Manifolds of General Type

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