Consider a compact Kähler manifold $(X,ω)$ and the space $\cal E(X,ω)=\cal E$ of $ω$--plurisubharmonic functions of full Monge--Ampère mass on it. We introduce a quantity $ρ[u,v]$ to measure the distance between $u, v\in\cal E$; $ρ[u,v]$ is not a number but rather a decreasing function on a certain interval $(0,V)\subset\mathbb R$. We explore properties of $ρ[u,v]$, and using them we study Lagrangians and associated energy spaces of $ω$--plurisubharmonic functions. Many results here generalize Darvas's findings about his metrics $d_χ$.
The goal of this short note is to point out that every Fano manifold with a nef tangent bundle possesses an almost K{ä}hler-Einstein metric, in a weak sense. The technique relies on a regularization theorem for closed positive (1, 1)-currents. We also discuss related semistability questions and Chern inequalities.
In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation. Optimal Liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established. The results are th…
Using the theory of totally real number fields we construct a new class of compact complex non-K{ä}hler manifolds in every even complex dimension and study their analytic and geometric properties.
A hypercomplex manifold is a manifold equipped with a triple of complex structures $I, J, K$ satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics, and prove a quaternionic analogue of A.D. Aleksandrov and Chern-Levine-Nirenberg theorems.
We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.
We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. Our method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry as we explain in this article. In a sequel we shall explain how this approach also applies to the hermitian setting producing new relative a priori bounds, as well as existence results.
We show that if $\{M_t\}_{t\in Δ}$ is a polarized family of compact Kähler manifolds over the open unit disk $Δ$, if $N$ is a Riemannian manifold of nonpositive complexified sectional curvature, and if $\{ϕ_t:M_t\to N\}_{t\in Δ}$ is a smooth family of pluriharmonic maps, then the Dirichlet energy $E(ϕ_t)$ is a subharmonic function of $t\inΔ$.
A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian is nonnegative. We show that the associated Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kahler manifolds.
We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel \cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In \cite{GL21b} we have studied the behavior of Monge-Ampère volumes on hermitian manifolds. We extend here the techniques of \cite{GL21a} to the hermitian setting and use the bounds established in \cite{GL21b}, producing new …
We provide a sufficient condition for open sets W and X such that a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain W to a complex manifold X holds.
This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence {u_j} of smooth, strictly J-plurisubharmonic functions point-wise decreasing down to u. On any almost complex manifold (X,J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes loca…
We prove that the maximum of two smooth strictly plurisubharmonic functions on an almost complex manifold can be uniformly approximated by smooth strictly plurisubharmonic functions.
We show that a disc functional on a complex manifold has a plurisubharmonic envelope if all its pullbacks by holomorphic submersions from domains of holomorphy in affine space do and it is locally bounded above and upper semicontinuous in a certain weak sense. For naturally defined classes of disc functionals on manifolds, this result reduces a property somewhat stronger than having a plurisubharmonic envelope to the affine case. The proof uses a recent Stein neighbourhood construction of Rosay, who proved the plurisubharmonicity of the Poisson envelope on all manifolds. As a consequence, the …
We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact K{ä}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{ä}hler-Ricci flows on compact K{ä}hler manifolds of general type as well as on stable varieties.
The Jensen envelope $Jϕ$ of an upper semicontinuous function $ϕ$ on a complex manifold X is defined at $x\in X$ as the infimum of $μ(ϕ)$ over all Jensen measures $μ$ centred at x. The Poisson envelope $Pϕ$ is defined by using only the boundary measures of analytic discs centred at x. One of the main open problems in the theory of disc functionals is whether the Poisson envelope is plurisubharmonic on an arbitrary manifold. This is equivalent to the two envelopes being equal, so plurisubharmonicity of $Jϕ$ is a necessary condition for $Pϕ$ to be plurisubharmonic. We prove that the Jensen envelo…
In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. This paper investigates, in depth, questions of: pseudo-convexity and cores, positive phi-currents, Duval-Sibony Duality, and boundari…
In this paper, we prove the $C^{1, 1}$-regularity of the plurisubharmonic envelope of a $C^{1,1}$ function on a compact Hermitian manifold. We also present examples to show this regularity is sharp.
We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.
We introduce a wide subclass ${\cal F}(X,ω)$ of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well-defined and the convergence theorem is valid. We also prove that ${\cal F}(X,ω)$ is a convex cone and includes all quasi-plurisubharmonic functions which are in the Cegrell class.
Let $M$ be a complex manifold and $PSH^{cb}(M)$ be the space of bounded continuous plurisubharmonic functions on $M$. In this paper we study when functions from $PSH^{cb}(M)$ separate points. Our main results show that this property is equivalent to each of the following properties of $M$: (1) the core of $M$ is empty. (2) for every $w_0\in M$ there is a continuous plurisubharmonic function $u$ with the logarithmic singularity at $w_0$. Moreover, the core of $M$ is the disjoint union of 1-pseudoconcave in the sense of Rothstein sets $E_j$ with the following Liouville property: every function f…
Given a compact complex manifold $Y$ with a negative line bundle $L\rightarrow Y$, we study the Schwarz-type symmetrization on the total space of $L$. We prove that this symmetrization does not increase the Monge-Ampère energy for the fibrewise $S^{1}$-invariant plurisubharmonic functions in the "unit ball" under some assumptions. As an application we generalize the sharp Moser-Trudinger inequality on the unit ball.
