Mihai Păun

Bio: Mihai Păun is an academic researcher from University of Strasbourg. The author has contributed to research in topics: Serre duality & K3 surface. The author has an hindex of 1, co-authored 1 publications receiving 421 citations. Previous affiliations of Mihai Păun include Nancy-Université.

The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

Abstract: We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of " movable curves " , which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

Monge-Ampère equations in big cohomology classes

Abstract: 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.

Asymptotic invariants of base loci

Abstract: 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.

On Shokurov's rational connectedness conjecture

Abstract: We prove the rational connectedness conjecture of V. V. Shokurov in [20] which, in particular, implies that the fibres of a resolution of a variety with divisorial log terminal singularities are rationally chain connected

Monge-Amp\`ere equations in big cohomology classes

Abstract: 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.

Restricted volumes and base loci of linear series

Abstract: 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.