An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class of the functions u from M to a positive bounded interval which are plurisubharmonic in M and such that u(p) = 0 for each p in V. If the class admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E(V,M) on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge-Ampere models introduced by Lempert and Szoke in [16].
In this note we study the plurifinely locally maximal plurifinely plurisubharmonic functions and improve some known results on these functions. We prove in particular that any locally bounded plurifinely locally maximal plurifinely plurisubharmonic function is maximal, and that any finite plurisubharmonic function on an open subset of $\mathbb{C}^n$ is maximal if and only if it is plurifinely locally maximal.
In this paper, we prove that a continuous $\mathcal F$-plurisubharmonic functions defined in an $\mathcal F$-open set in $\mathbb C^n$ is $\mathcal F$-maximal if and only if it is $\mathcal F$-locally $\mathcal F$-maximal.
The main purpose of this paper is to introduce and study the notion of plurifinely-maximal plurifinely plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to a plurifine domain of C^n in a natural way. Our main result is that a finite plurifinely plurisubharmonic function u on a plurifine domain U satisfies (dd^c u)^n=0 if and only if u is plurifinely-locally plurifinely-maximal outside some pluripolar set. In particular, a finite plurifinely-maximal plurisubharmonic function u satisfies (dd^c u)^n=0.
Let $u$ be a maximal plurisubharmonic function in a domain $Ω\subset\mathbb{C}^n$ ($n\geq 2$). It is classical that, for any $U\SubsetΩ$, there exists a sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$ satisfying the property: $(dd^c u_j)^n$ is weakly convergent to $0$ as $j\rightarrow\infty$. In general, this property does not hold for arbitrary sequence. In this paper, we show that for any sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$, $(|u_j|+1)^{-a} (dd^cu_j)^n$ is weakly convergent to $0$ as $j\rightarrow\infty$, where $a>n-1$. We also ge…
In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact Kähler manifold. We prove that a precise bound on the complex Monge-Ampère mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Ampère measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Ampè…
Local-to-global maximality for plurisubharmonic functions with locally analytic singularities
Institute for Artificial Intelligence, VNU University of Engineering and Technology（越南国立大学工程技术大学人工智能研究所）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
We study continuity properties of generalized Monge-Ampère operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a formula for the total mass of the Monge-Ampère measure of such a function on a compact Kähler manifold.
Comparison principles for local plurisubharmonic potentials on complex manifolds and applications
Institute for Artificial Intelligence, VNU University of Engineering and Technology（越南国立大学工程技术大学人工智能研究所）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
We show how the formalism of Levi currents on complex manifolds, as introduced by Sibony, can be used to study the analytic structure of singular sets associated to families of plurisubharmonic functions, in the sense of Slodkowski.
Given a polarized projective variety (X,L) over any non-Archimedean field, assuming continuity of envelopes, we show that the space of finite-energy metrics on L is a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous.
We study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual Monge--Ampère masses. They are characterized in terms of Green functions for Demailly's approximations, relative types, and valuations. Furthermore, the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals.
We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold $M$. Applications are given to the Hausdorff measure of the intersection of the variety with $M$ and the exponential integrability of plurisubharmonic functions on $M$.
We show that the monodromy theorem holds on arbitrary connected free sets for noncommutative free analytic functions. Applications are numerous-- pluriharmonic free functions have globally defined pluriharmonic conjugates, locally invertible functions are globally invertible, and there is no nontrivial cohomology theory arising from analytic continuation on connected free sets. We describe why the Baker-Campbell-Hausdorff formula has finite radius of convergence in terms of monodromy, and solve a related problem of Martin-Shamovich. We generalize the Dym-Helton-Klep-McCullough-Volcic theorem--…
Suppose that $X$ is an analytic subvariety of a Stein manifold $M$ and that $φ$ is a plurisubharmonic (psh) function on $X$ which is dominated by a continuous psh exhaustion function $u$ of $M$. Given any number $c>1$, we show that $φ$ admits a psh extension to $M$ which is dominated by $cu$ on $M$. We use this result to prove that any $ω$-psh function on a subvariety of the complex projective space is the restriction of a global $ω$-psh function, where $ω$ is the Fubini-Study Kähler form.
