Celebratio Mathematica — Thomas

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E. Rees and E. Thomas: “Complex cobordism and intersections of projective varieties,” pp. 168–178 in Variétés analytiques compactes [Compact analytic varieties] (Nice, 19–23 September 1977). Edited by Y. Hervier and A. Hirschowitz. Lecture Notes in Mathematics 683. Springer (Berlin), 1978. MR 517524 Zbl 0427.57016 incollection

Complete intersections are, in many ways, the best understood and simplest projective varieties. So, it is interesting to know the extent to which a given projective variety differs from a complete intersection. An obvious first step is to ask whether the given variety \( X_n\subset P_{n+k} \) is the transverse intersection of a hypersurface \( H \) with some smooth variety \( Y_{n+1}\subset P_{n+k} \). In this paper we use complex cobordism to study this question for smooth complex varieties. We obtain integrality conditions on the various transverse intersections of \( X_n \) with the linear subspaces of \( P_{n+k} \). In the case of high codimension, these conditions can be stated very simply in terms of complex cobordism. When the codimension is lower (\( n\geq k+2 \)) there are some extra, unstable, restrictions which are analogous to those that we have already studied in [Rees and Thomas 1978]; for the case \( n=k+2 \) of our present problem we study them here. The precise relationship between these unstable restrictions and the problem studied in [Rees and Thomas 1978] will be discussed at the end of §3. As well as obtaining these (necessary) integrality conditions we show that they are also sufficient for one to be able to find a “complex normal” submanifold \( Y \) such that \( Y \cap H = X \). In this introduction we have stressed this particular application, however it is a consequence of a much more general result that we discuss in §2.

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