähler manifolds.
method Formulation of a correspondence and application to r-maps in string theory.
result One-parameter deformations of projective special Kähler manifolds correspond to perturbative α'-corrections.
The paper proves rigidity properties of holomorphic isometries into homogeneous Kähler manifolds.
problem Rigidity of holomorphic isometries into homogeneous Kähler manifolds.
method Analyzing Kähler-Ricci solitons, flat spaces, and homogeneous bounded domains.
result Strong extensions of rigidity results in previous studies.
We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form ω. This generalises Freed's definition …
We give an intrinsic definition of the special geometry which arises in global N=2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kahler manifold is so related to an integrable system. The cotangent bundle of a special …
Study on integrals on non-Kähler manifolds, focusing on special metrics.
problem Understanding Morse-type integrals on non-Kähler manifolds.
method Conjecture and proof for semipositive classes and special metrics.
result The conjecture holds for semipositive classes and certain special metrics.
Construct quaternionic-Kähler metrics from special Kähler manifolds with specific BPS structure variations.
problem Construct quaternionic-Kähler metrics from special Kähler manifolds with mutually local variations of BPS structures.
method Construct quaternionic-Kähler metrics from a conical special Kähler manifold with a certain type of mutually-local variation of BPS structures. Provide global and local explicit formulas for the quaternionic-Kähler metric.
result Construct quaternionic-Kähler metrics that are positive-definite and deform the 1-loop corrected Ferrara-Sabharval metric.
The study classifies special manifolds leading to quaternionic Kähler manifolds.
problem Classifying special projective real manifolds and their quaternionic Kähler counterparts.
method Classification through projective special real manifolds and supergravity maps.
result Found a series of complete quaternionic Kähler manifolds of co-homogeneity one.
Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.
problem Understanding the structure of holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds.
method Analyzing the properties of Lagrangian fibrations and special Kähler structures.
result Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.
The variation of Hodge structure of a Calabi-Yau 3-fold induces a canonical Kähler metric on its Kuranishi moduli space, known as the Weil-Petersson metric. Similarly, special pseudo Kähler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classific…
New homogeneous special Lagrangian submanifolds discovered in nearly Kähler CP3.
problem Exploring special Lagrangian submanifolds in nearly Kähler CP3.
method Intrinsically and extrinsically using moving frame and moment-type maps.
result Classification of totally geodesic special Lagrangian submanifolds and homogeneity of special Lagrangians.
The study explores special Ricci-Hessian equations on Kähler manifolds and identifies three types of solutions.
problem Exploring special Ricci-Hessian equations on Kähler manifolds.
method Using the Cartan-Kähler theorem and analyzing specific cases.
result Three types of solutions are identified for special Ricci-Hessian equations on Kähler manifolds.
A special Kähler-Ricci potential on a Kähler manifold is any nonconstant C∞ function τ such that J(∇τ) is a Killing vector field and, at every point with dτ=0, all nonzero tangent vectors orthogonal to ∇τ and J(∇τ) are eigenvectors of both ∇dτ and the Ricci tensor. For instan…
This work is devoted to the study of Einstein equations with a special shape of the energy-momentum tensor. Our results continue Stepanov's classification of Riemannian manifolds according to special properties of the energy-momentum tensor to Kähler manifolds. We show that in this case the number of classes reduces.
Identifies special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
problem Determining special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
method Algebraic equations, invariant distributions, deformation theory, SYZ mirror symmetry.
result Existence of topologically distinct SLags and non-Kähler SYZ mirrors.
We determine all complete projective special real surfaces. By the supergravity r-map, they give rise to complete projective special Kähler manifolds of dimension 6, which are distinguished by the image of their scalar curvature function. By the supergravity c-map, the latter manifolds define in turn complete quaternio…
Under the action of the c-map, special Kahler manifolds are mapped into a class of quaternion-Kahler spaces. We explicitly construct the corresponding Swann bundle or hyperkahler cone, and determine the hyperkahler potential in terms of the prepotential of the special Kahler geometry.
H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with nega…
We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler m…
Study classifies special 4D shapes with certain curvature.
problem Classifying specific types of 4D shapes.
method Classifying compact almost-Kähler four manifolds with nonnegative biorthogonal curvature.
result Classified compact almost-Kähler four manifolds with nonnegative biorthogonal curvature.
New non-Kähler complex manifolds found with special metrics.
problem Finding non-Kähler compact complex manifolds with specific metrics.
method Constructing manifolds with balanced and astheno-Kähler metrics.
result Non-Kähler compact complex manifolds with balanced and astheno-Kähler metrics exist for every dimension n≥4. The Euler number of special symplectic hyperbolic manifolds is positive.
problem Understanding the Euler number of symplectic hyperbolic manifolds.
method Study L2-harmonic forms on the universal covering space and prove the Singer conjecture. result The Euler number of a special symplectic manifold satisfies (−1)nχ(X)>0. Study on special Kähler metrics with specific asymptotic properties.
problem Finding complete constant scalar curvature Kähler metrics with Poincaré-Mok-Yau asymptotic property.
method Use of Calabi's ansatz and moment construction.
result Provide examples of such metrics on non-compact Kähler manifolds.
Existence proven for special metrics on certain complex manifolds.
problem Existence of constant scalar curvature Kähler metrics.
method Compact Kähler manifolds with semi-ample canonical bundles.
result Existence of constant scalar curvature Kähler metrics proven.
