Sebastien Boucksom, Charles Favre, Mattias Jonsson
备注：
37 pages, new version. Changed the terminology from convex fonctions to formal psh functions. Corrected statement about the continuity of formal psh functions. Added a proof of the continuity of psh envelopes (theorem 5.13). Clarified the exposition (removed a section on the action of finite maps on formal psh functions). Added new references
We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the last two authors. Following Kontsevich and Soibelman we describe the geometry of the space V of all normalized valuations on C[x_1,...,x_n] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C^n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we define a mixed Monge-Ampere operator which reflects the intersection theory of divisors above the origin of C^n. This operator associates to any (n-1)-tuple of formal psh functions a positive measure of finite mass on V. Next, we show that the collection of Lelong numbers of a given germ u of a psh function at all infinitely near points induces a formal psh function u' on V called its valuative transform. When ϕis a psh Holder weight in the sense of Demailly, the generalized Lelong number nu_ϕ(u) equals the integral of u' against the Monge-Ampere measure of the valuative transform of ϕ. In particular, any generalized Lelong number is an average of valuations. We also show how to compute the multiplier ideal of u and the relative type of u with respect to ϕin the sense of Rashkovskii, in terms of the valuative transforms of u and ϕ.
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.
A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.
A variant of Siu's analyticity theorem is proved for relative types of plurisubharmonic functions. Some results on propagation of plurisubharmonic singularities and maximality of pluricomplex Green functions with analytic singularities are derived.
We prove several results showing that plurisubharmonic functions with various bounds on their Monge-Ampere masses on a bounded hyperconvex domain always admit global plurisubharmonic subextension with logarithmic growth at infinity.
We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are a formula for the complex integrability exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of the openness conjecture by Demailly and Kollar. Our technique also yields new proofs of two recent results: one on the structure of the set of complex singularity exponents for holomorphic functions; the other by Lipman and Watanabe on…
This is a survey of results, both classical and recent, on behaviour of plurisubharmonic functions near their $-\infty$-points, together with the related topics for positive closed currents.
A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those determined by generic holomorphic mappings) in terms of decomposability of certain convex sets in $\Rn$. Another class of extreme singularities is presented by means of a notion of relative type.
In this article, we will construct a plurisubharmonic function whose jumping coefficients have a cluster point. We also give a class of plurisubharmonic functions which cannot be equisingular to any plurisubharmonic function with generalized analytic singularities.
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 construct examples of plurisubharmonic functions with isolated singularities at $0\in \mathbb{C}^n$, whose residual Monge-Ampère masses at the origin can not be approximated by masses of analytic approximations obtained via multiplier ideals. This answers negatively a conjecture of Demailly.
We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set V of valuations on R and--by way of a natural Laplace operator defined in terms of the tree structure on V--a positive measure on V. This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampere mass of two pluri…
It is a natural question to ask which plurisubharmonic functions admit a 'nice' approximation in the sense of a decreasing equisingular approximation with analytic singularities. For arbitrary toric plurisubharmonic functions, we give a criterion for admitting a nice approximation with toric approximants. Our results are motivated by a recent result of Guan for toric plurisubharmonic functions of the diagonal type.
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 study the complex Monge-Ampère operator in bounded hyperconvex domains of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Ampère energy. They generalize the classes introduced by U.Cegrell, and give a stratification of the space of (almost) all unbounded plurisubharmonic functions. We give an interpretation of these classes in terms of the speed of decreasing of the Monge-Ampère capacity of sublevel sets and solve associated complex Monge-Ampère equations.
By comparing Green functions of multi-circled plurisubharmonic singularities in the n-domensional complex space to their indicators, we obtain formulas for the higher Lelong numbers and integrability index for such singularities and extend Howald's result on multiplier ideals for monomial ideals to multi-circled singularities. This also leads to an elementary proof of the relations between the k-th Lelong numbers and the integrability index. For k=1 and arbitrary plurisubharmonic functions the inequality is due to Skoda, and for k=n and any plurisubharmonic function with isolated singularity t…
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 properties of relative types of plurisubharmonic functions with respect to maximal plurisubharmonic weights. It is shown that they give a general form for upper semicontinuous, tropically additive functionals on plurisubharmonic singularities. We consider some extremal problems whose solutions are Green-like functions that give best possible bounds on a plurisubharmonic function, given the values of its types relative to some of (or all) the weights. An analyticity theorem is proved for the upperlevel sets for the types with respect to exponentially Hölder continuous weights, which le…
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…
The goal of this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, and its applications to theory of valuations on convex sets and HKT-geometry (HyperKähler with Torsion). The exposition follows the articles math.CV/0104209, math.CV/0208005, math.MG/0401219 by the author and math.CV/0510140 by M. Verbitsky and the author.
