Related papers: Extremal cases for the log canonical threshold
We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\bar\Omega$, with functions that are plurisubharmonic on (shrinking) neighborhoods…
We consider mixed Monge-Amp\`ere products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one parameter limits of mixed Monge-Amp\`ere products of smooth functions,…
We study the complex Monge-Amp\` ere operator on compact K\"ahler manifolds. We give a complete description of its range on the set of $\omega-$plurisubharmonic functions with $L^2$ gradient and finite self energy, generalizing to this…
We prove some basic properties of quasinearly subharmonic functions and quasinearly subharmonic functions in the narrow sense.
We show the existence and uniqueness of bounded solutions to the degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds. We also study the asymptotics of these solutions. As applications, we give partial answers to…
We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Amp\`ere flow on a compact K\"ahler manifold, with the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere…
A PDE proof is provided for the sharp $L^\infty$ estimates for the complex Monge-Amp\`ere equation which had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics.…
In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of…
Generalized Lelong numbers of plurisubharmonic functions with respect to plurisubharmonic weights (due to Demailly) are specified for weights with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the…
We review recent advances in the numerical analysis of the Monge-Amp\`ere equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods…
On smooth threefolds, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. We reduce it to the case when the boundary is the product of a…
We undertake a preliminary step towards studying non-Archimedean pluripotential theory on polarized affine cones over a trivially valued field. We study plurisubharmonic functions and the Monge--Amp\`ere operator defined on the finite…
We recall known and establish new properties of the Dieudonn\'e and Moore determinants of quaternionic matrices.Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. Then we…
Let $(X, \omega)$ be a compact K\"ahler manifold of complex dimension n and $\theta$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class $\{ \theta \}\in H^{1,1}(X, \mathbb{R})$ is pseudoeffective. Let $\varphi$ be a…
We consider the pointwise approximation of a subharmonic function by the logarithm of the modulus of an entire function up to a bounded quantity. In the case of finite order an estimate from below of the planar Lebesgue measure of an…
In this note we find optimal one-sided majorants of exponential type for the signum function subject to certain monotonicity conditions. As an application, we use these special functions to obtain a simple Fourier analysis proof of the…
Let X,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\limsup_n (n\log\log n)^{-1}|\sum_{1<=…
We express the Segre class of a monomial scheme in projective space in terms of log canonical thresholds of associated ideals. Explicit instances of the relation amount to identities involving the classical polygamma functions.
Given a psh function $\varphi\in\mathcal{E}(\Omega)$ and a smooth, bounded $\theta\geq 0$, it is known that one can solve the Monge-Amp\`{e}re equation $\mathrm{MA}(\varphi_\theta)=\theta^n\mathrm{MA}(\varphi)$, with some form of Dirichlet…
We give characterizations of (quasi-)plurisubharmonic functions in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic functions.