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A Brody theorem for orbifolds. (English) Zbl 1162.14012

A (geometric) orbifold is a pair \((X, \Delta)\) where \(X\) is an irreducible complex space and \(\Delta\) is a \(\mathbb Q\)-Weil divisor \(\Delta=a_i Z_i\) such that the \(a_i\) are of the form \(1-\frac{1}{m_i}\) where \(m_i\) (the so-called multiplicity of \(Z_i\)) is an element of \(\mathbb N \cup \infty\). An orbifold is compact if \(X\) is compact and the orbifold divisor \(\Delta\) has no component of multiplicity \(\infty\). The concept of orbifolds was introduced by F. Campana [Ann. Inst. Fourier 54, No. 3, 499–630 (2004; Zbl 1062.14014)] in order to develop a classification theory of projective varieties that takes into account multiple fibres of fibrations and is (at least conjecturally) related to arithmetic and hyperbolic properties.
In the paper under review the authors study basic hyperbolicity properties of orbifolds: a holomorphic map \(h\) from the unit disc \(\mathbb D\) to an orbifold \((X, \Delta)\) is a (classical) orbifold morphism if \(h(D)\) is not contained in the support of \(\Delta\) and if the multiplicity of the pull-back \(h^* Z_i\) is a multiple of \(m_i\). In analogy to the usual Kobayashi pseudodistance, the (classical) orbifold Kobayashi pseudodistance of an orbifold \((X, \Delta)\) is then the largest pseudodistance such that every classical orbifold morphism is distance-decreasing. The main theorem states if \((X, \Delta)\) is a compact orbifold such that the classical orbifold Kobayashi pseudodistance is not a distance, then there exists a non-constant holomorphic map \(f: \mathbb C \rightarrow X\) with bounded derivative which is either a classical orbifold morphism or has its image in the support of \(\Delta\). This generalises Brody’s fundamental theorem saying that on a compact complex manifold the Kobayashi pseudodistance is not a distance if and only if \(X\) contains entire curves. As a corollary, one dimensional hyperbolic orbifolds are classified.
An important feature of Campana’s theory is the notion of non-classical multiplicities: a non-classical orbifold morphism \(h\) from the unit disc \(\mathbb D\) to an orbifold \((X, \Delta)\) is a map such that \(h(D)\) is not contained in the support of \(\Delta\) and the multiplicity of the pull-back \(h^* Z_i\) is at least \(m_i\). Using this more general notion, one can define a non-classical orbifold Kobayashi pseudodistance and ask for a non-classical hyperbolicity theory. The paper discusses in detail a number of examples and additional difficulties in the non-classical case.

MSC:

14E99 Birational geometry
14E22 Ramification problems in algebraic geometry
14G05 Rational points
14J40 \(n\)-folds (\(n>4\))
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables

Citations:

Zbl 1062.14014
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References:

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