arXiv research

Let $(M, \bar g)$ be an $n$-dimensional complete Riemannian manifold. In this paper, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $Ω$ with $\partial Ω$, can one find a conformal metric $g$ whose scalar curvature $R[g]\ge R[\bar g]$ on $Ω$ and the mean curvature $…

We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let $B_1$ be an open ball in $\mathbb{R}^n$ and $B_0$ be a ball contained in $B_1$. Let $ν$ be the outward unit normal on $\partial B_1$. Then the first eigenvalue o…

We adapt the results of Part 1 to include the unit ball in the Heisenberg group, the model domain with characteristic boundary points. In particular, we construct function spaces on which the Kohn Laplacian with the \bar{\partial}_b-Neumann boundary conditions is an isomorphism. As an application, we establish sharp re…

New geometric conditions ensure compactness of $\bar{\partial}$-Neumann problem.

We establish sharp regularity and Fredholm theorems for the \bar{\partial}_b-Neumann problem on domains satisfying some non-generic geometric conditions. We use these domains to construct explicit examples of bad behaviour of the Kohn Laplacian: it is not always hypoelliptic up to the boundary, its partial inverse is n…

We introduce an effective method to solve the $\bar\partial$-harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE system to countably many linear ODE systems. By solving a fundamental problem on line…

Let $Ω$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator$N$exists for$(p,q)$-forms on$Ω$. Furthermore, there exists a$t_0>0$such that the operators$N$,$\bar\partial^*N$,$\bar\partial N$ and the Bergman projection are regular in the Sobolev …

Paper introduces a new operator and solves equations on higher-dimensional almost Kähler manifolds.

The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.

On a compact $\partial\bar\partial$-manifold $X$, one has the Hodge decomposition: the de Rham cohomology groups split into subspaces of pure-type classes as $H_{dR}^k (X)=\oplus_{p+q=k}H^{p,\,q}(X)$, where the $H^{p,\,q}(X)$ are canonically isomorphic to the Dolbeault cohomology groups $H_{\bar\partial}^{p,\,q}(X)$. F…

The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.

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

3-manifold curvature comparison with rotationally symmetric bodies.

We study the $\bar{\partial}_b$-Neumann problem for domains $Ω$ contained in a strictly pseudoconvex manifold M^{2n+1} whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts of a single CR function w. When the Kohn Laplacian is a priori known to have closed r…

For a smooth strictly plurisubharmonic function $u$ on a open set $Ω\subset\mathbb{C}^{n}$ and $F$ a $C^{1}$ nondecreasing function on $\mathbf{R}^{*}_{+}$, we investigate the complex partial differential equations $Δ_{g}\log\det(u_{i\bar j})=F(\det(u_{i\bar j}))\Vert\nabla_{g}\log\det(u_{i\bar j})\Vert_{g}^{2},$ whe…

New theorems on Hodge numbers and Kähler structures derived from complex differential forms.

We extend the notion of the symmetric signature $σ(\bar{M},r)$ in L^n(R) for a compact n-dimensional manifold M without boundary, a reference map r from M to BG and a homomorphism of rings with involutions from ZG to R to the case with boundary $\partial M$, where $(\bar{M},\bar{\partial M}) \to (M,\partial M)$ is the …

We consider the relative canonical line bundle $K_{\mathcal{X}/\mathcal{T}}$ and a relatively ample line bundle $(L, e^{-φ})$ over the total space $\mathcal{X}\to \mathcal{T}$ of fibration over the Teichmüller space by Riemann surfaces. We consider the case when the induced metric $\sqrt{-1}\partial\bar{\partial}φ|_{\…

Let $(X^{n},g_+)$ $(n\geq 3)$ be a Poincaré-Einstein manifold which is $C^{3,α}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=ρ^2g_+)$ via some boundary defining function $ρ$, there are two types of Yamabe constants: $Y(\overline{X},\pa…

Extends deformation theory to higher-page analogues of manifolds.

