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464 PHILLIP A. GRIFFITHS and, for any affine n-plane L in N (3.13) n! <strong>Euclid</strong>ean measure on (to see this choose orthonormal coordinates so that L is a translate of ([;’9. Two important consequences of (3.13) are: (i) For any complex manifold M (3.14) n! b ’ vol (M) is the <strong>Euclid</strong>ean volume of M (Wirtinger theorem), and (ii) for any a A"( there is an estimate in the sense that for all complex n-planes L where the function on the Grassmannian is bounded. The fact that (3.12) defines a current is then equivalent to the finiteness of the volume n Cl B[Z,] < of analytic varieties in the e-ball B[z, e] around singular points z V,. Similarly, Stokes’ theorem follows from the usual version for manifolds together with the fact that the (2n 1)-dimensional area of the boundary 0 T,(V,) of the e-tube around the singular points tends to zero as e 0. Later on we shall be examining more delicate integrals / where is not jv the restriction of a form in U (such as a curvature integral), or may be the restriction of a form but one having singularities on V. To obtain some feeling for these We shall examine one of the latter types here. With the notations d c we shall prove the formula ( -1 4 w d d c log ilzll {z e v: ilzll r} o, d r]- o], o r, (3.15) r Oe Geometrically, if we consider the residual map [rl [O] [O,rl CN {0} IP N m.
CURVATURE AND COMPLEX SINGULARITIES 465 defined by z 0z, then to is the pullback of the standard Kfihler form on pNand as such has a singularity at origin. To prove (3.15) we note that If we define d c log I[zll z d 2 (dz, dz) =dd ilzll 4 {(z, dz)-dz, z) { (z,z)(dz, dz)- (dz, z)(z, dz) ) (3.16) r d log I1 11 then d ton, and by Stokes’ theorem the right hand side of (3.15) is| Now, for any fixed t, on O V[t] we have which implies that on 0 V[t] Then t2n 0 (dz, z) + (z, dz) "= r." 9V[t] V[t] t2n fV[tl proving (3.15). As a consequence the function (V, r) is an increasing function of r, and the limit d II=ll A vol V[r] /,(V) lim /x(V, r) r-+0
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 37: 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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