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 $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ is positive, (the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ need not be closed here). The conjecture is trivial if $u$ is of class ${\cal C}^2$. The result is elementary in the complex integrable case because the operator $i\pa…
Let $(M,ω)$ be a Kahler manifold. An integrable function on M is called $ω^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $ω^q$-plurisubharmonic function is q-convex. A continuous $ω^q$-plurisubharmonic function admits a local approximation by smooth, $ω^q$-plurisubharmonic functions. For any closed subvariety $Z\subset M$, $\dim Z < q$, there exists a strictly $ω^q$-plurisubharmonic function in a neighbourhood of $Z$ (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony's lemma on integrabi…
We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Amère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, $p>1$ densities, or moderate measures in the sense of Dinh-Nguyen-Sibony.
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 give a geometric condition on a compact subset of a complex manifold which is necessary and sufficient for the existence of a smooth strictly plurisubharmonic function defined in a neighbourhood of this set.
We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log terminal singularities.
Let $X$ be a compact Kähler manifold and $θ$ a smooth closed $(1,1)$-real form representing a big cohomology class $α\in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that any $θ$-plurisubharmonic function $\f$ can be approximated from above by a decreasing sequence of continuous $θ$-plurisubharmonic functions with minimal singularities, assuming that there exists a single such function.
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.
We prove that a relatively compact pseudoconvex domain with smooth boundary in an almost complex manifold admits a bounded strictly plurisubharmonic exhaustion function. We use this result for the study of convexity and hyperbolicity properties of these domains and the contact geometry of their boundaries.
Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,ϕ). In particular, on X there exist ϕ-plurisubharmonic functions, ϕ-convex domains, ϕ-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sense, the plurisubharmonic functions are the polar duals of the ϕ-submanifolds, or more generally, the ϕ-currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the ϕ-convex hull of a compa…
We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain $W$ in a Stein manifold to a larger domain $X$ under suitable conditions on $W$ and $X$. We introduce a related equivalence relation on the space of analytic discs in $X$ with boundary in $W$. The quotient, if it is Hausdorff, is a complex manifold with a local biholomorphism to $X$. We use our disc formula to generalise Kiselman's minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.
In this note, we will present global equisingular approximations of quasi-plurisubharmonic functions with stable analytic pluripolar sets on compact complex manifolds.
We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. In particular, the notion of pseudo-convexity for a calibrated manifold (X,ϕ) is introduced and studied. Analogues of totally real submanifolds are also int…
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.
We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Ampère equation on a compact Hermitian manifold for a very general measre on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen.
A version of Gromov's cup product lemma in which one factor is the (1,0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kaehler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a 2-dimensional complex analytic set at some point has the Bochner-Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.
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…
For a bounded domain $D$ and a real number $p>0$, we denote by $A^p(D)$ the space of $L^p$ integrable holomorphic functions on $D$, equipped with the $L^p$- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular $n=m$) if there is a linear isometry between $A^p(D_1)$ and $A^p(D_2)$ for some $0<p<2$. The same result holds for $p>2, p\neq 2,4,\cdots$, provided that the $p$-Bergman kernels on $D_1$ and $D_2$ are exhaustive. With this as a motivation, we show that, for all $p>0$, the $p$-Bergman kernel on a strongly pseu…
We give a sufficient condition on a sequence of uniformly bounded $ω$-plurisubharmonic functions, $ω$ being a Hermitian metric, for which the sequence of associated Monge-Ampère measures converges weakly. This criterion can be used to obtained a bounded $ω$-plurisubharmonic solution to the Monge-Ampère equation.
We give new characterizations of plurisubharmonic functions and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics. As applications, we present a different method to prove plurisubharmonic variation of generalized Bergman kernel metrics and Griffiths positivity of the direct images of twisted relative canonical bundles associated to holomorphic families of Kähler manifolds.
In this paper we derive formulas for the Monge-Ampère measures of functions of the form $\log|Φ|_c$, where $Φ$ is a holomorphic map on a complex manifold $X$ of dimension $n$ with values in $\mathbb{C}^{n+1}\setminus\{0\}$ and $|\cdot|_c$ is the Lie norm on $\mathbb{C}^{n+1}$.
We generalize the Poletsky disc envelope formula for the function $\sup \{u \in \PSH(X,ω) ; u\leq ϕ\}$ on any complex manifold $X$ to the case where the real (1,1)-current $ω=ω_1-ω_2$ is the difference of two positive closed (1,1)-currents and $ϕ$ is the difference of an $ω_1$-upper semicontinuous function and a plurisubharmonic function.
The complex Monge-Ampère operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded quasi-plurisubharmonic functions by various authors. As this operator is not continuous for the $L^{1}$-topology, several stronger topologies have been introduced over the last decades to remedy this, while maintaining efficient compactness criteria. The purpose of this note is to show that these stronger topologies are essentially equivalent to the natural quas…
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.
On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions
CAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院、中国科学院吴文俊数学重点实验室）
We show that any fibration of a 'special' compact K{ä}hler manifold X onto an Abelian variety has no multiple fibre in codimension one. This statement strengthens and extends previous results of Kawamata and Viehweg when $κ$(X) = 0. This also corrects the proof given in [2], 5.3 which was incomplete.
In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C^{1,1} regularity. We construct examples which prove that the global C^{1,1} regularity fails both in the real and complex case in contrast to the Kähler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions.
Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. I…
We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous Monge-Ampère equations. This extends to a new family of mixed type examples various classical results on parabolic spaces and complexifications of symmetric spaces. Rigidity results on complex spaces modeled on such new examples are given.
We show that the orbifold fundamental group of an effective compact K{ä}hler orbifold with nef anticanonical bundle has polynomial growth, which generalizes M.P ăun's results for manifolds [P ău97, Theorem 1,Theorem 2]
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…
