Emil J. Straube, Marcel K. Sucheston
We show that the approaches to global regularity of the d-bar Neumann problem via the methods listed in the title are equivalent when the conditions involved are suitably modified. These modified conditions are also equivalent to one that is relevant in the context of Stein neighborhood bases and Mergelyan type approximation.
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We show that a Skoda complex with a general plurisubharmonic weight function is exact if its 'degree' is sufficiently large. This answers a question of Lazarsfeld and implies that not every integrally closed ideal is equal to a multiplier ideal even if we allow general plurisubharmonic weights for the multiplier ideal, extending the result of Lazarsfeld and Lee \cite{LL}.
Necessary conditions for a domain $Ω\subset \mathbb C^n$ admitting a local plurisubharmonic defining function on the boundary are given. In tandem, we give an algorithm to construct a local plurisubharmonic defining function on the boundary when one exists. In some cases we show that the necessary conditions are also sufficient.
Let $Ω=\{r<0\}\subset\mathbb C^2$, with $r$ plurisubharmonic on $bΩ=\{r=0\}$. Let $ρ$ be another defining function for $Ω$. A formula for the determinant of the complex Hessian of $ρ$ in terms of $r$ is computed. This formula is used to give necessary and sufficient conditions that make $ρ$ (locally) plurisubharmonic.
We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in $\mathbb{C}^2$. In particular, we construct a family of simple counterexamples to the existence of plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains.
Let D be a smoothly bounded domain in C^2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.
Let D be a smoothly bounded domain in complex space of dimension larger than 2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.
We show that every bounded domain $D$ in $\mathbb R^n$ with smooth $p$-convex boundary for $2\le p < n$ admits a smooth defining function $ρ$ which is $p$-plurisubharmonic on $\overline D$; if in addition $bD$ has no $p$-flat points then $ρ$ can be chosen strongly $p$-plurisubharmonic on $D$. If $bD$ is $2$-convex then for any open connected conformal surface $M$ and conformal harmonic map $f:M\to \overline D$, either $f(M)\subset D$ or $f(M)\subset bD$. In particular, every conformal harmonic map $\mathbb D^*\to D$ from the punctured disc extends to a conformal harmonic map $\mathbb D\to D$.
We show that every strictly pseudoconvex domain $Ω$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a smooth plurisubharmonic function $φ\colon U \to \mathbb R$ defined on an open neighbourhood $U \subset \mathcal{M}$ of $\overlineΩ$ such that $Ω= \{φ< 0\}$, $dφ\neq 0$ on $bΩ$ and $φ$ is strictly plurisubharmonic near $bΩ$. We then introduce the notion of the core $\mathfrak{c}(Ω)$ of an arbitrary domain $Ω\subset \mathcal{M}$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $Ω$ fails to be strictl…
Local-to-global maximality for plurisubharmonic functions with locally analytic singularities
本文证明了具有局部解析奇点的局部极大多次调和函数在满足解析扩张界时即为极大，并通过Monge--Ampère测度消失条件刻画极大性。
In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $Ω$ in $\mathbb{CP}^n$ there exists a Lipschitz defining function $ρ$ and an exponent $0<η<1$ such that $-(-ρ)^η$ is strictly plurisubharmonic on $Ω$. This generalizes a result of Ohsawa and Sibony for $C^2$ domains. In contrast to the Ohsawa-Sibony result, we provide a counterexample demonstrating that we may not assume $ρ=-δ$, where $δ$ is the geodesic distance function for the boundary of $Ω$.
Let D be a smoothly bounded domain in a complex vector space of dimension n. Suppose that D has a smooth defining function, such that the sum of any q eigenvalues of its complex Hessian are non-negative on the closure of D. We show that this implies global regularity of the Bergman projection on (0,j)-forms for j larger or equal to q-1.
A theory of global regularity of the d-bar-Neumann operator is developed which unifies the two principal approaches to date, namely the one via compactness due to Kohn-Nirenberg and Catlin, and the one via plurisubharmonic defining functions and/or vector fields that commute approximately with d-bar due to Boas and the author.
The Minkowski function is a crucial tool used in the study of balanced domains and, more generally, quasi-balanced domains in several complex variables. If a quasi-balanced domain is bounded and pseudoconvex then it is well-known that its Minkowski function is plurisubharmonic. In this short note, we prove that under the additional assumption of smoothness of the boundary, the Minkowski function of a quasi-balanced domain is in fact smooth away from the origin. This allows us to construct a smooth plurisubharmonic defining function for such domains. Our result is new even in the case of balanc…
Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the $\bar\partial_b$-equation on $M$ can be solved in $C^\infty(M)$. (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$-forms, then $G_q$ is a continuous opera…
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity of solutions of complex Monge-Ampere equation and the reflection principle, enables us to give a new proof of the Fefferman mapping theorem.
$L^2$ extension of holomorphic functions and log canonical places
本文指出在 Demailly 的 $L^2$ 延拓定理及 Zhou-Zhu 的推广中，存在许多非解析奇点定义函数（如环面奇点）使得仅有零函数具有有限的 Ohsawa 范数，导致 $L^2$ 延拓结论在这些情况下变得空泛。
This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\overline{\partial}$--Neumann operators on $Ω$ preserve the Sobolev spaces $W^{s}_{(0,q)}(Ω)$, $s\geq 0$. The same authors then proved a further regularity result and made explicit the role of D'Angelo forms for regularity (\cite{BoasStraube93}). A few years later, Kohn (\cite{Kohn99}) initiated a quantitative study of the re…
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