Laszlo Lempert
We prove that the equation d-bar u = f can be solved on a ball B(R) of radius R in the Banach space l^1 if f is a closed Lipschitz continuous (0,1) form on B(R). We also present examples of closed (0,1) forms f of various regularities on the spaces l^p that are not exact. In particular, in the first result above, it is not enough to assume that f is merely continuous, rather than Lipschitz continuous.
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For a large class of separable Banach spaces, we prove the real analytic Dolbeault Isomorphism Theorem for open subsets.
We show that any Dolbeault cohomology group $H^{p,q}(D)$, $p\ge0$, $q\ge1$, of an open subset $D$ of a closed finite codimensional complex Hilbert submanifold of $\ell_2$ is either zero or infinite dimensional. We also show that any continuous character of the algebra of holomorphic functions of a closed complex Hilbert submanifold $M$ of $\ell_2$ is induced by evaluation at a point of $M$. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of $\ell_1$ admits a nowhere critical holomorphic function.
The loop space LP_1 of the Riemann sphere is an infinite dimensional complex manifold consisting of maps (loops) from S^1 to P_1 in some fixed C^k or Sobolev W^{k,p} space. In this paper we compute the Dolbeault cohomology groups H^{0,1}(LP_1).
We study Dolbeault harmonic $(1,1)$-forms on compact quotients $M=Γ\backslash G$ of $4$-dimensional Lie groups $G$ admitting a left invariant almost Hermitian structure $(J,ω)$. In this case, we prove that the space of Dolbeault harmonic $(1,1)$-forms on $(M,J,ω)$ has dimension $b^-+1$ if and only if there exists a left invariant anti self dual $(1,1)$-form $γ$ on $(G,J)$ satisfying $id^cγ=dω$. Otherwise, its dimension is $b^-$. In this way, we answer to a question by Zhang.
We prove that the dimension $h^{1,1}_{\overline\partial}$ of the space of Dolbeault harmonic $(1,1)$-forms is not necessarily always equal to $b^-$ on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally conformally almost Kähler, almost Hermitian structures on compact 4-manifolds with $h^{1,1}_{\overline\partial}=b^-+1$. This answers to a question by Holt.
Let $X$ be a complex space of pure dimension. We introduce fine sheaves $\A^X_q$ of $(0,q)$-currents, which coincides with the sheaves of smooth forms on the regular part of $X$, so that the associated Dolbeault complex yields a resolution of the structure sheaf $\hol^X$. Our construction is based on intrinsic and quite explicit semi-global Koppelman formulas.
The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the smooth and the $L^2$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.
We define the relative Dolbeault homology of a complex manifold with currents via a Čech approach and we prove its equivalence with the relative Čech-Dolbeault cohomology as defined in [T. Suwa, Čech-Dolbeault cohomology and the $\overline\partial$-Thom class, {\em Singularities---Niigata---Toyama 2007}, 321--340, Adv. Stud. Pure Math., \textbf{56}, Math. Soc. Japan, Tokyo, 2009. ]. This definition is then used to compare the relative Dolbeault cohomology groups of two complex manifolds of the same dimension related by a suitable proper surjective holomorphic map. Finally, an application to bl…
It is conjectured that the Dolbeault cohomology of a complex nilmanifold $X$ is computed by left-invariant forms. We prove this under the assumption that $X$ is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six. Our approach generalises previous methods, where the existence of a holomorphic fibration was a crucial ingredient.
Let ${\cal F}$ be a complex affine Reeb foliation of dimension $1$ on the Hopf manifold ${\Bbb S}^{n+1}\times {\Bbb S}^1$. We prove that its foliated Dolbeault cohomology in degree $1$ is isomorphic to ${\Bbb C}$ by giving an explicit generator.
We consider a bounded open Stein subset $Ω$ of a complex Stein manifold $X$ of dimension $n$. We prove that if $f$ is a current on $X$ of bidegree $(p,q+1)$, $\bar\partial$-closed on $Ω$, we can find a current $u$ on $X$ of bidegree $(p,q)$ which is a solution of the equation $\bar\partial u=f$ in $Ω$. In other words, we prove that the Dolbeault complex of temperate currents on $Ω$ (i.e. currents on $Ω$ which extend to currents on $X$) is concentrated in degree $0$. Moreover if $f$ is a current on $X= C^n$ of order $k$, then we can find a solution $u$ which is a current on $C^n$ of order $k+2n…
We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we consider the relations between Morse-Novikov cohomology and Dolbeault-Morse-Novikov cohomology, moreover, investigate stabilities of their dimensions under the deformations of complex structures. In some aspects, Morse-Novikov and Dolbeault-Morse-Novikov cohomology behave similarly with…
In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformations by using the power series method. We also prove a simple relation between the existence of deformations and the varying of the dimensions of Dolbeault cohomology. The deformations of $(p,q)$-forms is shown to be unobstructed under some mild conditions. By analyzing Nakamura's e…
The classical result of J.J. Kohn asserts that over a relatively compact subdomain $D$ with $C^\infty$ boundary of a Hermitian manifold whose Levi form has at least $n-q$ positive eigenvalues or at least $q+1$ negative eigenvalues at each boundary point, there are natural isomorphisms between the $(p,q)$ Dolbeault cohomology groups defined by means of $C^\infty$ up to the boundary differential forms on $D$ and the (finite-dimensional) spaces of harmonic $(p,q)$-forms on $D$ determined by the corresponding complex Laplace operator. In the present paper, using Kohn's technique, we give a similar…
Let $G$ be a simply connected solvable Lie group with a lattice $Γ$ and the Lie algebra $\g$ and a representation $ρ:G\to GL(V_ρ)$ whose restriction on the nilradical is unipotent. Consider the flat bundle $E_ρ$ given by $ρ$. By using "many" characters $\{α\}$ of $G$ and "many" flat line bundles $\{E_α\}$ over $G/Γ$, we show that an isomorphism \[\bigoplus_{\{α\}} H^{\ast}(\g, V_α\otimes V_ρ)\cong \bigoplus_{\{E_α\}} H^{\ast}(G/Γ, E_α\otimes E_ρ)\] holds. This isomorphism is a generalization of the well-known fact:"If $G$ is nilpotent and $ρ$ is unipotent then, the isomorphism $H^{\ast}(\g, V_…
We give an example of a pseudoconvex domain in a complex manifold whose $L^2$-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The domain is a smoothly bounded Levi-flat domain in a two complex-dimensional compact complex manifold. The domain is biholomorphic to a product domain in $\mathbb{C}^2$, hence Stein. This implies that for $q>0$, the usual Dolbeault cohomology with respect to smooth forms vanishes in degree $(p,q)$. But the $L^2$-Cauchy-Riemann operator on the domain does not have closed range on $(2,1)$-forms and consequently its $L^2$-Dolbeault cohomology is not Hausd…
Let $X$ be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of $\overline{\partial}$-equation on $X$ and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves $\mathcal{A}_X^q$ of $(0,q)$-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf $\mathscr{O}_X$. Our construction is based on intrinsic semi-global Koppelman formulas on $X$.
