The values of the Milnor genus on smooth projective connected complex varieties. (English) Zbl 1046.57025
The algebraic structure of the complex cobordism ring was determined by 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
(a) all even integers less than or equal to \(2\) if \(n=1\);
(b) all even integers divisible by \(p\) if \(n=p^r-1\) for some prime \(p\);
(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 work. A central rôle is played by a fundamental result on genera of blow-ups originally due to I. R. Porteous [Proc. Camb. Philos. Soc. 56, 118–124 (1960; Zbl 0166.16701)], and the author rederives this using topological ideas; the reviewer feels this may help make this important result more accessible and better known to topologists.
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
(a) all even integers less than or equal to \(2\) if \(n=1\);
(b) all even integers divisible by \(p\) if \(n=p^r-1\) for some prime \(p\);
(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 work. A central rôle is played by a fundamental result on genera of blow-ups originally due to I. R. Porteous [Proc. Camb. Philos. Soc. 56, 118–124 (1960; Zbl 0166.16701)], and the author rederives this using topological ideas; the reviewer feels this may help make this important result more accessible and better known to topologists.
Reviewer: Andrew Baker (Glasgow)
MSC:
| 57R77 | Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism) |
| 55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
| 57R20 | Characteristic classes and numbers in differential topology |
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