Hyperbolic Complex Spaces pp 343-392 | Cite as

# Manifolds of General Type

Chapter

## Abstract

In general, given a topological space
, where

*X*with a pseudo-distance*d*and a non-negative real number*k*, the*k*-dimensional**Hausdorff measure***m*_{ k }is defined as follows. For a subset*E*⊂*X*, we set$${m_k}(E) = \mathop {\sup }\limits_{\varepsilon >0} \inf \left\{ {\sum\limits_{i = 1}^\infty {{{(\delta ({E_i}))}^k};E = \bigcup\limits_{i = 1}^\infty {{E_i},\delta ({E_i})} } < \varepsilon } \right\}$$

*δ*(*E*_{ i }) denotes the diameter of*E*_{ i }. If*X*is a complex space, then the pseudo-distances*c*_{ x }and*d*_{ x }induce Hausdorff measures on*X*. Since every holomorphic map is distance-decreasing with respect to these intrinsic pseudo-distances, it is also measure-decreasing with respect to the Hausdorff measures they define. There are other intrinsic measures on complex spaces. For a systematic study of intrinsic measures on complex manifolds, see Eisenman [1]. In Section 2 we shall discuss the intrinsic mesaures which may be considered as direct generalizations of*c*_{ x }and*d*_{ x }In this section we discuss their infinitesimal forms.## Keywords

Line Bundle General Type Complex Space Ample Line Bundle Smooth Projective Variety
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 1998