The values of the Milnor genus on smooth projective connected complex varieties.
type: Item
instance of: scholarly article
title: The values of the Milnor genus on smooth projective connected complex varieties.
zbMATH Open document ID: 1046.57025
author: Bryan Johnston
published in: Topology and its Applications
publication date: 2004-03-14T00:00:00Z
review text: The algebraic structure of the complex cobordism ring was determined by \textit{J. W. Milnor} [Am. J. Math. 82, 505--521 (1960; Zbl 0095.16702)]. The definition of this ring involves compact stably almost complex manifolds of which complex manifolds provide a special case, and it was also known long ago that every element of this ring is representable by a disjoint union of smooth projective complex varieties. The old question (due to Hirzebruch in the early 1960's) of which cobordism classes are representable by smooth projective complex varieties is the main result of the present paper; we emphasize that complex varieties are connected in the analytic topology. To explain the result proved in this elegant paper, we recall that the cobordism ring \(\Omega_*=\Omega^{-*}\) is the \(\mathbb Z\)-graded polynomial ring \(\mathbb Z[x_n:n\geqslant1]\) with generators \(x_n\in\Omega_{2n}\). By a criterion due to Milnor and Novikov, the generators \(x_n\) can be taken to have Milnor genus \(s_n(x_n)\) given by \[ s_n(x_n)= \begin{cases} \pm p&\text{ if }n=p^r-1\text{ for some prime }p, \\ \pm 1&\text{ otherwise}. \end{cases} \] We remark that the Milnor genus is an additive cobordism invariant. The main result (Theorem~1.1) proved shows that the set of values of the Milnor genus attained on smooth projective complex varieties consists of \newline (a) all even integers less than or equal to \(2\) if \(n=1\); \newline (b) all even integers divisible by \(p\) if \(n=p^r-1\) for some prime \(p\); \newline (c) all integers otherwise. As a corollary, it is possible to explicitly construct smooth projective complex varieties \(X_n\) realising the generators \(x_n\). The methods used involve a careful application of algebraic geometric constructions, in particular blowing-up along subvarieties. In essence, the techniques used were all available in the 1960's, although it seems that more recent work on elliptic genera and related topics has provided an important stimulus for this wo…
reviewed by: Andrew Baker
Mathematics Subject Classification ID: 57R77
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 2056826
zbMATH Keywords: complex cobordism
zbMATH Keywords: Chern numbers
zbMATH Keywords: Milnor genus
zbMATH Keywords: projective variety
zbMATH Keywords: blow-up
MaRDI profile type: Publication
full work available at URL: j.topol.2003.09.001
OpenAlex ID: W1980263522
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DOI: 10.1016/J.TOPOL.2003.09.001
description: scientific article; zbMATH DE number 2056826