The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.
problem Characterizing normal subgroups of Kähler groups.
method Analyzing embeddings and conjugation actions of surface groups and one-ended hyperbolic groups.
result Restrictions on normal subgroups of Kähler groups, including virtual direct products and surface group properties.
Computes Weyl group of Kähler toric manifold isometries.
problem Computing the Weyl group of Kähler toric manifold isometries.
method Analyzes the group of holomorphic isometries of a Kähler toric manifold with real analytic Kähler metric.
result Computed the Weyl group of the group of holomorphic isometries.
Study shows Kahler extensions of abelian groups are products or related to mapping class groups.
problem Characterizing Kahler extensions of abelian groups.
method Surface topology and restrictions on Kahler groups.
result Kahler extensions are virtually products or related to mapping class groups.
The BNS invariant is applied to Kähler groups in new proofs and results.
problem Understanding properties of Kähler groups through the BNS invariant.
method Applications of the Bieri-Neumann-Strebel invariant on Kähler groups.
result Amenable Kähler groups have an empty complement of the BNS invariant.
The study examines when braid groups of manifolds are Kähler.
problem When are braid groups of manifolds Kähler?
method The approach uses previous results on Kähler groups and homological properties of braid groups.
result With some exceptions, the pure braid group of a Riemann surface with at least 2 strands is never Kähler.
New Kähler groups found from surface groups.
problem Understanding Kähler subgroups of direct products of surface groups.
method Construction of infinite classes of Kähler groups.
result New classes of irreducible, coabelian Kähler subgroups of direct products of surface groups.
The study explores the Dehn functions of Kähler groups and their properties.
problem Which functions can arise as Dehn functions of Kähler groups?
method Analyzes examples of Kähler groups with various Dehn functions and proves the existence of a Kähler group with a cubic bounded Dehn function.
result There exists a Kähler group with a cubic bounded Dehn function and an exponential upper bound.
Research explores Kähler and semi-para-Kähler structures on specific Lie groups.
problem Existence of Kähler and semi-para-Kähler structures on six-dimensional unsolvable Lie groups.
method Examines four specific Lie algebras and their structures.
result One Lie algebra admits Kähler metrics, others admit semi-para-Kähler and semi-Kähler structures.
Explicit isometry groups found for nearly Kähler manifolds.
problem Understanding the symmetries of nearly Kähler manifolds.
method Alternative, less algebraic approach to find isometry groups.
result Explicit expression for isometry groups of six-dimensional nearly Kähler manifolds.
Constructs special Kähler structures on Lie groups.
problem Creating special Kähler structures on Lie groups.
method Introducing twisted cartesian product and double extension process.
result Characterizes left invariant flat special Kähler structures.
We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral sym…
Study of Sasaki groups vs Kähler groups, showing distinct behaviors.
problem Comparing fundamental groups of Sasaki and Kähler manifolds.
method Analysis of properties and behavior of Sasaki groups compared to Kähler groups.
result Sasaki groups exhibit distinct behavior from Kähler groups, not closed under direct products, and have upper bounds on dimension.
The paper shows how certain Kähler groups are uniquely determined by their profinite completions.
problem Understanding the uniqueness of Kähler groups within residually finite groups.
method Holomorphic fibrations and profinite completions of fundamental groups.
result Aspherical smooth projective varieties are determined by their algebraic fundamental groups.
The paper describes geodesics on a Kähler cone of the Heisenberg group.
problem Understanding geodesics on the Kähler cone of the Heisenberg group.
method Analyzing the Heisenberg group to describe geodesics.
result It is not a complete manifold.
The paper explores p-Kähler structures on Lie group quotients.
problem Existence of p-Kähler structures on compact quotients of Lie groups. method Analyzes p-Kähler structures on nilmanifolds and almost abelian solvmanifolds. result Proves that (n−2)-Kähler almost abelian solvmanifolds of complex dimension n≥3 are Kähler. Introduces new deformation classes in generalized Kähler geometry.
problem No specific problem stated; focuses on new concepts.
method Uses Courant symmetry group to introduce deformation classes.
result Generalized Kähler cone is preserved by the generalized Kähler-Ricci flow.
