Special Varieties and classification Theory
By Frédéric Campana. 2001
87 citations recorded by the source corpus.
A new class of compact K\"ahler manifolds, called special, is defined, which are the ones having no surjective meromorphic map to an orbifold of general type. The special manifolds are in many respect higher-dimensional generalisations of rational and elliptic curves. For example, we show that being rationally connected or having vanishing Kodaira dimension implies being special. Moreover, for any compact K\"ahler $X$ we define a fibration $c_X:X\to C(X)$, which we call its core, such that the general fibres of $c_X$ are special, and every special subvariety of $X$ containing a general point of $X$ is contained in the corresponding fibre of $c_X$. We then conjecture and prove in low dimensions and some cases that: 1) Special manifolds have an almost abelian fundamental group. 2) Special manifolds are exactly the ones having a vanishing Kobayashi pseudometric. 3) The core is a fibration of general type, which means that so is its base $C(X)$,when equipped with its orbifold structure coming from the multiple fibres of $c_X$. 4) The Kobayashi pseudometric of $X$ is obtained as the pull-back of the orbifold Kobayashi pseudo-metric on $C(X)$, which is a metric outside some proper algebraic subset. 5) If $X$ is projective,defined over some finitely generated (over $\Bbb Q$) subfield $K$ of the complex number field, the set of $K$-rational points of $X$ is mapped by the core into a proper algebraic subset of $C(X)$. These two last conjectures are the natural generalisations to arbitrary $X$ of Lang's conjectures formulated when $X$ is of general type.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/25247472951b6748c540cc181dab8313c8449eaa
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Relative adjoint transcendental classes and Albanese maps of compact Kaehler manifolds with nef Ricci curvature (2012, unknown): “[2], [3], [4], [6], [14], [17], [18], [21], [22], [23], [24], [25], [28], [34], [35], [39], [40], [41] to quote only a few.”
- Rational connectedness of log Q-Fano varieties (2004, background): “Remark: The notion of special varieties was introduced and studied by F. Campana in [ 20 ].”
- Boundedness of pluricanonical maps of varieties of general type (2005, background): “is weakly positive, where �h consists of those components of � which dominate Y . In fact both [ 4 ], (4.13) and [15], (9.8) prove that this sheaf is weakly positive, as soon as m�h is integral.”