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512 PHILLIP A. GRIFFITHS
One way in which this makes sense is for the individual forms qk to have a
singularity at the origin ofjust the right order to prevent (5.33) from holding (cf.
the proof of (3.17) in 3(b)), but in the alternating sum
k
(-1)
2(n k)
cancellation occurs in the highest order term. If this were the case, then there
would be a procedure for looking into the topological invariance of other curvature
integrals. We note in closing that the question is purely one on the singular
varieties V0,s and might be examined by blowing up the origin in a fashion similar
to how (3.17) was established.
Footnotes
1. cf. the references cited in footnote 2 of the introduction.
2. cf. the reference cited in footnote of the introduction.
3. There is a discussion of the Plticker formulas in the paper cited in footnote 4 of the introduction.
4. The motivation for this terminology is given by the discussion at the end of the preceeding
section 5(a).
5. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic
zero, Annals of Math., vol. 79 (1964).
6. B. Tessier, Introduction to equisingularity problems, Proc. Symposia in Pure Math., vol. 29
(1975), pp. 593-632; this paper contains an extensive bibliography on singularities.
7. Milnor numbers were introduced in Milnor’s book cited in footnote 5 in the introduction.
There is an extensive bibliography on material pertaining to Milnor numbers given in Tessier’s
article referred to in footnote 6 above.
8. Of course the classic source is S. Lefschetz, L’Analysis situs et la gomtrie algObrique,
Gauthier-Villars, Paris, (1924). A recent paper which in fact is pertinent to our present study is A.
Landman, On the Picard-Lefschetz transformations, Trans. Amer. Math. Soc., vol. 181 (1973), pp.
89-126.
9. As noted in 3(a) (cf. (3.7)) the differential forms cn- k(flv,) have the positivity property
which is consistent with the positivity of the right hand side of (5.22). We may ask if the more subtle
positivity
lim lim
--*0
(- 1)kC(k’ n)
is valid?
10. Of course it may happen that A is degenerate, but in view of (3.11) it would seem that this
should be considered an exceptional occurrence.
11. In this discussion, , does not denote the pullback of forms.
12. e.g., the degree will not, but the class will.
13. cf. the discussion in the reference cited in footnote 4 of the introduction.
14. G. Laumon, Degrd de la varidtd duale d’une hypersurface singularitids isolds, Bull. Soc.
Math. France (1976), pp. 51-63.
15. cf. The Fenchel-Fary-Milnor theorem in 2(b) the reference cited in Footnote 7 there.
DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS 02138
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