No abstract

Given a Fano manifold ( X , ω ) {(X,\omega)} , we develop a variational approach to characterize analytically the existence of Kähler–Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically. Moreover, we define a function α ω {\alpha_{\omega}} on the set of prescribed singularities which generalizes Tian’s α-invariant, showing that its upper lever set { α ω ⁢ ( ⋅ ) > n n + 1 } {\{\alpha_{\omega}(\,\cdot\,)>\frac{n}{n+1}\}} produces a subset of the Kähler–Einstein locus, i.e. of the locus given by all prescribed singularities that admit Kähler–Einstein metrics. In particular, we prove that many K-stable manifolds admit all possible Kähler–Einstein metrics with prescribed singularities. Conversely, we show that enough positivity of the α-invariant function at nontrivial prescribed singularities (or other conditions) implies the existence of genuine Kähler–Einstein metrics. Finally, through a continuity method we also prove the strong continuity of Kähler–Einstein metrics on curves of totally ordered prescribed singularities when the relative automorphism groups are discrete.

On a compact Kähler manifold (X, ω), we study the strong continuity of solutions with prescribed singularities of complex Monge-Ampère equations with convergent integrable Lebesgue densities. Then we address the strong continuity of solutions when the right hand sides are modified to includes all (log-)Kähler Einstein metrics with prescribed singularities. This leads to the closedness of a new continuity method when the densities are modified together with the prescribed singularities setting. For Monge-Ampère equations of Fano type, we also prove an openness result when the singularities decrease. Finally we deduce a strong stability result for (log-)Kähler Einstein metrics on semi-Kähler classes given as modifications of {ω}.

On (X, ω) compact Kähler manifold, given a model type envelope ψ ∈ P SH(X, ω) (i.e. a singularity type) we prove that the Monge-Ampère operator is an homeomorphism between the set of ψ-relative finite energy potentials and the set of ψ-relative energy measures endowed with their strong topologies given as the coarsest refinements of the weak topologies such that the relative energies become continuous. Moreover, given a totally ordered family A of model type envelopes with positive total mass representing different singularities types, the sets XA, YA given respectively as the union of all ψ-relative finite energy potentials and of all ψ-relative finite energy measures varying ψ ∈ A have two natural strong topologies which extends the strong topologies on each component of the unions. We show that the Monge-Ampère operator produces an homeomorphism between XA and YA.As an application we also prove the strong stability of a sequence of solutions of prescribed complex Monge-Ampère equations when the measures have uniformly L p -bounded densities for p > 1 and the prescribed singularities are totally ordered.

Starting from the data of a big line bundle L on a projective manifold X with a choice of N ≥ 1 different points on X we give a new construction of N Okounkov bodies which encodes important geometric features of (L → X; p1, . . . , pN ) such as the volume of L, the (moving) multipoint Seshadri constant of L at p1, . . . , pN , and the possibility to construct Khler packings centered at p1, . . . , pN . Toric manifolds and surfaces are examined in detail.