We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain in C^n is an obstruction to compactness of the d-bar-Neumann operator on the domain, provided that at some point of M, the Levi form has the maximal possible rank n-1-dim(M) (i.e. the domain is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue. We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmon…
In the first part of this paper, we study the properties of some particular plurisubharmonic functions, namely the toric ones. The main result of this part is a precise description of their multiplier ideal sheaves, which generalizes the algebraic case studied by Howald. In the second part, almost entirely independent of the first one, we generalize the notion of the adjoint ideal sheaf used in algebraic geometry to the analytic setting. This enables us to give an analogue of Howald's theorem for adjoint ideals attached to monomial ideals. Finally, using the local Ohsawa-Takegoshi-Manivel theo…
A weakly complete space is a complex space admitting a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such exhaustion function can be chosen real analytic: they can be modifications of Stein spaces or proper over a non compact (possibly singular) complex curve or foliated with real analytic Levi-flat hypersurfaces which in turn are foliated by dense complex leaves (these we call surfaces of Grauert type). In the last case, we also show that such Levi-flat hypersurfaces are in fact level sets of a global proper plurihar…
We show by an example that the (equivalence class of) singularity of a plurisubharmonic function cannot be determined by the data of its Lelong numbers, in a nontrivial sense. Such an example is provided by Siu-type singular hermitian metrics associated to an effective line bundle. We also show that a Siu-type metric has analytic singularities if and only if the section ring of the line bundle is finitely generated.
The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces (\emph{not} necessarily normal) via Ohsawa's extension measure, as a special case of which, it turns out to be the so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As applications, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to the analytic setting and establish some extension theorems on Kähler mani…
We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Ampère densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw sever…
The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes are locally bounded outside an analytic set of singularities. It is furthermore shown that a parametrized version of this result yields a priori inequalities for the solution of the Dirichlet problem for a degenerate Monge-Ampere operator; applications to geodesics in the space o…
We extend the study of jumping numbers of multiplier ideals due to Ein-Lazarsfeld-Smith-Varolin from the algebraic case to the case of general plurisubharmonic functions. While many properties from Ein-Lazarsfeld-Smith-Varolin are shown to generalize to the plurisubharmonic case, important properties such as periodicity and discreteness do not hold any more. Previously only two particular examples with a cluster point (i.e. failure of discreteness) of jumping numbers were known, due to Guan-Li and to Ein-Lazarsfeld-Smith-Varolin respectively. We generalize them to all toric plurisubharmonic fu…
The notion of the projective hull of a compact set in a complex projective space was introduced by Harvey and Lawson in 2006. In this paper we describe the projective hull by Poletsky sequences of analytic discs, in analogy to the known descriptions of the holomorphic and the plurisubharmonic hull.
We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a noncommutative complex hessian that is positive semidefinite when evaluated on all tuples of n x n matrices for every size n. In this paper, we show that the symmetric noncommutative plurisubharmonic polynomials are precisely the noncommutative convex polynomials with a noncommutative analytic change of variables; i.e., a noncommutative symmetric polynomial…
In this paper, we first establish a Poincaré-Lelong type formula in the almost complex setting. Then, after introducing the notion of $J$-analytic subsets, we study the restriction of a closed positive current defined in an almost complex manifold $(M,J)$ on a $J$-analytic subset. Finally, we prove that the Lelong numbers of a plurisubharmonic current defined on an almost complex manifold are independent of the coordinate systems.
A complex manifold $X$ is \emph{weakly complete} if it admits a continuous plurisubharmonic exhaustion function $ϕ$. The minimal kernels $Σ_X^k, k \in [0,\infty]$ (the loci where are all $\mathcal{C}^k$ plurisubharmonic exhaustion functions fail to be strictly plurisubharmonic),introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far $X$ is from being Stein. We compare these notions, prove that all Levi currents are supported by all the $Σ_X^k$'s, and give sufficient conditions for points in $Σ_X^k$ to be in the support of so…
Using a recent result of Lárusson and Poletsky regarding plurisubharmonic subextensions we prove a disc formula for the quasiplurisubharmonic global extremal function for domains in complex projective space. As a corollary we get a characterization of the projective hull for connected compact sets in complex projective space by the existence of analytic discs.
Let $X$ be a complex space and $H$ a positive homogeneous plurisubharmonic function $H$ on $X\times\C^m$. Consider the Hartogs-type domain $Ω_{H}(X):=\{(z,w)\in X\times \C^m:H(z,w)<1 \}$. Let $S$ be an analytic subset of $X$. We give necessary and sufficient conditions for hyperbolicity and tautness modulo $S\times \C^m$ of $Ω_{H}(X)$, with the obvious corollaries for the special case of Hartogs domains.
We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the function of the Hausdorff dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we provide an extensive collection of classes of examples of analytic fam…
Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f: W:=(A\times Y) \cup (X\times B)\longrightarrow Z$ extends to a holomorphic mapping $\hat{f}$ on $\hat{W}:=\left\lbrace(z,w)\in X\times Y:\ \widetildeω(z,A,X)+\widetildeω(w,B,Y)<1 \right\rbrace$ such that $\hat{f}=f$ on $W\cap \hat{W},$ where $\widetildeω(\cdot,A,…
We study extensions of Wermer's maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold.
