Valuative analysis of planar plurisubharmonic functions

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 of valuations on R and – by way of a natural Laplace operator defined in terms of the tree structure on – a positive measure on . 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-Ampère mass of two plurisubharmonic functions. As a consequence, any generalized Lelong number can be interpreted as an average of valuations. Using our machinery we also show that the singularity of any positive closed (1,1) current T can be attenuated in the following sense: there exists a finite composition of blowups such that the pull-back of T decomposes into two parts, the first associated to a divisor with normal crossing support, the second having small Lelong numbers. Mathematics Subject Classification (1991) 32U25, 13A18, 13H05     

A sharp lower bound for the log canonical threshold

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function ${\varphi}$ with an isolated singularity at 0 in an open subset of ${\mathbb{C}^n}$ . This threshold is defined as the supremum of constants c > 0 such that ${e^{-2c\varphi}}$ is integrable on a neighborhood of 0. We relate ${c(\varphi)}$ to the intermediate multiplicity numbers ${e_j(\varphi)}$ , defined as the Lelong numbers of ${(dd^c\varphi)^j}$ at 0 (so that in particular ${e_0(\varphi)=1}$ ). Our main result is that ${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$ . This inequality is shown to be sharp; it simultaneously improves the classical result ${c(\varphi)\geqslant 1/e_1(\varphi)}$ due to Skoda, as well as the lower estimate ${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.     

Poincaré–Lelong formula, J-analytic subsets and Lelong numbers of currents on almost complex manifolds

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 on an almost complex manifold (M,J)$(M,J)$ 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.     

Approximating minimum-cost edge-covers of crossing biset-families

Part of this paper appeared in the preliminary version [16]. An ordered pair ? = (S, S ₊) of subsets of a groundset V is called a biset if S ? S+; (V S ⁺;V S) is the co-biset of ?. Two bisets \(\hat X,\hat Y\) intersect if ^X X ∩ Y ≠ \(\not 0\) and cross if both X ∩ Y \(\not 0\) and X ⁺ ∪ Y ⁺ ≠= V. The intersection and the union of two bisets \(\hat X,\hat Y\) are defined by \(\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )\) and \(\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )\) . A biset-family \(\mathcal{F}\) is crossing (intersecting) if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) that cross (intersect). A directed edge covers a biset ? if it goes from S to V S ⁺. We consider the problem of covering a crossing biset-family \(\mathcal{F}\) by a minimum-cost set of directed edges. While for intersecting \(\mathcal{F}\) , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing \(\mathcal{F}\) is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family \(\mathcal{F}\) is k-regular if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing \(\mathcal{F}\) if in addition both \(\mathcal{F}\) and the family of co-bisets of \(\mathcal{F}\) are k-regular, our ratios are: \(O\left( {\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) , and \(O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) . Using these generic algorithms, we derive for some network design problems the following approximation ratios: \(O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right) \) for k-Connected Subgraph, and O(logk) \(\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} \) for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.     

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k ∈ $\mathbb{T}^d $ , where $\mathbb{T}^d $ is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ?^d, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ $\mathbb{T}^d $ if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H₊(k), k ∈ $\mathbb{T}^d $ , corresponding to a two-particle system with repulsive interaction.     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

О разложениях в ряд ы о бобщенно-абсолютно-м онотонных функций

In an earlier paper [1] the notion of the so-called 〈?, GL_J>-absolutely monotonie functions was introduced, where ?≧1, {λ_{k k=0} ^∞ is an arbitrary non-increasing sequence of positive numbers. It was found that the condition \(\sum\limits_{\lambda _{k > 0} } {\lambda _k^{ - 1} } = + \infty \) is necessary in order to have the series expansion for any function f(x)∈〈?, λ_j). HereL ^k/? f/(x) are special integro-differential operators of fractional order, is a system of functions associated with the Mittag-Leffler type functions \(E_\varrho (z;\mu ) = \sum\limits_{k = 0}^\infty {z^n /\Gamma (\mu + \kappa /\varrho )} \) and with the sequence {λ_k}. In the present paper it is proved (in particular, see Theorem 3.2) that the expansion (*) is valid almost everywhere on (0,l) if ∑ λ _k ^?1 =+∞. This result contains, as a special case (when ?=1 and λ_k=0,k≧0) the known theorem of S. N. Bernstein on absolutely monotonic functions.     

Closed Range Property for Holomorphic Semi-Fredholm Functions

Given Banach spaces X and Y, we show that, for each operator-valued analytic map ${\alpha \in \mathcal O (D,\mathcal L(Y,X))}$ satisfying the finiteness condition ${\dim (X/\alpha (z)Y) < \infty}$ pointwise on an open set D in ${\mathbb {C}^n}$ , the induced multiplication operator ${\mathcal O(U,Y) \stackrel{\alpha}{\longrightarrow} \mathcal O (U,X)}$ has closed range on each Stein open set ${U \subset D}$ . As an application we deduce that the generalized range ${{\rm R}^{\infty}(T) = \bigcap_{k \geq 1}\sum_{| \alpha | = k} T^{\alpha}X}$ of a commuting multioperator ${T \in \mathcal L(X)^n}$ with ${\dim(X/\sum_{i=1}^n T_iX) < \infty}$ can be represented as a suitable spectral subspace.     

Expansions in Series of Suitable Systems of Functions

The paper is devoted to the study of expansions of functions ?, which are holomorphic in a circle Ω around 0, in a series of the form where the given system of functions (f_k) _k=0 ^? has the property that each function f_k has a representation as a power series of the form with a ₀ ^(k) = 1. It is the question if and where an expansion of the form (*) holds. We present sufficient conditions for the sequence (a _n ^(k) ) _k,n=0 ^∞ , under which there exists a subset Ω₀ of Ω such that for arbitrary ? the expansion (*) holds on Ω₀ and the series converges absolutely uniformly on compact subsets of Ω₀. The obtained results can be applied to prove expansions of analytic functions in a series of a suitable system of m-fold products of Bessel functions. Expansions of this type have been treated by many authors in special cases, but we are able to state a general expansion theorem.     

Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator

We consider two-particle Schrödinger operator H(k) on a three-dimensional lattice ? ³ (here k is the total quasimomentum of a two-particle system, $k \in \mathbb{T}^3 : = \left( { - \pi ,\pi ]^3 } \right)$ . We show that for any $k \in S = \mathbb{T}^3 \backslash ( - \pi ,\pi )^3$ , there is a potential $\hat v$ such that the two-particle operator H(k) has infinitely many eigenvalues z_n(k) accumulating near the left boundary m(k) of the continuous spectrum. We describe classes of potentials W(j) and W(ij) and manifolds S(j) ? S, i, j ∈ {1, 2, 3}, such that if k ∈ S(3), (k ₂ , k ₃ ) ∈ (?π,π) ² , and $\hat v \in W(3)$ , then the operator H(k) has infinitely many eigenvalues z_n(k) with an asymptotic exponential form as n → ∞ and if k ∈ S(i) ∩ S(j) and $\hat v \in W(ij)$ , then the eigenvalues z_nm(k) of H(k) can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.     

Super currents and tropical geometry

We introduce the formalism of positive super currents on ${\mathbb{R}^{n}}$ , in strong analogy with the theory of positive currents in ${\mathbb{C}^{n}}$ . We consider intersection of currents and Lelong numbers, and as an application we show that the formalism can be used to describe tropical varieties. This is similar in spirit to the fact that in complex analysis the current of integration of an analytic variety can be identified with a closed, positive current.     

Uniqueness of diffusion operators and capacity estimates

Let Ω be a connected open subset of R ^d . We analyse L ₁-uniqueness of real second-order partial differential operators ${H = - \sum^d_{k,l=1} \partial_k c_{kl} \partial_l}$ and ${K = H + \sum^d_{k=1}c_k \partial_k + c_0}$ on Ω where ${c_{kl} = c_{lk} \in W^{1,\infty}_{\rm loc}(\Omega), c_k \in L_{\infty,{\rm loc}}(\Omega), c_0 \in L_{2,{\rm loc}}(\Omega)}$ and C(x) = (c _kl (x)) > 0 for all ${x \in \Omega}$ . Boundedness properties of the coefficients are expressed indirectly in terms of the balls B(r) associated with the Riemannian metric C ^?1 and their Lebesgue measure |B(r)|. First, we establish that if the balls B(r) are bounded, the Täcklind condition ${\int^\infty_R dr r({\rm log}|B(r)|)^{-1} = \infty}$ is satisfied for all large R and H is Markov unique then H is L ₁-unique. If, in addition, ${C(x) \geq \kappa (c^{T} \otimes c)(x)}$ for some ${\kappa > 0}$ and almost all ${x \in \Omega}$ , ${{\rm div} c \in L_{\infty,{\rm loc}}(\Omega)}$ is upper semi-bounded and c ₀ is lower semi-bounded, then K is also L ₁-unique. Secondly, if the c _kl extend continuously to functions which are locally bounded on ?Ω and if the balls B(r) are bounded, we characterize Markov uniqueness of H in terms of local capacity estimates and boundary capacity estimates. For example, H is Markov unique if and only if for each bounded subset A of ${\overline\Omega}$ there exist ${\eta_n \in C_c^\infty(\Omega)}$ satisfying , where ${\Gamma(\eta_n) = \sum^d_{k,l=1}c_{kl} (\partial_k \eta_n) (\partial_l \eta_n)}$ , and for each ${\varphi \in L_2(\Omega)}$ or if and only if cap(?Ω) = 0.     

Jacobians and differents of projective varieties

Let X be a reduced and irreducible projective variety of dimension d. Let π:X→Y be a separable noetherian normalization of X and ? the canonical morphism Ω^d _X/k→Ω^d _L/k. From our main result: $$J_X \varphi (\pi ^* \Omega ^d _{Y/k} ) = \theta _k (X/Y)\varphi (\Omega ^d _{X/k} )$$ we deduce relations among: complementary module C(X/Y), Kähler different θ_k(X/Y), Dedekind different θ_D(X/Y), jacobian ideal J_K and ω-jacobian ideal \(\tilde J_X\) .     

On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation

One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of Kähler manifolds with nonnegative holomorphic bisectional curvature, $\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of K?hler manifolds with nonnegative holomorphic bisectional curvature, for some a > 67(n + 4)². We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincaré inequality.     

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

Lelong numbers with respect to regular plurisubharmonic weights

Generalized Lelong numbers v(T, φ) due to Demailly are specified for the case of positive closed currents T = dd ^cu and plurisubharmonic weights φ with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the directional Lelong numbers of the functions u and the Newton diagrams of φ. An extension of Demailly’s approximation theorem is proved as well.     

Factorial functions and stirling numbers of fractional orders

The aim of this paper is to study the binomial coefficients ( _n ^x ), the factorial polynomials [x]_n and [x]ⁿ, the Stirling numbers of first and second kind, namely s(n,k) and S(n,k), in the case that n ∈ ? is replaced by real α ∈ ?. In the course of the paper, the Vandermonde convolution formula is presented in an infinite series frame, the binomial coefficient function ( _a ^x ), α ∈ ?, is sampled in terms of the binomial coefficients ( _k ^x ) for k ∈ ?_o, Bell numbers of fractional orders are introduced. Emphasis is placed on the fractional order Stirling numbers s(α,k) and S(α,k), first studied here. Some applications of the S(α,k) are given.