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We prove functorial weak factorization of projective birational morphisms of regular quasi-excellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces., Comment: 45 pages

A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$. Under two geometric assumptions on the web-structure, the pairwise non-integrability condition and the bracket-generating condition, we prove that the local differential geometry determines the global algebraic geometry of $X$, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X \subset {\bf P}^N$ is a Fano submanifold of Picard number 1, and the family of lines covering $X$ becomes a web. In this special case, we have a stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X, X' \subset {\bf P}^N, \dim X' \geq 3,$ are two such Fano manifolds of Picard number 1, then any surjective morphism $f: X \to X'$ is an isomorphism., Comment: 35 pages, to appear in Duke Math. J

We find good dynamical compactifications for arbitrary polynomial mappings of C^2 and use them to show that the degree growth sequence satisfies a linear integral recursion formula. For maps of low topological degree we prove that the Green function is well behaved. For maps of maximum topological degree, we give normal forms., Comment: 31 pages. To appear in Ann. of Math

International audience; This article is a new presentation of the main results of D. Mathieu [M.00] on meromorphic equivalence relations. We introduce the space of finite type cycles (closed analytic cycles with finitely many irreducible components) of a given finite dimensional complex space and a natural topology on this space, in order to avoid the "regularity" condition for analytic families of cycles introduced in loc. cit. and also the two notions of "escape to infinity" which are here incoded in a natural way in our framework. Then the results are slightly stronger and much simple to state and to use. This contains, in an other language, a clean and more general version of the works of H. Grauert [G.83] and [G.86] and of B. Siebert [S.93] and [S.94] on meromorphic equivalence relations. G.83 Grauert, H. Set theoretic equivalence relations, Math. Ann. 265 (1983), p.137-148. G.86 Grauert, H. On meromorphic equivalence relations, Aspect Math. E9 (1986), p.115-147. M.00 Mathieu, D.Universal reparametrization..., Ann. Inst. Fourier (Grenoble) t.50 fasc.4 (2000), p.1155-1189 S.93 Siebert, B. Fiber cycles of holomorphic maps I. Local flattening, Math. Ann. 296 (1993), p.328-370. S.94 Siebert, B. Fiber cycle space and canonical flattening II, Math. Ann. 300 (1994), p.243-271.

In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings of C^m into P^n with (3n+1) moving targets and truncated multiplicities., Final version, published in Nagoya J. Math

In this paper we give some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space X over the complex m-space into a projective algebraic manifold and give their applications. We study this problem under a condition on the existence of meromorphic mappings separating the fibers of X.

We study the dynamics of meromorphic maps for a compact Kaehler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then, we study the particular case of a family of generic birational maps of P^k for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the measure is hyperbolic., 107 pages