On the singularity type of full mass currents in big cohomology classes

Let $X$ be a compact K\"ahler manifold and $\{\theta\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $\theta$-plurisubharmonic functions with full mass are the same as those of the current with minimal singularities. Second, given another big and nef class $\{\eta\}$, we show the inclusion $\mathcal{E}(X,\eta) \cap {PSH}(X,\theta) \subset \mathcal{E}(X,\theta).$ Third, we characterize big classes whose full mass currents are "additive". Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.Comment: v2. Theorem 1.1 updated to include statement about multiplier ideal sheaves. Several typos fixed. v3. we make our arguments independent of the regularity results of Berman-Demaill

L^1 metric geometry of big cohomology classes

Suppose $(X,\omega)$ is a compact K\"ahler manifold of dimension $n$, and $\theta$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,\theta)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog from the K\"ahler case, as it only relies on pluripotential theory, with no reference to infinite dimensional $L^1$ Finsler geometry. Lastly, by adapting the results of Ross and Witt Nystr\"om to the big case, we show that one can construct geodesic rays in this space in a flexible manner

Monge-Ampère measures on contact sets

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 $\theta$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $\omega$ such that $\varphi \leq f.$ Then the non-pluripolar measure $\theta_\varphi^n:= (\theta + dd^c \varphi)^n$ satisfies the equality: $${\bf{1}}_{\{ \varphi = f \}} \ \theta_\varphi^n = {\bf{1}}_{\{ \varphi = f \}} \ \theta_f^n,$$ where, for a subset $T\subseteq X$, ${\bf{1}}_T$ is the characteristic function. In particular we prove that \[ \theta_{P_{\theta}(f)}^n= { \bf {1}}_{\{P_{\theta}(f) = f\}} \ \theta_f^n\qquad {\rm and }\qquad \theta_{P_\theta[\varphi](f)}^n = { \bf {1}}_{\{P_\theta[\varphi](f) = f \}} \ \theta_f^n. \

Uniqueness and short time regularity of the weak K\"ahler-Ricci flow

Let $X$ be a compact K\"ahler manifold. We prove that the K\"ahler-Ricci flow starting from arbitrary closed positive $(1,1)$-currents is smooth outside some analytic subset. This regularity result is optimal meaning that the flow has positive Lelong numbers for short time if the initial current does. We also prove that the flow is unique when starting from currents with zero Lelong numbers.Comment: 33 page

Entropy for Monge-Amp\`ere Measures in the Prescribed Singularities Setting

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Amp\`ere mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, \theta, \phi)$ (provided $n>1$), while they have the same singularities of $\phi$ when $n=1$

Log-concavity of volume and complex Monge-Amp\`ere equations with prescribed singularity

Let $(X,\omega)$ be a compact K\"ahler manifold. We prove the existence and uniqueness of solutions to complex Monge-Amp\`ere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the K\"ahler case as well. As an application we confirm a conjecture by Boucksom-Eyssidieux-Guedj-Zeriahi concerning log-concavity of the volume of closed positive $(1,1)$-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn-Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and $P$-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.Comment: v2. Examples added, no other significant changes v.3 Small changes in presentation. Final versio