Harmonic integrals on algebraic varieties

Extract

In my book The Theory and Applications of Harmonic Integrals (hereinafter referred to as H. I.), Chapter iv is devoted to the application of the theory to algebraic varieties. In order to introduce harmonic integrals on an algebraic variety I have to assign a metric to the variety; this metric is not related to the variety in any invariant sense, and indeed its introduction is extremely artificial. Nevertheless it turns out that we can deduce from the properties of the harmonic integrals a number of properties of the variety which are birationally invariant, that is, do not depend on the choice of metric. This remarkable fact seems to indicate either that the invariant properties in question should be obtainable directly without the introduction of the harmonic integrals, or that the metric introduced is not so artificial as it first appeared to be. This note is intended to examine this question.