Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of intensive study quite in their own right. Lazarsfeld's book "Positivity in Algebraic Geometry" contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples.

A primer on Seshadri constants

Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced in [Dem92] by Demailly. The original hope of using them towards a proof of t ...

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Let X be a canonically polarized variety, i.e. a complex projective variety such that its canonical class K (X) defines an ample -line bundle, and satisfying the conditions G (1) and S (2). Our main result says that X admits a Kahler-Einstein metric iff X has semi-log canonical singularities i.e. iff X is a stable variety in the sense of Kollar-Shepherd-Barron and Alexeev (whose moduli spaces are known to be compact). By definition a Kahler-Einstein metric in this singular context simply means a Kahler-Einstein on the regular locus of X with volume equal to the algebraic volume of K (X) , i.e. the top intersection number of K (X) . We also show that such a metric is uniquely determined and extends to define a canonical positive current in c (1)(K (X) ). Combined with recent results of Odaka our main result shows that X admits a Kahler-Einstein metric iff X is K-stable, which thus confirms the Yau-Tian-Donaldson conjecture in this general setting of (possibly singular) canonically polarized varieties. More generally, our results are shown to hold in the setting of log minimal varieties and they also generalize some prior results concerning Kahler-Einstein metrics on quasi-projective varieties.

Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs

https://pure.mpg.de/pubman/item/item_3117749_1/component/file_3117750/Meng_Building%20blocks_oa_2018.pdf

Building blocks of polarized endomorphisms of normal projective varieties

Let X be a normal projective variety. A surjective endomorphism $$f{:}X\rightarrow X$$ is int-amplified if $$f^*L - L =H$$ for some ample Cartier divisors L and H. This is a generalization of the so-called polarized endomorphism which requires that $$f^*H\sim qH$$ for some ample Cartier divisor H and $$q>1$$. We show that this generalization keeps all nice properties of the polarized case in terms of the singularity, canonical divisor, and equivariant minimal model program.

Building blocks of amplified endomorphisms of normal projective varieties

We explain how to deduce from recent results in the Minimal Model Program a general uniruledness theorem for base loci of adjoint divisors. As a special case, we recover previous results by Takayama.

Uniruledness of stable base loci of adjoint linear systems via Mori theory

Let $X$ be a normal variety such that $K_X$ is $\mathbb {Q}$-Cartier, and let $f:X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism $f$. As a consequence we prove that if $X$ is projective and $f$ polarised, then $X$ has at most log-canonical singularities.

Singularities of varieties admitting an endomorphism

We compute in this note the Seshadri constants of the anticanonical bundle at every point of Del Pezzo surfaces. During the proof, we enlight the role of rational curves in our computations. We present then two exemples where the positivity of the anticanonical bundle cannot be detected using rational curves.

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Constantes de Seshadri du diviseur anticanonique des surfaces de del Pezzo

We explain how to deduce from recent results in the Minimal Model Pro- gram a general uniruledness theorem for base loci of adjoint divisors. We also show how to recover special cases by extending a technique introduced by Takayama.

Uniruledness of stable base loci of adjoint linear systems with and without mori theory

We discuss the Calabi-Yau type structure of normal projective surfaces and Mori dream spaces admitting non-trivial polarized endomorphisms.

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Amaël Broustet

Papers

Uniruledness of stable base loci of adjoint linear systems via Mori theory

Singularities of varieties admitting an endomorphism

Constantes de Seshadri du diviseur anticanonique des surfaces de del Pezzo

Uniruledness of stable base loci of adjoint linear systems with and without mori theory

Remarks on Log Calabi-Yau Structure of Varieties Admitting Polarized Endomorphisms