The Seiberg-Witten family of elliptic curves defines a Jacobian rational elliptic surface $\Z$ over $\mathbb{C}\mathrm{P}^1$. We show that for the $\bar{\partial}$-operator along the fiber the logarithm of the regularized determinant $-1/2 \log \det' (\bar\partial^* \bar\partial)$ satisfies the anomaly equation of the …

Compute Dolbeault and Bott-Chern cohomologies of complex solvmanifolds.

Let $π:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $π_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for la…

We give a simple proof of a result on the $\partial\bar{\partial}$-lemma property under a blow-up transformation by Deligne--Griffiths--Morgan--Sullivan's criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of…

Proves a general ∂∂̄-lemma and applies it to a Fujino conjecture.

In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifold $M$ satisfies the $\bar{\pa…

New complex non-Kähler manifolds with specific properties are constructed.

The limiting behavior of the normalized Kähler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi K-energy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabu…

This is a large audience version of our previous work (see math.AG/0301146) in which we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of $\bar{\partial}$-coherent sheaves. We also include here the complete proof of our main Theorem.

The purpose of this paper is to study the bimeromorphic invariants of compact complex manifolds in terms of Bott-Chern cohomology. We prove a blow-up formula for Bott-Chern cohomology. As an application, we show that for compact complex threefolds the non-Kählerness degrees, introduced by Angella-Tomassini [Invent. Mat…

Holomorphic bundles on complex manifolds with boundary are studied, extending results from Donaldson's work.

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=…

Study on harmonic forms on almost Hermitian 4-manifolds, calculating dimensions and invariants.

In this paper we study complex symplectic manifolds, i.e., compact complex manifolds $X$ which admit a holomorphic $(2, 0)$-form $σ$ which is $d$-closed and non-degenerate, and in particular the Beauville-Bogomolov-Fujiki quadric $Q_σ$ associated to them. We will show that if X satisfies the $\partial \bar{\partial}$-l…

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

In this paper, we prove that there exists a dimensional constant $δ> 0$ such that given any background Kähler metric $ω$, the Calabi flow with initial data $u_0$ satisfying \begin{equation*} \partial \bar \partial u_0 \in L^\infty (M) \text{ and } (1- δ)ω< ω_{u_0} < (1+δ)ω, \end{equation*} admits a unique short time so…

We prove that for any open orientable surface $S$ of finite topology, there exist a Riemann surface $\mathcal{M},$ a relatively compact domain $M\subset\mathcal{M}$ and a continuous map $X:\bar{M}\to\mathbb{C}^3$ such that: $\mathcal{M}$ and $M$ are homeomorphic to $S,$ $\mathcal{M}-M$ and $\mathcal{M}-\bar{M}$ contain…

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…

Preserves positive intermediate curvature on manifolds.

The procedure to remove double intersections called the Whitney trick is one of the main tools in the topology of manifolds. The analogues of Whitney trick for $r$-tuple intersections were `in the air' since 1960s. However, only recently they were stated, proved and applied to obtain interesting results. Here we prove …

Let $X$ be a compact complex manifold with trivial canonical bundle and satisfying the $\partial\bar{\partial}$-Lemma. We show that the Kuranishi space of $X$ is a smooth universal deformation and that small deformations enjoy the same properties as $X$. If, in addition, $X$ admits a complex symplectic form, then the l…

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied.…

Proves Simpson's conjecture about Higgs bundles and moduli spaces.

Associated with a smooth, $d$-closed $(1, 1)$-form $α$ of possibly non-rational De Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0, 1)$-connections $\bar\partial_k$ for which $\bar\partial_k^2\neq 0$. Their study was…

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

If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the fact that the $(1,1)$-cur…

In this paper, we solve a logarithmic $\bar{\partial}$-equation on a compact Kähler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at $E_1$, and we al…

We construct a simply-connected compact complex non-Kähler manifold satisfying the $\partial\bar\partial$-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the $\partial\bar\partial$-Lemma under modifications of compact complex m…