Manifolds of General Type

  • Shoshichi Kobayashi
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 318)


In general, given a topological space 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 EX, we set
$${m_k}(E) = \mathop {\sup }\limits_{\varepsilon >0} \inf \left\{ {\sum\limits_{i = 1}^\infty {{{(\delta ({E_i}))}^k};E = \bigcup\limits_{i = 1}^\infty {{E_i},\delta ({E_i})} } < \varepsilon } \right\}$$
, where 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.


Line Bundle General Type Complex Space Ample Line Bundle Smooth Projective Variety 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Shoshichi Kobayashi
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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