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.
A notion of local indicator for a plurisubharmonic function is introduced. The indicator is a certain plurisubharmonic function in the unit polydisc, which controls the behavior of the considered function near a fixed point of its singularity, as well as the residual Monge-Ampere mass of the function at this point. A pluricomplex Green function with respect to given indicators is constructed and applied to the Dirichlet problem for the Monge- Ampere operator in the class of plurisubharmonic functions with isolated singularities.
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.
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）
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…
Let $u$ be a plurisubharmonic function, defined on a neighbourhood of a point $x,$ such that the complex Monge-Ampère operator is well-defined on $u.$ Suppose also that $u$ has a weak singularity, in the sense that the Lelong number of $u$ at $x$ vanish. Is it true that the residual mass of the measure $(dd^c{u})^n$ vanish at $x$? To our knowledge there is no known example that falsifies the posed question. In this paper some partial results are obtained. We find that for a significant subset of plurisubharmonic functions with well defined Monge-Ampère mass vanishing Lelong number does implies…
On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $ϕ$ such that $ϕ^*=ϕ_*$ on the closure of the domain. For $ϕ$ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron-Bremermann envelope $P ϕ$ is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge--Ampère equation \[ (dd^cu)^n = μ\] being uniquely solvable for continuous boundary data implies that …
We study geodesics for plurisubharmonic functions from the Cegrell class ${\mathcal F}_1$ on a bounded hyperconvex domain of ${\mathbb C}^n$ and show that, as in the case of metrics on Kähler compact menifolds, they linearize an energy functional. As a consequence, we get a uniqueness theorem for functions from ${\mathcal F}_1$ in terms of total masses of certain mixed Monge-Ampère currents. Geodesics of relative extremal functions are considered and a reverse Brunn-Minkowski inequality is proved for capacities of multiplicative combinations of multi-circled compact sets. We also show that fun…
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 observe that a slight adjustment of a method of Caffarelli, Li, and Nirenberg yields that plurisubharmonic functions extend across subharmonic singularities as long as the singularities form a closed set of measure zero. This solves a problem posed by Chirka.
We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization (Y, 0) of (X, 0), centered at the point 0 of Y. If (X, 0) is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is an algebra of finite type over the field of complex numbers, namely the coordinate ring of a non necessarily normal affine tor…
First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball $\B \sub \C^n$ with its relative logarithmic capacity in $\C^n$ with respect to the same ball $\B$. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of $\C^n$ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at in…
We study properties of a Green function G_A with singularities along a complex subspace A of a complex manifold X. It is defined as the largest negative plurisubharmonic function u satisfying locally u\leq \log|ψ|+C, where ψ=(ψ_1, ...,ψ_m), ψ_1, ...,ψ_m are local generators for the ideal sheaf I_A of A, and C is a constant depending on the function u and the generators. A motivation for this study is to estimate global bounded functions from the sheaf I_A and thus proving a ``Schwarz Lemma'' for I_A.
We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact Kähler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on Kähler manifolds. We then go on to prove the same regularity for envelopes when the manifold is assumed to have boundary. As an application, we answer affirmatively a question of Ross--Witt-Nyström concerning the Hele-Shaw flow on an arbitrary Reimann surface.
We provide some conditions for the graph of a Hoelder-continuous function on \bar{D}, where \bar{D} is a closed disc in the complex plane, to be polynomially convex. Almost all sufficient conditions known to date --- provided the function (say F) is smooth --- arise from versions of the Weierstrass Approximation Theorem on \bar{D}. These conditions often fail to yield any conclusion if rank_R(DF) is not maximal on a sufficiently large subset of \bar{D}. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also a…
Let $(X,ω)$ be a compact Kähler manifold. We introduce and study the largest set $DMA(X,ω)$ of $ω$-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\om)$ of all $\om$-psh functions. We prove that certain twisted Monge-Ampère operators are well defined for all $ω$-psh functions. As a consequence, any $\om$-psh function with slightly attenuated singularities has finite weighted Monge-Ampère energy.
Given a domain $Ω\subset \mathbf C^n$ we introduce a class of plurisubharmonic (psh) functions $\mathcal G(Ω)$ and Monge-Ampère operators $u\mapsto [dd^c u]^p$, $p\leq n$, on $\mathcal G(Ω)$ that extend the Bedford-Taylor-Demailly Monge-Ampère operators. Here $[dd^c u]^p$ is a closed positive current of bidegree $(p,p)$ that dominates the non-pluripolar Monge-Ampère current $\langle dd^c u\rangle^p$. We prove that $[dd^c u]^p$ is the limit of Monge-Ampère currents of certain natural regularizations of $u$. On a compact Kähler manifold $(X, ω)$ we introduce a notion of non-pluripolar energy and…