Study semi-Kähler structures on specific Lie groups without symplectic structures.
problem Exploring structures on Lie groups without symplectic structures.
method Defined semi-Kähler and almost para-semi-Kähler structures on specific Lie groups.
result Geometric properties of these structures are studied.
Study Kähler groups from orbifold compactifications of curve moduli.
problem Shafarevich conjecture on holomorphic convexity.
method Quantum representations of orbifold compactifications and mapping class groups.
result Construct interesting Kähler groups and settle most conjectures.
Existence of Kähler-Einstein metrics on compactifications of Lie groups.
problem Existence of Kähler-Einstein metrics on Q-Fano compactifications of Lie groups. method Proving existence through compactifications of Lie groups.
result Classification of Q-Fano compactifications of SO4(C) with Kähler-Einstein metrics. Study symplectically aspherical Kähler manifolds with unique properties.
problem Existence and properties of symplectically aspherical Kähler manifolds.
method Detailed study and analysis of geometric and topological features.
result Existence of symplectically aspherical Kähler manifolds with large fundamental groups.
Study para-complex structures on specific Lie groups, finding explicit forms and properties.
problem Characterizing para-Kähler structures on six-dimensional nilpotent Lie groups.
method Examined left-invariant para-complex structures on six-dimensional nilpotent Lie groups, obtained explicit expressions and investigated curvature properties.
result Para-complex structures are nilpotent and para-Kähler metrics are Ricci-flat.
The paper studies Kähler groups and their virtual algebraic fibrations, finding implications for their fundamental groups.
problem Understanding the structure of Kähler groups and their finite index subgroups.
method Analyzing the virtual algebraic fibrations and Albanese dimension of Kähler groups.
result Groups with virtual Albanese dimension ≤ 1 have strong relations with surface groups and coincide with Green--Lazarsfeld sets.
Study unipotent group actions on Kähler manifolds using moment maps.
problem Analyzing unipotent group actions on compact Kähler manifolds.
method Moment map techniques, stability conditions, symplectic reduction.
result Geometric quotients with compactifiable Kähler structures.
Study of M5-brane flows leading to Kähler-Einstein metrics.
problem Understanding Kähler uniformization in M-theory.
method Holographic renormalization group flows in seven-dimensional gauged supergravity.
result Reach a Kähler-Einstein metric at the infrared fixed point.
Paper proves Kähler-Ricci solitons on specific compactifications.
problem Existence of Kähler-Ricci solitons on compact manifolds.
method Continuity method applied to wonderful group compactifications.
result Existence of Kähler-Ricci solitons on certain compactifications.
New structure found on Lie group tangent bundle.
problem Finding new structures on Lie group tangent bundles.
method Analyzing left invariant structures on Lie groups.
result Tangent bundle of Lie group admits a left-invariant nearly pseudo-Kähler structure.
Flat Hermitian Lie algebras are always Kähler.
problem Classifying Lie groups with Hermitian structures that are flat.
method Analysis on the Hermitian geometry of 2-step solvable Lie groups.
result Flat Hermitian Lie algebras are Kähler.
New groups without finite classifying spaces found in Kodaira fibrations.
problem Finding groups without finite classifying spaces.
method Constructing Kähler groups as fundamental groups of Kodaira fibrations.
result Found Kähler groups that are not commensurable to surface groups and do not have finite classifying spaces.
The paper presents a classification theorem for the class of flat connections with triangular (0,1)-components on a topologically trivial complex vector bundle over a compact Kahler manifold. As a consequence we obtain several results on the structure of Kähler groups, i.e., the fundamental groups of compact Kahler man…
Develops scalar curvature in generalized Kahler geometry and shows constant scalar curvature on compact Lie groups.
problem Defines scalar curvature in generalized Kahler geometry.
method Introduces scalar curvature in terms of pure spinors formalism and develops a moment map framework.
result Scalar curvature is given by the moment map, generalizing results from ordinary Kahler geometry.
Study shows complex affine transformations index is at most 2 for Kähler manifolds.
problem Understanding transformations on Kähler manifolds.
method Analyzing groups of transformations on Kähler manifolds.
result Establishes a stronger version of Yano-Obata conjecture for complete Kähler manifolds.