On any pure $n$-dimensional, possibly non-reduced, analytic space $X$ we introduce the sheaves $\mathscr{E}_X^{p,q}$ of smooth $(p,q)$-forms and certain extensions $\mathscr{A}_X^{p,q}$ of them such that the corresponding Dolbeault complex is exact, i.e., the $\bar\partial$-equation is locally solvable in $\mathscr{A}_X$. The sheaves $\mathscr{A}_X^{p,q}$ are modules over the smooth forms, in particular, they are fine sheaves. We also introduce certain sheaves $\mathscr{B}_X^{n-p,n-q}$ of currents on $X$ that are dual to $\mathscr{A}_X^{p,q}$ in the sense of Serre duality. More precisely, we s…
About top-degree $L^2$- and $L^{2,\mathrm{loc}}$-Dolbeault cohomologies of complex spaces with pseudoconvex boundary
本文证明在具有 $\mathscr{C}^3$ 光滑边界的伪凸相对紧域中，$L^2$- 和 $L^{2,\mathrm{loc}}$-Dolbeault $(n,q)$-上同调群在 $q>0$ 时消失，并推广到 Nakano 半正定全纯向量丛取值的情形，进而证明由严格多重次调和函数定义的域中两类上同调群的等价性。
Left-invariant almost complex structures on the higher dimensional Kodaira-Thurston manifolds
针对广义Kodaira-Thurston流形上的左不变殆复结构，提出计算方法以计算其Kodaira维数与Dolbeault调和形式空间维数。
The loop space of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop space as an infinite dimensional complex Lie group with Lie algebra the first Dolbeault group. The group of Mobius transformations $G$ and its loop group $LG$ act on this loop space. We prove that an element of the Picard group is $LG$-fixed if it is $G$-fixed; thus completely answer the question by Millson and Zombro about $G$-equivariant projective embedding …
We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action $Γ$, and obtain an asymptotic estimate for the $Γ$-dimension of the harmonic space with respect to the tensor times $k$ in the holomorphic line bundle $L^{k}\otimes E$ and the type $(n,q)$ of the differential form, when $L$ is semipositive. In particular, we estimate the $Γ$-dimension of the corresponding reduced $L^2$-Dolbeault cohomology group. Essentially, we obtain a local estimate of the pointwise norm of harmonic forms with valued in semipositive line bundles over Hermitian manif…
Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2…
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 equality in such an inequality à la Frölicher characterizes the validity of the so-called cohomological property of satisfying the $\partial\overline{\partial}$-Lemma. As an application, we study cohomological properties of compact either complex, or symplectic, or, more in general, …
In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof of Serre duality on any reduced pure $n$-dimensional paracompact complex space $X$. At the core of the paper is the introduction of concrete fine sheaves $\mathscr{B}_X^{n,q}$ of certain currents on $X$ of bidegree $(n,q)$, such that the Dolbeault complex $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ becomes, in a certain sense, a dualizing complex. In particular, if $X$ is Cohen-Macaulay (e.g., Gorenstein or a comple…
The $k$-Cauchy-Fueter operator $ D_0^{(k) } $ on one dimensional quaternionic space $\mathbb{H}$ is the Euclidean version of helicity $\frac k 2$ massless field operator on the Minkowski space in physics. The $k$-Cauchy-Fueter equation for $k\geq 2$ is overdetermined and its compatibility condition is given by the $k$-Cauchy-Fueter complex. In quaternionic analysis, these complexes play the role of Dolbeault complex in several complex variables. We prove that a natural boundary value problem associated to this complex is regular. Then by using the theory of regular boundary value problems, we …
In this paper we investigate complex dynamics in infinite dimensions.
We show that the group of holomorphic automorphisms of a Stein manifold X of dimension greater than 1 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group.
It is proved that the dimension of the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. is infinite.
In New York J. Math. 17 (2011), 41--49, Li has obtained an analogue of the Jørgensen inequality in the infinite-dimensional Möbius group. We show that this inequality is strict.
Construction of an infinite-dimensional family of exact solutions of a three-dimensional biharmonic equation by the hypercomplex method
本研究采用超复方法，成功构建出三维双调和方程的无穷维精确解族，为该方程的求解研究提供了新的精确解结果。
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