Study properties of pointwise k-slant submanifolds in Kähler manifolds.
problem Characterize the integrability of component distributions in Kähler manifolds.
method Characterization through integrability and totally geodesic cases.
result Characterize the integrability of component distributions in Kähler manifolds.
Given a special Kahler manifold M, we give a new, direct proof of the relationship between the quaternionic structure on its cotangent bundle and the variation of Hodge structures on the complexification of TM.
We proved a conjecture of D. Freed that there are no non-trivial complete special Kaehler manifolds.
Formula for tube volume on special geometric manifolds.
problem Calculating volumes of geometric structures.
method Using Einstein constant and polarization volume.
result Formula for renormalized tube volume.
The paper proves rigidity results for manifolds with special holonomy.
problem Proving rigidity results for compact Riemannian manifolds with special holonomy.
method Using divergence free Weyl tensors and curvature operators, the paper proves similar results for manifolds with special holonomy.
result The paper proves that manifolds with special holonomy are locally symmetric or conformally equivalent to a quotient of the sphere.
Classifies Kähler manifolds with special gradient vector fields.
problem Characterizing Kähler manifolds with geodesic holomorphic gradients.
method Classification based on specific vector fields and integrability conditions.
result All such manifolds are biholomorphic to bundles of complex projective spaces.
Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.
problem Understanding the Kobayashi-Hitchin correspondence for specific sheaves.
method Using Hermitian-Yang-Mills flow on Kähler manifolds with simple normal crossing divisors.
result Established the correspondence for saturated reflexive parabolic sheaves.
The paper explores Hamiltonian stationary Lagrangian fibrations in various geometries.
problem Understanding HSLAG submanifolds and fibrations in different geometries.
method Deforming toric Kähler metrics into non-toric almost Kähler metrics to find HSLAG submanifolds.
result A large class of Hamiltonian stationary Lagrangian fibrations are found.
The article confirms Joyce's examples of G2-holonomy are formal spaces.
problem Formality of special holonomy manifolds and nearly Kähler manifolds.
method Analyzing rational homotopy theory and cohomology algebra.
result Joyce's examples of G2-holonomy are formal spaces.
We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N=TM carries a canonical structure of (affine) special Kähler manifold. This gives a…
Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.
problem Stability and rigidity of scalar curvature in quaternion-Kähler manifolds.
method Analysis of stability and rigidity conditions using Einstein manifold properties.
result Quaternion-Kähler manifolds of negative scalar curvature are stable and scalar curvature rigid.
Study on special metrics and deformations of solvmanifolds.
problem Existence and properties of Kähler metrics on solvmanifolds.
method Investigation of strong Kähler with torsion metrics and balanced metrics on deformations of specific solvmanifolds.
result Non-existence of certain metrics on specific solvmanifolds.
Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.
problem Existence of Poisson metrics on flat vector bundles over noncompact Riemannian manifolds.
method Generalization of Corlette-Donaldson-Hitchin-Simpson's nonabelian Hodge correspondence to noncompact Kähler manifolds.
result Existence of Poisson metrics on Higgs bundles over noncompact Kähler manifolds.
The study finds infinitely many special Lagrangian submanifolds in certain complex manifolds.
problem Existence of special Lagrangian submanifolds in log Calabi-Yau manifolds.
method Analyzes the existence of special Lagrangian submanifolds in log Calabi-Yau manifolds using Ricci-flat Kähler metrics.
result Proves existence of infinitely many disjoint special Lagrangian submanifolds in certain log Calabi-Yau manifolds.
Formulae for special almost-complex structures on Vogan diagrams.
problem Existence of special almost-complex structures on almost-Kähler manifolds.
method Combinatorics of Vogan diagrams for classical semisimple Lie groups.
result Explicit formulae for special almost-complex structures.
We study complex non-Kähler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal Kähler metrics, too.
Study on special Hermitian metrics on cohomogeneity one manifolds.
problem Characterizing and constructing Hermitian metrics on cohomogeneity one manifolds.
method Investigation of geometry of Hermitian manifolds with compact Lie group action by holomorphic isometries.
result Construction of new examples of cohomogeneity one Hermitian metrics solving specific equations.
The study generalizes twistor spinors to Kähler manifolds and finds bilinear form equations.
problem Generalizing twistor spinors to Kähler manifolds.
method Finding differential equations and reducing them to conformal Killing-Yano equations.
result Bilinear forms of Kählerian twistor spinors reduce to Kählerian conformal Killing-Yano equations under certain conditions.
We give an explicit formula for the quaternionic Kähler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic Kähler. A similar explicit formula is given for the analogous (K/K) correspondenc…
Study shows special Kähler geometry on base of holomorphic Lagrangian fibrations implies projective space.
problem Understanding the geometry of holomorphic Lagrangian fibrations.
method Using special Kähler geometry and parallel splitting of tangent bundles.
result Holomorphic Lagrangian fibrations over non-projective spaces are impossible.
The term "special biconformal change" refers, basically, to the situation where a given nontrivial real-holomorphic vector field on a complex manifold is a gradient relative to two Kähler metrics, and, simultaneously, an eigenvector of one of the metrics treated, with the aid of the other, as an endomorphism of the tan…
We show that the contact reduction can be specialized to Sasakian manifolds. We link this Sasakian reduction to Kähler reduction by considering the Kähler cone over a Sasakian manifold. We present examples of Sasakian manifolds obtained by S1 reduction of standard Sasakian spheres.