In this paper we apply the hyper-Kähler quotient construction to Lie groups with a left invariant hyper-Kähler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-Kähler metrics can be recovered in this way by considering different Lie grou…
Complete classification of para-Kähler structures on Lie groups.
problem Classifying para-Kähler structures on Lie groups.
method Complete classification via automorphism consideration.
result Classification of para-Kähler structures on four-dimensional Lie groups.
Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
Counterexamples found for Torelli groups of Kähler and hyper-Kähler manifolds.
problem Torelli groups of Kähler and hyper-Kähler manifolds are not always finite.
method Construction of homomorphisms and analysis of diffeomorphisms' actions.
result Found counterexamples showing Torelli groups are not always finite.
The Hodge theorem connects cohomology groups on compact Kähler manifolds.
problem Establishing a relationship between cohomology groups on compact Kähler manifolds.
method Proving the Hodge decomposition theorem on compact d-Kähler manifolds.
result Hodge decomposition theorem on compact d-Kähler manifolds.
Left-invariant metrics force 2-step nilpotent groups, preserving Kähler-like conditions.
problem Existence of left-invariant pluriclosed Hermitian metrics on Lie groups.
method Analyzing left-invariant metrics on unimodular Lie groups with abelian complex structures.
result Pluriclosed flow preserves Strominger Kähler-like conditions on 2-step nilpotent Lie groups.
The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.
problem Classifying Q-Fano compactifications of semisimple groups with Kähler-Einstein metrics. method Proving finiteness through classification of compactifications.
result There are only finitely many Q-Fano compactifications of semisimple groups with Kähler-Einstein metrics. The paper proves uniqueness and existence of Kähler-Einstein metrics on certain compactifications.
problem Existence and uniqueness of Kähler-Einstein metrics on Q-Fano group compactifications. method Analyzes Q-Fano group compactifications, proving uniqueness and existence of Kähler-Einstein metrics. result Proves the existence and uniqueness of Kähler-Einstein metrics on Q-Fano group compactifications. Researchers find symplectic and Kähler structures on biquotients.
problem Constructing and investigating symplectic and Kähler structures on biquotients.
method Constructing symplectic structures on biquotients and finding Kähler structures.
result Another Kähler structure found on a biquotient SU(4)//T3. The subject of this paper is six-dimensional nearly (para-)Kähler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly Kähler manifold and prove applications to the automorphism group of a nearly (para-)Kähler structure. Secondly…
We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical …
Fundamental groups of certain Kähler orbifolds have polynomial growth.
problem Understanding the fundamental groups of specific types of orbifolds.
method Analyzing the orbifold fundamental group with respect to the nef anticanonical bundle.
result The orbifold fundamental group has polynomial growth.
The study examines Kähler structures on coadjoint orbits of Lie groups using coherent and squeezed states.
problem Does the coadjoint orbits of Lie groups support a Kähler structure?
method Examined three Lie groups: Weyl-Heisenberg, SU(2), and SU(1,1). Used coherent and squeezed states to explore Kähler structures.
result Coherent states provide Kähler embeddings, while squeezed states only symplectic embeddings.
The fundamental group of a Riemannian manifold with δ-pinched negative curvature, δ>1/4, cannot be the fundamental group of a quasicompact Kähler manifold. The proof also implies that a non-uniform lattice in F4(−20) cannot be the fundamental group of a quasicompact Kähler manifold. We also construct examples …
Study shows Kähler gradient Ricci solitons have limited symmetry groups.
problem Understanding the symmetry groups of Kähler gradient Ricci solitons.
method Connection to almost contact metric structure.
result Group of isometries is at most n^2, with equality characterized.
Study of Hermitian structures on Lie groups with 2D commutator subgroups.
problem Classifying Hermitian structures on Lie groups with specific commutator subgroups.
method Explicit classification of Type I and Type II structures, computation of Bismut connections, and examples of Kahler structures.
result Classification of Kahler structures within Type I and Type II structures.
Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.
problem Constructing non-homogeneous quaternionic Kähler manifolds with finite volume ends.
method Using cohomogeneity one deformation of symmetric spaces, the researchers constructed manifolds with specific fundamental groups.
result The constructed manifolds are aspherical and have finite volume ends, not locally homogeneous.