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The Kobayashi pseudometric on algebraic manifold and a canonical fibration | | Steven Shin-Yi Lu
; | | Date: |
18 Jun 2002 | | Subject: | Algebraic Geometry MSC-class: 14C30, 14E30, 14D06, 14J10, 32J27, 32Q57 | math.AG | | Abstract: | Given a compact complex manifold X of dimension n, we define a bimeromorphic invariant $kappa_+(X)$ as the maximum p for which there is a saturated line subsheaf L of the sheaf of holomorphic p forms whose Kodaira dimension $kappa (L)$ equals p. We call X special if $kappa_+(X)=0$. We give some evidence that this condition characterizes X to have identically vanishing Kobayashi pseudometric. We use the well-known construction of F. Campana to give, for each projective X a canonical fibration $f: X o Y$ holomorphic outside a proper subvariety of Y and whose general fibers are special. We show that the inherited orbifold structure on Y defined via the minimum multiplicity of those of the components of each fiber does not admit positive dimensional special sub-orbifolds through the general points of Y. We note that the Iitaka fibration or any rationally connected fibration of X is a natural factor of f and we show that this solves a general conjecture in Mori’s classification program of algebraic varieties, namely, that an algebraic variety is either of general type, or (birationally) has a canonical fibration with positive dimensional special type fibers that factors through the Iitaka and rationally connected fibrations of X. | | Source: | arXiv, math.AG/0206170 | | Services: | Forum | Review | PDF | Favorites | |
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