Related papers: Volumes of components of Lelong upper level sets I…

We prove an upper bound for the volume of maximal analytic sets on which the generic Lelong number of a closed positive current is positive. As a particular case, we give a uniform upper bound on the volume of the singular locus of an…

Let $T$ be a positive closed current of bidegree $(1,1)$ on a multiprojective space $X={\mathbb P}^{n_1}\times\ldots\times{\mathbb P}^{n_k}$. For certain values of $\alpha$, which depend on the cohomology class of $T$, we show that the set…

Using $\delta$-invariants and Newton--Okounkov bodies, we derive the optimal volume upper bound for K\"ahler manifolds with positive Ricci curvature, from which we get a new characterization of the complex projective space.

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on $\mathbb P^2$ and $V_{\alpha}(T)$ be the upper level sets of Lelong numbers $\nu(T,x)$ of $T$. For any $\alpha\geq \frac{1}{3}$, we show that…

We prove a uniform local non-collapsing volume estimate for a large family of singular metrics in the big cohomology classes, which are K\"ahler on an open Euclidean subset of the manifold. The key ingredient is a generalization of a mixed…

We establish a relation between Lelong numbers and the full mass property of relative non-pluripolar products. We use it to show that if the restricted volume of a big cohomology class $\alpha$ in a compact K\"ahler $n$-dimensional manifold…

We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of…

It is a well-known fact that the non-pluripolar self-products of a closed positive (1,1)-current in a big nef cohomology class on a compact Kahler manifold are not of full mass in the presence of positive Lelong numbers of the current in…

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on the complex projective space $\mathbb P^2$. For $\alpha > 2/5$ and $\beta = (2-2\alpha)/3$ we show that if $T$ has four point with Lelong number greater than…

Let $X$ be a normal projective variety of dimension $d$ over an algebraically closed field and $f$ an automorphism of $X$. Suppose that the pullback $f^*|_{\mathsf{N}^1(X)_\mathbf{R}}$ of $f$ on the real N\'eron--Severi space…

Let f be a meromorphic correspondence on a compact Kahler manifold. We show that the topological entropy of f is bounded from above by the logarithm of its maximal dynamical degree. An analogous estimate for the entropy on subvarieties is…

Let $\{\alpha\}$ and $\{\beta\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional K\"ahler manifold $X$ such that the difference of intersection numbers $\{\alpha\}^n - n\,\{\alpha\}^{n-1}.\,\{\beta\}$ is…

We study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott-Chern cohomology in terms of Betti…

Let M be a compact, holomorphically symplectic Kahler manifold, and $\eta$ a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of $\eta$ is parabolic, that is, its top power vanishes. We prove that all…

Let $T$ be a positive closed current of bidimension $(p,p)$ with unit mass on the complex projective space $\mathbb P^n$. For certain values of $\alpha$ and $\beta = \beta(p, \alpha)$ we show that if $T$ has enough points where the Lelong…

Extending well-known results on surfaces, we give bounds on the cohomological action of automorphisms of compact K\"ahler threefolds. More precisely, if the action is virtually unipotent we prove that the norm of $(f^n)^*$ grows at most as…

Let $M$ be an $m$-dimensional compact Riemannian manifold with boundary. We obtain the upper bound of the harmonic mean of the first $m$ nonzero Neumann eigenvalues and Steklov eigenvalues involving the conformal volume and relative…

Let $T$ be a positive closed current of unit mass on the complex projective space $\mathbb P^n$. For certain values $\alpha<1$, we prove geometric properties of the set of points in $\mathbb P^n$ where the Lelong number of $T$ exceeds…

We show that the volume of transcendental big $(1,1)$-classes on compact K\"ahler manifolds can be realized by convex bodies, thus answering questions of Lazarsfeld-Musta\c{t}\u{a} and Deng. In our approach we use an approximation process…

In this paper, we study upper bounds for the topological complexity of the total spaces of some classes of fibre bundles. We calculate a tight upper bound for the topological complexity of an $n$-dimensional Klein bottle. We also compute…