Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

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Existence of minimal models for varieties of log general type

Dans son travail sur la log-concavite des multiplicites, Okounkov montre au passage que l'on peut associer un corps convexe a un systeme lineaire sur une variete projective, puis utiliser la geometrie convexe pour etudier ces systemes lineaires. Bien qu'Okounkov travaille essentiellement dans le cadre classique des fibres en droites amples, il se trouve que sa construction s'etend au cas d'un grand diviseur arbitraire. De plus, ce point de vue permet de rendre transparentes de nombreuses proprietes de base des invariants asymptotiques des systemes lineaires, et ouvre la porte a de nombreuses extensions. Le but de cet article est d'initier un developpement systematique de la theorie et de donner quelques applications et exemples.

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Convex bodies associated to linear series

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kahler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L1+e-density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kahler–Einstein volume forms with minimal singularities on varieties of general type.

/pdf/monge-ampere-equations-in-big-cohomology-classes-4q4n30hctb.pdf

Monge-Ampère equations in big cohomology classes

This volume is an expansion of lectures given by the author at the Park City Mathematics Institute (Utah) in 2008, and on other occasions. The purpose of this volume is to describe analytic techniques useful in the study of questions pertaining to linear series, multiplier ideals, and vanishing theorems for algebraic vector bundles. The author aims to be concise in his exposition, assuming that the reader is already somewhat acquainted with the basic concepts of sheaf theory, homological algebra, and complex differential geometry. In the final chapters, some very recent questions and open problems are addressed--such as results related to the finiteness of the canonical ring and the abundance conjecture, and results describing the geometric structure of Kahler varieties and their positive cones.

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Analytic Methods in Algebraic Geometry

The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.

Analytic Functions of Several Complex Variables

Le but de cet article est de definir et d'etudier systematiquement quelques invariants asymptotiques associes aux lieux de base des fibres en droites sur les varietes projectives lisses. Le comportement fonctionnel de ces invariants est lie au comportement ensembliste des lieux de base.

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Asymptotic invariants of base loci

We introduce and study the restricted volume of a divisor along a subvariety. Our main result is a description of the irreducible components of the augmented base locus by the vanishing of the restricted volume.

Restricted volumes and base loci of linear series

The purpose of this paper is to define and study systematically some asymptotic invariants associated to base loci of line bundles on smooth projective varieties. We distinguish an open dense subset of the real big cone, called the stable locus, consisting of the set of classes on which the asymptotic base locus is locally constant. The asymptotic invariants define continuous functions on the big cone, whose vanishing characterizes, roughly speaking, the unstable locus. We show that for toric varieties at least, there exists a polyhedral decomposition of the big cone on which these functions are polynomial.

We introduce the notion of Mukai regularity ($M$-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others are the subject of upcoming sequels.

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Regularity on abelian varieties I

In this paper we use the geometry of curves lying on K3 surfaces in order to obtain a number of results about effective divisors on the moduli space of stable curves Mg. We begin by showing two statements on the slopes of such divisors: first that the HarrisMorrison Slope Conjecture fails to hold on M10 and second, that in order to compute the slope of Mg for g ≤ 23, one only has to look at the coefficients of the classes λ and δ0 in the standard expansion in terms of the generators of the Picard group. The proofs are based on a general result providing inequalities between the first few coefficients of effective divisors on Mg. We then describe in detail the divisor K on M10 consisting of smooth sections of K3 surfaces, and its compactification K in the moduli space M10 (which is the counterexample to the conjecture mentioned above). As far as we know this is the first intersection theoretic analysis of a geometric subvariety on Mg which is not of classical Brill-Noether-Petri type, that is, a locus of curves carrying an exceptional linear series gd. Along the way, various other results on the Kodaira-Iitaka dimension of distinguished linear series on certain moduli spaces of (pointed) stable curves are obtained. We give the technical statements in what follows. 1

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Mihnea Popa

Papers

Asymptotic invariants of base loci

Restricted volumes and base loci of linear series

Asymptotic invariants of base loci

Regularity on abelian varieties I

Effective divisors on \overline{ℳ}_{ℊ}, curves on 3 surfaces, and the slope conjecture