Abstract
Given a complex space X, we shall define a new infinitesimal form [Formula: see text] of what is known as the Kobayashi pseudo-distance dx. At each point x of X, [Formula: see text] defines a pseudo-norm in the tangent space TxX; it satisfies the usual conditions for a norm except that [Formula: see text] may vanish for a nonzero vector υ. It turns out that the new pseudo-metric is the double dual of the old pseudo-metric Fx defined in [2] and [4]. This means that the indicatrix of [Formula: see text] is the convex hull of the indicatrix of Fx. In particular, [Formula: see text]. Advantages of [Formula: see text] over Fx are two-fold. First, [Formula: see text] satisfies the usual convexity condition, i.e., [Formula: see text]. (In [3] Lang calls Fx a semi-length function since it does not, in general, satisfy the convexity condition.) Second, it is defined on Zariski tangent spaces. It can be easily shown that [Formula: see text] is upper semicontinuous at nonsingular points of X. The upper semicontinuity for Fx is known also only in the nonsingular case. Although [Formula: see text], it can be shown, at least when X is nonsingular, that [Formula: see text] induces dx.
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