arXiv research

Paper constructs solutions to a system using Aeppli class without auxiliary gauge connection.

The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.

The study explores polarized deformations of SKT Calabi-Yau manifolds using Aeppli classes.

Let $(M,J,g,ω)$ be a complete Hermitian manifold of complex dimension $n\ge2$. Let $1\le p\le n-1$ and assume that $ω^{n-p}$ is $(\partial+\overline{\partial})$-bounded. We prove that, if $ψ$ is an $L^2$ and $d$-closed $(p,0)$-form on $M$, then $ψ=0$. In particular, if $M$ is compact, we derive that if the Aeppli class…

The paper explores moduli space of heterotic system using two deformation paths.

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Her…

Study canonical deformations of complex forms and their cohomology properties.

We propose the study of a Monge-Ampère-type equation in bidegree $(n-1,\,n-1)$ rather than $(1,\,1)$ on a compact complex manifold $X$ of dimension $n$ for which we prove uniqueness of the solution subject to positivity and normalisation restrictions. Existence will hopefully be dealt with in future work. The aim is to…

Study bigraded formality and Aeppli-Bott-Chern-Massey products on complex manifolds.

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the $(1,1)$ curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. …

Study $L^2$ Hilbert complexes on complex manifolds.

In studying the Bott-Chern and Aeppli cohomologies for q-complete manifolds, we introduce the class of cohomologically Bott-Chern q-complete manifolds.

A notion of geometric formality in the context of Bott-Chern and Aeppli cohomologies on a complex manifold is discussed. In particular, by using Aeppli-Bott-Chern-Massey triple products, it is proved that geometric Aeppli-Bott-Chern formality is not stable under small deformations of the complex structure.

We give the complete Bott-Chern-Aeppli cohomology for compact complex 3-folds in terms of Dolbeault, Frolicher, a bi-degree DeRham-like type of cohomology, $K^{p,q}$, defined as $$ K^{p,q}=\frac{ker( \partial ) \cap ker( {\bar{\partial}}) }{im( \partial )\cap ker( {\bar{\partial}} )+im( {\bar{\partial}})\cap ker( \part…

Introduces new cohomologies on complex manifolds, extending classical Bott-Chern and Aeppli.

Study primitive decompositions for harmonic forms on almost Kähler manifolds.

New cohomological obstruction found for astheno-Kahler metrics.

Inequalities for symplectic cohomology groups are derived.

The paper introduces new cohomologies and studies harmonic forms on almost complex manifolds.

The study proves stability of a flow on specific Lie groups.

The article studies critical points of a new energy functional in higher dimensions.

We define Aeppli and Bott-Chern cohomology for bi-generalized complex manifolds and show that they are finite dimensional for compact bi-generalized Hermitian manifolds. For totally bounded double complexes $(A, d', d'')$, we show that the validity of $d'd''$-lemma is equivalent to having the same dimension of several …

New findings on complex manifold properties under deformations.

We prove that, for some classes of complex nilmanifolds, the Bott-Chern cohomology is completely determined by the Lie algebra associated to the nilmanifold with the induced complex structure. We use these tools to compute the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, com…

The paper characterizes when the $\partial \overline{\partial}$-lemma holds for twistor spaces.

We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality à la Frölicher relating the dimensions of the Bott-Chern and Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove that the equa…

Survey on Strominger system and Ricci flow in non-Kähler geometry.

By comparing Deligne complex and Aeppli-Bott-Chern complex, we construct a differential cohomology $\widehat{H}^*(X, *, *)$ that plays the role of Harvey-Lawson spark group $\widehat{H}^*(X, *)$, and a cohomology $H^*_{ABC}(X; \Z(*, *))$ that plays the role of Deligne cohomology $H^*_{\mathcal{D}}(X; \Z(*))$ for every …

Paper solves a metric-independent problem on almost Kähler 4-manifolds.

Classifies and computes cohomologies of complex structures on Lie groups.

We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-…

The study examines harmonic forms on almost Hermitian manifolds and complex surfaces.

The solution of the Calabi Conjecture by Yau implies that every Kähler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\textrm{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^{1,1}(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-Kähle…

The study examines stability of fibres on Hopf surfaces as harmonic maps and minimal surfaces.

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Che…

We compute the double complex of smooth complex-valued differential forms on projective bundles over and blow-ups of compact complex manifolds up to a suitable notion of quasi-isomorphism. This simultaneously yields formulas for 'all' cohomologies naturally associated with this complex (in particular, de-Rham, Dolbeaul…

The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.

Decomposes harmonic forms on almost Kähler manifolds, revealing non-trivial structure.

This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kähler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to exp…

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the $\partial\overline\partial$-Lemma, is characterized in terms of the strongly Gauduchon cone and of the first $\partial\overl…

Study cohomologies of complex manifolds with symplectic forms and their stability.

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=…

New insights prevent certain types of metrics on compact spaces.

We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent useful tools in studying non Kähler geometry. We give an overview on the comparis…

Proves Kato manifolds satisfy Hodge decomposition.

Study on geometrically formal metrics on complex manifolds.

The paper computes non-trivial triple Massey products on specific non-Kähler solvmanifolds.

In this note we study the optimal dividend problem for a company whose surplus process, in the absence of dividend payments, evolves as a generalized compound Poisson model in which the counting process is a generalized Poisson process. This model including the classical risk model and the Polya-Aeppli risk model as sp…