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462 PHILLIP A. GRIFFITHS de1 o)11 el (-012 e2 de2 o)21 el 0922 e2 -I- 092a e3. As before, from o h21a091 -t- h22a092 and h21a h123 0 we have oa t9 oz. From exterior differentiation of 0)13 0, 0 &o 13 (.012/ 0)23 which implies that o o-oz. The fibres of 3’ are then defined by (02 0, and along these curves dz o)1 el, de o11 e as before. Thus in both cases the fibres of the Gauss map are lines, and consequently S is a ruled surface. As such, it may be given parametrically by (t, x) v(t) + Xw(t) where v(t), w(t) are holomorphic vector-valued functions and h is a linear parameter on the line Lt. This representation is unique up to a substitution as depicted by The Gauss map is f(t) v(t) + h(t)w(t) (t) (t) w(t) y(t, X) (v’ + hw’)/k w and this map is degenerate exactly when v’/ w and w’/ w are proportional. If w’/ w 0 so that w’ cw, then i,’ ’w + gaw and we may make ri,’ 0 by solving ’ + c 0. In this case S is a cone (including the special case of a cylinder). If w’/ w 0, then v’ / w r w’ / w. Under the above substitution (and taking # w),
CURVATURE AND COMPLEX SINGULARITIES 463 3’A= v’Aw+ hw’Aw (X+r)w’Aw, so that setting X + r 0 gives v’ A w 0. Thus w =/3v’ and taking =/3 -1 we may assume w v’. Then S is given by (t, X) v(t) + hv’(t), and is a developable ruled surface. Q.E.D. (b) Remarks on integration over analytic varieties. We now discuss some matters related to integration on analytic varieties. Let V be an n-dimensional analytic variety defined in an open set U C N. The set of singular points V forms a proper analytic subvariety, and the complement V* V V is a complex manifold which is open and dense in V. We denote by A(U) the C differential forms of degree q having compact support in U. The basic fact is that the linear function (3.12) Tv(a) [ a, A"(U), 3v defines a closed, positive current of type (n, n). Essentially this means that the integral (3.12) is absolutely convergent, and that Stokes’ theorem v d O, an- l( U), is valid. We shall also use the result that if {Vt} is a family of varieties depending holomohically on parameters (in a sense to be made precise in 5b when needed), then also depends holomohically on (all we use is that it depends continuously). In somewhat more detail, if 2 & A d is the standard Kfihler form on g, and if for any index set I {i, ., i} we denote by * 2 (&l A dq) A. A 2 the <strong>Euclid</strong>ean measure on the corresponding C CN, then n #i= n (&*n A
Page 1 and 2: Vol. 45, No. 3 DUKE MATHEMATICAL JO
Page 3 and 4: CURVATURE AND COMPLEX SINGULARITIES
Page 11 and 12: we deduce that CURVATURE AND COMPLE
Page 13 and 14: so that CURVATURE AND COMPLEX SINGU
Page 35: CURVATURE AND COMPLEX SINGULARITIES
Page 41 and 42: while by Stokes’ theorem CURVATUR
Page 45 and 46: we conclude that CURVATURE AND COMP
Page 63 and 64: For k we have (4.33) CURVATURE AND
Page 79 and 80: As a consequence of (5.15), CURVATU

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