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472 PHILLIP A. GRIFFITHS
Combining this with (3.21) and (3.22) gives our result.
Now suppose that V C N is an entire analytic set. It is by now well-estabfished
that the function/z0(V, r) gives the basic analytic measure of the growth
of V, playing to some extent a role analogous to the degree of an algebraic
variety in projective space. 11) It may be that in more refined questions the
function k(V, r) should also be used as growth indicators, especially since they
appear in the growth of the currents T,(V) obtained by the standard smoothing
of the current Tv defined by (3.12).
Footnotes
1. A good reference for general material on complex manifolds is S. S. Chern, Complex Manifolds
Without Potential Theory, van Nostrand, 1968.
2. cf. the last section in S. S. Chern, Characteristic classes of Hermitian manifolds, Ann. of
Math., vol. 47 (1946), pp. 85-121.
3. This is just another way of saying that the n x (N n) matrix of forms {o,} gives a basis for
the (1, 0) tangent space to G(n, N).
4. A ruled surface is the locus of1 straight lines Lt in n / /
2; the tangent lines to a curve in
form a developable ruled surface; and finally a cone consists of 1 lines through a fixed point, "
possibly at infinity. We note the following global corollary of (3.11), which was pointed out to us by
Joe Harris: If S C IPN is a smooth algebraic surface which is not a plane, then the Gaussian image of
S is two-dimensional.
5. P. Lelong, Fonctions plurisousharmoniques et formes differentielles positiv, Paris, Gordon
Breach, 1968.
6. This is the basic integral formula in holomorphic polar coordinates (these coordinates essentially
amount to the standard Hopf bundle over IPN- 1).
7. P. Thie, The Lelong number ofpoints of a complex analytic set, Math. Ann., vol. 172 (1967),
pp. 269-312.
8. cf. the reference cited in footnote (7) of the introduction.
9. Recall that V* V Vs is the manifold of smooth points on V.
10. By closer examination of the behaviour near a singularity it seems likely that this remains
true with no assumptions about the singularities of V.
11. cf. H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans (, Bull. Soc. Math.
France, vol. 100 (1972), pp. 353-408.
4. Hermitian integral geometry
(a) The elementary version of Crofton’s formula. We shall first take up the
complex analogue of (2.1). Let C be an analytic curve defined in some open set
in 2. Denote by (, n) the Grassmannian of complex affine k-planes in n, so
that (1, 2) is the space of complex lines in the plane. For each line L the
analytic intersection number #(L, C) is defined and is a non-negative integer.
By a basic fact in complex-analytic geometry, this is also the geometric number
of intersections n(L fq C) of the line L with the curve C. The point is that since
complex manifolds are naturally oriented the geometric and topological intersection
numbers coincide. We will prove that
(4.1) n(L f) C)dL vol (C)