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

/pdf/existence-of-minimal-models-for-varieties-of-log-general-43vm28b5my.pdf

Existence of minimal models for varieties of log general type

This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.

/pdf/kahler-einstein-metrics-on-fano-manifolds-i-approximation-of-3crqnpu6tk.pdf

Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities

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

/pdf/kahler-einstein-metrics-on-fano-manifolds-iii-limits-as-cone-rurql5tjx0.pdf

Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2 and completion of the main proof

/pdf/kahler-einstein-metrics-on-fano-manifolds-ii-limits-with-58i5d1zlwn.pdf

Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2

We study degenerate complex Monge-Ampere equations of the form $(\omega+dd^c\f)^n = e^{t \f}\mu$ where $\omega$ is a big semi-Kahler form on a compact Kahler manifold $X$ of dimension $n$, $t \in \R^+$, and $\mu=f\omega^n$ is a positive measure with density $f\in L^p(X,\omega^n)$, $p>1$. We prove the existence and unicity of continuous $\o$-plurisubharmonic solutions. In case $X$ is projective and $\omega=\psi^*\omega'$, where $\psi:X\to V$ is a proper birational morphism to a normal projective variety, $[\omega']\in NS_{\R} (V)$ is an ample class and $\mu$ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kahler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

/pdf/singular-kahler-einstein-metrics-29u5pmduik.pdf

Singular Kähler-Einstein metrics

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 a closed positive current in given big cohomology class. The solution is shown to have minimal singularities in the sense of Demailly if the measure is regular enough. These results are combined with a fixed point argument to construct singular Kaehler-Einstein volume forms with minimal singularities on varieties of general type.

Monge-Amp\`ere equations in big cohomology classes

We prove the existence and uniqueness of Kahler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kahler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kahler-Ricci flow provides weak convergence independently of Perelman's celebrated estimates.

Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties

We prove the existence and uniqueness of K\"ahler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized K\"ahler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the K\"ahler-Ricci flow provides weak convergence independently of Perelman's celebrated estimates.

/pdf/k-ahler-einstein-metrics-and-the-k-ahler-ricci-flow-on-log-3iijs7jnvo.pdf

Philippe Eyssidieux

Papers

Monge-Ampère equations in big cohomology classes

Singular Kähler-Einstein metrics

Monge-Amp\`ere equations in big cohomology classes

Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties

K\"ahler-Einstein metrics and the K\"ahler-Ricci flow on log Fano varieties