Complex Monge-Amp{è}re equations with boundary values: a variational approach
Université Assane Seck de Ziguinchor（齐古尼奥尔阿萨内塞克大学）；Institut Universitaire de France（法国高等研究院）；Univ Angers（昂热大学）；CNRS（法国国家科学研究中心）；Université Gaston Berger de Saint-Louis（圣路易加斯顿伯杰大学）
Let f_λ be a family of holomorphic functions in the unit disk, holomorphic in parameter λ\in U\subset\C^{n}. We estimate the number of zeros of f_λ in a smaller disk via some characteristic of the ideal generated by Taylor coefficients of f_λ. Our estimate is locally sharp and improve the previous estimate obtained by Roytwarf and Yomdin.
For an analytic function $f$ defined on the unit disk $|z|<1$, let $Δ(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<r\le 1$. In 1990, Yamashita conjectured that $Δ(r,z/f)\le πr^2$ for convex functions $f$ and it was finally settled in 2013 by Obradović and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation $zf'(z)/f(z)\prec (1+(1-2β)αz)/(1-αz)$ for $0\le β<1$ and $0<α\le 1$. We prove Yamashita's conjecture problem for functions in this class, which solves a partial solution to an open problem …
One of the classical problems concerns the class of analytic functions $f$ on the open unit disk $|z|<1$ which have finite Dirichlet integral $Δ(1,f)$, where $$Δ(r,f)=\iint_{|z|<r}|f'(z)|^2 \, dxdy \quad (0<r\leq 1). $$ The class ${\mathcal S}^*(A,B)$ of normalized functions $f$ analytic in $|z|<1$ and satisfies the subordination condition $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$ in $|z|<1$ and for some $-1\leq B\leq 0$, $A\in {\mathbb C}$ with $A\neq B$, has been studied extensively. In this paper, we solve the extremal problem of determining the value of $$\max_{f\in {\mathcal S}^*(A,B)}Δ(r,z/f)$$ a…
Let M be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of $TM$. We prove here that the only real analytic Zoll metric on the $n$-sphere with an entire adapted complex structure on $TM$ is the round sphere. Using similar ideas, we answer a special case of an algebraization question raised by the first author, characterizing some Stein manifolds as affine algebraic in ter…
It is shown that for any non-decreasing, continuous and unbounded doubling function $\om$ on $[0,1)$, there exist two analytic infinite products $f_0$ and $f_1$ such that the asymptotic relation $|f_0(z)| + |f_1(z)| \asymp \om(|z|)$ is satisfied for all $z$ in the unit disc. It is also shown that both functions $f_j$ for $j=0,1$ satisfy $T(r,f_j)\asymp\logω(r)$, as $r\to1^-$, and hence give examples of analytic functions for which the Nevanlinna characteristic admits the regular slow growth induced by $ω$.
We prove that a maximal surface in Lorentz-Minkowski space $\Bbb L^3$ can be extended analytically along its boundary if the boundary lies in a plane meeting the surface at a constant angle.
We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari--Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke…
A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can arise in more than one way from this construction. We show that given a compact M and a finite exhaustion, the underlying Riemannian structure is unique. The proof uses the technique of holomorphic disks spanning two exact Lagrangian submanifolds of the cotangent bundle of M, and S…
We strengthen, in various directions, the theorem of Garnett that every $σ$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some ${\mathbb C}^N$ so that $\hat K \setminus K$ contains a …
We proved recently math.CA/0510403 that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may be square integrable on the circle. Here we prove that it can even be infinitely differentiable, and we characterize precisely the possible degree of smoothness in terms of the rate of decrease of the Fourier coefficients. This sharp condition might be viewed as a "new quasi-analyticity".
We study the real-analytic continuation of local real-analytic solutions to the Cauchy problems of quasi-linear partial differential equations of first order for a scalar function. By making use of the first integrals of the characteristic vector field and the implicit function theorem we determine the maximal domain of the analytic extension of a local solution as a single-valued function. We present some examples including the scalar conservation laws that admit global first integrals so that our method is applicable.
In this paper we give precise characterizations of the relation between the Nevanlinna counting function and pull-back measure of an analytic self-map of the unit disk near the boundary. We show that it is quite worth considering these two concepts on the maximal ideal space of the bounded analytic functions.
The right invariant metric on the analytic automorphism group of the unit open disk induced by maximal modulus
School of Mathematical Science, Heilongjiang University（黑龙江大学数学科学学院）；Sixth High School of CAIDA（CAIDA第六中学）；School of Mathematics, Jilin University（吉林大学数学学院）
We show that the maximal Nevanlinna counting function and the Carleson function of analytic self-maps of the unit disk are equivalent, up to constants.
