Marco Gualtieri
In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the $dd^c$-lemma of Kähler geometry.
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Hodge decomposition for Kato manifolds
本文证明Kato流形满足Hodge分解，并研究其Bott-Chern和Aeppli上同调与Dolbeault上同调的关系，支持局部拟正交Kähler流形的Hodge分解猜想。
Topology-preserving Hodge Decomposition in the Eulerian Representation
针对欧拉表示中带边界流形的霍奇分解难题，提出一种统一的5分量保持拓扑的霍奇分解方法，并通过数值实验验证了其有效性和严格正交性。
The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for manifold with boundary is used. The identification allows the derivation of governing differential equations for variational problems on the space of conformal vector fields. Several examples are given. In addition, the paper also gives a review, in full detail, of already known vector field decompositions involving subalgebras of volume preserving and symp…
Hodge decomposition theorem on compact $d$-Kähler manifolds
本文研究 $d$-复流形及其微分算子，证明紧 $d$-Kähler 流形上的 Hodge 分解定理，揭示 de Rham 与 Dolbeault 上同调群间的关键关系。
The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and $L^2$ forms. We further extend the Hodge decomposition to the Sobolev space $H^1$ for general $k$-forms on non-compact manifolds of nonpositive constant sectional curvature. As a result, we also obtain a decomposition on $\mathbb R^N$.
A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bi-complex where one of the two operators is the classical $\overline\partial$-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex m…
Let $G=\C^{n}\ltimes_ϕ \C^{m}$ with a semi-simple action $ϕ: \C^{n}\to GL_{m}(\C)$ (not necessarily holomorphic). Suppose $G$ has a lattice $Γ$. Then we show that in some conditions on $G$ and $Γ$, $G/Γ$ admits a Hermitian metric such that the space of harmonic forms satisfies the Hodge symmetry and decomposition. By this result we give many examples of non-Kähler Hermitian solvmanifolds satisfying the Hodge symmetry and decomposition.
We compute the Dolbeault cohomology of geodesically convex domains contained in Cousin groups which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of Oeljeklaus-Toma manifolds and in particular the fact that the Hodge decomposition holds for their cohomology.
The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.
We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that the Chevalley-Eilenberg complex of an elliptic, q-convex Lie algebroid admits a Hodge decomposition in degree q. This generalizes the well-known Hodge decompositions for the exterior derivative on real manifolds and the delbar-operator on q-convex complex manifolds. We establish t…
A generalized complex manifold which satisfies the $\partial \overline{\partial}$-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in smooth and holomorphic families of generalized complex manifolds. In particular we define period maps, prove a Griffiths transversality theorem and show that for holomorphic families the period maps are holomorphic. Further results on the Hodge decomposition for various special cases including the generalized Kähler case are obtained.
We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.
Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds
本文研究闭近 Kähler 流形上的调和形式，证明近复结构在适度可积时具有 Hodge 分解，并导出纯满性质、HLC 及四维刚性结论。
$L^p$-Hodge Decomposition with Sobolev classes in Sub-Riemannian Contact Manifolds
本文在无边的次黎曼紧接触流形上，针对Rumin复形建立了$L^p$-Hodge分解定理，并利用Morrey方法得到属于适当Sobolev类的更高正则性原项。
Pluripotential theory and special holonomy, I: Hodge decompositions and $\partial\bar\partial$ lemmata
本文证明了紧致无挠 $G_2$ 7-流形与 Calabi-Yau 6-流形上 $dd^ϕ$ 算子的 Hodge 分解定理及 $\partial\bar\partial$ 引理，定义了类 Bott-Chern 上同调空间，并在 $G_2$ 情形下给出了上同调类的几何解释。
We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, in electro-magnetic theory for exterior domains. In the appendix we translate our results to the classical framework of vector analysis.
In this article we use the combinatorial and geometric structure of manifolds with embedded cylinders in order to develop an adiabatic decomposition of the Hodge cohomology of these manifolds. We will on the one hand describe the adiabatic behaviour of spaces of harmonic forms by means of a certain Čech-de Rham complex and on the other hand generalise the Cappell-Lee-Miller splicing map to the case of a finite number of edges, thus combining the topological and the analytic viewpoint. In parts, this work is a generalisation of works of Cappell, Lee and Miller in which a single-edged graph is c…
For a certain class of complexes of pre-Hilbert $A$-modules, we prove that their cohomology groups equipped with a canonical quotient structure are again pre-Hilbert $A$-modules and derive the Hodge decomposition for them. We call these complexes self-adjoint parametrix possessing. We show that $A$-elliptic complexes of differential operator acting on sections of finitely generated projective $A$-Hilbert bundles over compact manifolds have this property if the images of certain extensions of their Laplacians are closed.
Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry
本文在 Finsler 流形上定义了水平微分、散度与 $p$-调和形式，证明了 Hodge 型定理，提出了调和向量场的新定义，并导出了基于调和 Ricci 标量的 Bochner-Yano 型分类定理。
$L^p$ cohomology and Hodge decomposition for ALE manifolds
研究ALE流形\(L^p\)约化上同调空间维数与衰减调和形式空间维数关系，证明特定\(k\)值下维数与\(p\)的关联及跳跃情况，还给出\(k\)形式的\(L^p\)霍奇分解及替代方法。
We investigate properties of the Hodge metric of a mixed period domain. In particular, we calculate its curvature and the curvature of the Hodge bundles. We also consider when the pull back metric via a period map is Kähler. Several applications in cases of geometric interest are given, such as for normal functions and biextension bundles.
The Quantization of Maxwell Theory in the Cauchy Radiation Gauge: Hodge Decomposition and Hadamard States
论文提出Cauchy辐射规范并借助完备Riemann流形上Sobolev空间k-形式的新Hodge分解，证明任意全局双曲时空上Maxwell方程存在Hadamard态。
In this paper, we define and study the Weil-Petersson geometry. Under the framework of the Weil-Petersson geometry, we study the Weil-Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil-Petersson metric and the Ricci curvature of the Weil-Petersson metric for Calabi-Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.
We study solutions for the Hodge laplace equation $Δu=ω$ on $p$ forms with $\displaystyle L^{r}$ estimates for $\displaystyle r>1.$ Our main hypothesis is that $Δ$ has a spectral gap in $\displaystyle L^{2}.$ We use this to get non classical $\displaystyle L^{r}$ Hodge decomposition theorems. An interesting feature is that to prove these decompositions we never use the boundedness of the Riesz transforms in $\displaystyle L^{s}.$ These results are based on a generalisation of the Raising Steps Method to complete non compact riemannian manifolds.
Helmholtz-Hodge Decomposition on Graphs
该研究提出有限简单图上向量场旋度的新定义，验证其满足经典算子性质，推导得到图上Helmholtz-Hodge分解，证明相关定理的图论版本，指出该旋度为非局部算子。
The Hodge decomposition provides a very powerful mathematical method for the analysis of 2D and 3D vector fields. It states roughly that any vector field can be $L^2$-orthogonally decomposed into a curl-free, divergence-free, and a harmonic field. The harmonic field itself can be further decomposed into three components, two of which are closely tied to the topology of the underlying domain. For practical computations it is desirable to find a discretization which preserves as many aspects inherent to the smooth theory as possible while at the same time remains computationally tractable, in pa…
Summary of main work 1999-2012
Nonlinear Hodge flows in symplectic geometry
本文提出非线性Hodge热流方法，用于研究辛类中辛形式的各向同性问题，并证明了流在接近几乎Kähler结构时的稳定性及收敛性。
Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz-Hodge decompositions
提出一种针对任意拓扑三角曲面上H(div)协调Brezzi-Douglas-Marini有限元的离散Helmholtz-Hodge分解，将无散BDM子空间分解为连续流函数空间的旋转梯度与有限维调和场空间，消除压力和鞍点结构，确保精确切向性、逐点无散性和压力鲁棒性。
A Note on Hodge theoretic anabelian geometry
本文提出一个基于Hodge理论的anabelian猜想，通过非阿贝尔Hodge理论中的ℂ×作用替代Galois作用，并证明了双曲曲线和复双曲流形的类比定理。
We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holonomy group causes a decomposition of the twisted cohomology. For SKT manifolds this decomposition is accompanied by an identity between different Laplacian operators and forces the collapse of a spectral sequence at the first page. Further we study the deformation theory of SKT st…
Carrollian $\mathbb{R}^\times$-bundles III: The Hodge Star and Hodge--de Rham Laplacians
利用主丛工具在Carrollian $\mathbb{R}^\times$-丛上构造Hodge星算子和Hodge--de Rham拉普拉斯算子，填补Carrollian几何与伪黎曼几何之间的空白，并应用于Schwarzschild黑洞视界及Carrollian电磁学。
Hodge Laplacian and geometry of Kuranishi family of Fano manifolds
本文利用 Bochner-Kodaira 公式估计 Fano 流形上 Hodge Laplacian 的特征值，并据此分析其形变 Kuranishi 族，证明 Kähler 形式的保持性、给出 Ricci 势公式且重新解释 Donaldson-Fujiki 图景。
Cohomology of semisimple local systems and the Decomposition theorem
本文以经典霍奇理论思想研究半单局部系统上同调，推广霍奇-黎曼双线性关系并证明整体不变闭链定理，还给出Sabbah分解定理的新几何证明。
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