Celebratio Mathematica

R. Hartshorne, E. Rees, and E. Thomas: “Nonsmoothing of algebraic cycles on Grassmann varieties,” Bull. Am. Math. Soc. 80 : 5 (September 1974), pp. 847–851. MR 0357402 Zbl 0289.14011 article

By a cycle \( Z \) of dimension \( r \) on a nonsingular algebraic variety \( X \), we mean a formal linear combination \( Z = \sum n_iY_i \) of irreducible subvarieties \( Y_i \) of dimension \( r \), with integer coefficients \( n_i \). The smoothing problem for cycles asks whether a given cycle \( Z \) is equivalent (for a suitable equivalence relation of cycles, such as rational equivalence or algebraic equivalence) to a cycle \( Z^{\prime} = \sum n^{\prime}_iY^{\prime}_i \), where the subvarieties \( Y^{\prime}_i \) are all nonsingular. Let \( X \) be a nonsingular projective variety of dimension \( n \) over \( \mathbf{C} \). Then for each cycle \( Z \) of dimension \( r \) on \( X \) we can assign a cohomology class \[ \delta(Z)\in H^{2n-2r}(X,\mathbf{Z}) .\] We say that two cycles \( Z \), \( Z^{\prime} \) are homologically equivalent if \( \delta(Z) = \delta(Z^{\prime}) \). Our main result is that there are cycles on certain Grassmann varieties which cannot be smoothed, even for homological equivalence, which is weaker than rational or algebraic equivalence.

Elmer Rees	Related
Robin Cope Hartshorne	Related

E. Rees and E. Thomas: “On the divisibility of certain Chern numbers,” Q. J. Math., Oxf. II. Ser. 28 : 4 (1977), pp. 389–401. MR 0467771 Zbl 0401.55009 article

E. Rees and E. Thomas: “Smoothings of isolated singularities,” pp. 111–117 in Algebraic and geometric topology (Stanford, CA, 2–21 August 1976), part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics 32. American Mathematical Society (Providence, RI), 1978. MR 520527 Zbl 0398.57001 incollection

Elmer Rees	Related
Richard James Milgram	Related

E. Rees and E. Thomas: “Cobordism obstructions to deforming isolated singularities,” Math. Ann. 232 : 1 (1978), pp. 33–53. MR 0500994 Zbl 0381.57011 article

If \( V \) is a projective complex variety with an isolated singularity at the point \( p \), it is natural to ask whether one can deform \( V \) slightly so that it becomes non-singular near \( p \). When \( V \) is a hypersurface (or a complete intersection) it is well known that such a deformation is possible — one alters the defining equations slightly. It is tempting to think that this can always be done, but R. Thom, in unpublished work (now reproduced in [Hartshorne 1974]), gave an example of a variety with an isolated singularity that is not smoothable. Thom’s method was to use cobordism to show that if one lets \( S \) denote the boundary of a small disc \( D \) centered at \( p \) and \( K=S\cap V \), then the manifold \( K \) does not bound a smooth manifold in \( D \). Of course if \( V \) can be deformed slightly to become nonsingular,the manifold \( K \) does bound in \( D \), thus Thom gives a necessary condition for an isolated singularity to be smoothable. In this paper we investigate Thom’s idea carefullly and we find that there are more delicate invariants than his which can nevertheless be calculated.

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.

Elmer Rees	Related
Yves Hervier	Related
Andras Hirschowitz	Related

E. Rees and E. Thomas: “Realizing homology classes by almost-complex submanifolds,” Math. Z. 172 : 2 (1980), pp. 195–201. MR 580860 Zbl 0443.14010 article

year	title	people

			clear

P. Emery Thomas

Complete Bibliography

Works connected to Elmer Rees

Filter the Bibliography List