Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.
problem Investigating volume invariants for Hermitian-symplectic metrics.
method Introducing a functional acting on metrics in Aeppli cohomology classes and proving critical points are Kähler.
result The volume invariant generalises the volume of a Kähler class and vanishing is a necessary condition for the existence of a Kähler metric.
A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form ω, for which the bilinear form ω(I⋅,⋅) is positive definite. In this work we prove ddc-lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifol…
Derivative estimates for pluriclosed flow control curvature and torsion.
problem Deriving derivative estimates for the pluriclosed flow.
method Control higher order derivatives of Chern curvature and torsion using Chern curvature; derive an estimate for torsion tensor using Chern Ricci curvature in dimension two; find a monotonic quantity in Hermitian-symplectic case.
result All Hermitian-symplectic solitons are Kähler Ricci solitons.
Degenerate twistor deformations of Kähler manifolds are also Kähler.
problem Understanding the Kähler structure of degenerate twistor deformations.
method Using positive currents, Hahn–Banach theorem, and Huybrechts's theorem.
result Degenerate twistor deformations of compact holomorphically symplectic Kähler manifolds are Kähler.
Deforms orbits in Lie algebras to Lagrangian submanifolds.
problem Deforming orbits in semisimple Lie algebras.
method Coadjoint orbit deformation and Hermitian symplectic form.
result Constructs Lagrangian submanifolds.
Research confirms Streets-Tian conjecture for 2-step solvmanifolds.
problem Streets-Tian conjecture for compact complex manifolds.
method Used special non-unitary frames to reveal hidden symmetries.
result Confirms conjecture for all 2-step solvmanifolds.
In this paper, we introduce the notions of p-Hermitian-symplectic and p-pluriclosed compact complex manifolds as generalisations for an arbitrary positive integer p not exceeding the complex dimension of the manifold of the standard notions of Hermitian-symplectic and SKT manifolds that correspond to the case $p=…
The study finds Lagrangian submanifolds in adjoint semisimple orbits for real forms.
problem Characterizing Lagrangian submanifolds in adjoint semisimple orbits.
method Analyzing real flags and orbits of real forms with respect to symplectic forms.
result Classification of infinitesimally tight Lagrangian submanifolds in the compact case and Lagrangian submanifolds in the complex case.
The Streets-Tian conjecture is confirmed for Lie algebras with specific abelian ideals.
problem The Streets-Tian conjecture on compact complex manifolds admitting Hermitian-symplectic metrics.
method Detailed case analysis of Lie algebras with abelian ideals of codimension 2, explicit construction of Hermitian-symplectic metrics and pathways to Kähler metrics.
result The Streets-Tian conjecture is confirmed for Lie algebras containing abelian ideals of codimension 2.
Derives criteria for Kähler structures on holomorphic submersions.
problem Criteria for Kähler structures on holomorphic submersions.
method Derives a criterion for Kähler structures using holomorphic submersions.
result Proves Kähler structures for certain holomorphic submersions.
The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.
problem Deformations of Calabi-Yau manifolds under co-polarised conditions.
method Analyzes local deformations of Calabi-Yau ∂∂ˉ-manifolds using Gauduchon metrics and constructs a new hp-HS form. result Proves the p-SKT h-∂∂ˉ-property is deformation open. The Streets-Tian conjecture is confirmed for specific types of Hermitian manifolds.
problem The Streets-Tian conjecture on compact Hermitian manifolds.
method Elementary approach, explicit descriptions, and pathways of deformation.
result The conjecture is confirmed for special types of compact Hermitian manifolds.
Study shows compact Vaisman manifolds cannot have certain special Hermitian metrics.
problem Compact Vaisman manifolds and their compatibility with special Hermitian structures.
method Proof of non-existence of specific Hermitian metrics on compact Vaisman manifolds.
result Compact Vaisman manifolds cannot admit special Hermitian metrics like special k-Gauduchon metrics or pluriclosed metrics. We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically H…
The paper explores spaces of Kähler and symplectic forms on 4-manifolds.
problem Investigating the properties of Kähler and symplectic forms on 4-manifolds.
method Analyzing the uniqueness, connectedness, and openness of spaces of Kähler forms and introducing holomorphically tamed symplectic forms.
result Formulated a parallel question for holomorphically tamed symplectic forms and related it to Kähler-type symplectic forms.
Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.
problem Computing invariants on six-dimensional solvmanifolds.
method Computed almost-complex and almost-Hermitian invariants on families of solvmanifolds.
result Provides obstructions to symplectic structures on compact almost-complex manifolds.
We study the stability of compact pseudo-Kähler manifolds, i.e. compact complex manifolds X endowed with a symplectic form compatible with the complex structure of X. When the corresponding metric is positive-definite, X is Kähler and any sufficiently small deformation of X admits a Kähler metric by a well-know…
Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form on M is closed, we study deformations of a compact coassociative submanifold N wi…
The article studies critical points of a new energy functional in higher dimensions.
problem Investigating critical points of a new energy functional in higher dimensions.
method Holomorphic deformations, closed and open properties, differential of the functional.
result Properties of critical points under holomorphic deformations are closed and open.
Introduces new cohomologies on complex manifolds, extending classical Bott-Chern and Aeppli.
problem Characterizing and studying cohomologies on complex manifolds.
method Introducing Er-Bott-Chern and Er-Aeppli cohomologies, extending classical cohomologies. result Provides analogues of Serre duality and characterizes page-(r−1)-∂∂ˉ-manifolds. In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler mani…