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5O8 PHILLIP A. GRIFFITHS pullback under yt of the standard Kfihler form on IP2, we infer as in (5.14) that (5.26) lim lim C t- KdA # (A, H), 0 0 Jct [d and this number 8 has the following interpretation: A generic pencil of plane sections Dt,x Ct f3 (h + H) represents Ct as a d mult0(C0)-sheeted covering of a disc in the )t-plane having 8 branch points. Now, assuming that Co is non-dengenerate, we may consider the second order Gauss map which assigns to each z Ct the osculating 2-plane. (11) As before we may define the 2nd order Pliicker defect A* by lim y(Ct) y(Co) + A*, t--0 and the letting Ct-e K*dA denote the pullback under y of the K/ihler form on IW* we infer the formula (5 27) lim lim C te K*dA # (A* L*) Jct[e] where L* C IP 2. is a generic line. The number 8" on the right hand side of (5.27) has the following geometric interpretation: Under a generic linear projection a 2 the curve Ct projects onto a plane curve C, and 8" is the number of flexes of C[e] as e, 0. Perhaps the general thrust of the discussion may be summarized as follows: In a family of global smooth algebraic varieties Vt c IP N tending to a singular variety V0, some of the projective characters may jump in the limit at 0(lz). This jump is measured by the intersection number of Schubert cycles with the Plticker defect A (and its higher order analogues A*, etc.), and by the analytic Pliicker formulas (13 may be expressed as a difference lim Iv {curvature form}-Iv {curvature form}. t--*O Moreover, in case V0 has an isolated singularity this whole process may be localized around the singular point, so that the correction factor which must be subtracted from the PlUcker formulas for Vt to obtain those for V0 is (a4 lira lim | {curvature form}. 0 JVt[e] Finally, and most importantly, certain combinations of these expressions will have intrinsic meaning for V0, and may even yield topological invariants of the singularity as was the case in (5.23).
CURVATURE AND COMPLEX SINGULARITIES 509 (iii) We will conclude with some observations about curvature integrals in the real and complex cases. In real differential geometry one generally encounters two types of curvature integrals, illustrated for a Riemannian surface by the two expressions (5.28) Is KdA, .ts IK]dA In general the first type includes characteristic classes and Weyl’s coefficients (1.10), and reflects intrinsic metric properties of the manifold which may even be of a topological nature. The second type usually describes extrinsic properties of the manifold in <strong>Euclid</strong>ean space. For a complex manifold M, C because of sign properties such as (_ 1)kcc(t) / n-k _> 0 the distinction between the two types of curvature integrals illustrated in (5.28) seems to disappear, and the situation may be said either to be simpler or more rigid, depending on one’s viewpoint. Finally, along similar lines it would seem interesting to try and draw conclusions on the curvature of real algebraic curves acquiring a singularity. For example, one might examine the real curves Ct,IR IR C where Ct c is defined by f(x, y) with f(x, y) being a weighted homogeneous polynomial having real coefficients. Superficial considerations suggest that the curvature of Ct,R in IR tends to , while on the Riemann surface Ct the geodesic curvature of the (purely imaginary) vanishing cycles remains bounded. (iv) This paragraph is an afterthought. Upon reviewing the preceeding discussion about "topologically invariant curvature integrals associated to a singularity" it seems to me that some clarification is desirable. To begin with let us consider the pedogogical question of how one might best prove the Gauss-Bonnet theorem (5.29) ct-e l KdM x(M) in a course on differential geometry. Chern’s intrinsic proof (cf. footnote (8) in section 1) is probably the quickest but in my experience leaves some mystery as to the origin of his formulas. The Allendoerfer-Weil proof (cf. footnote (5) in section 1) is intuitively appealing and explains the origin of the formula, but relies on either the Nash embedding theorem or an unpleasant construction to show that the left hand side of (5.29) is independent of the metric. Finally, the proof by characteristic classes, interpreting the Gauss-Bonnet integrand as a de Rham representative of the Euler class of the tangent bundle, is conceptually satisfying but involves establishing a fairly elaborate machine. The following proof, in four steps, represents a compromise" a) Prove (5.29) for oriented real hypersurfaces using the Hopf theorem, as
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