Toric polynomial generators of complex cobordism.

*(English)*Zbl 1350.57036The complex cobordism ring \(\Omega^U_*\) is isomorphic to the polynomial ring \(\mathbb{Z}[\alpha_1, \alpha_2, \cdots ]\), where each generator \(\alpha_n\) has complex dimension \(n\). Traditionally, these generators are chosen by taking products and disjoint unions of complex projective spaces and Milnor hypersurfaces; these result, though, in varieties that are smooth and algebraic but not necessarily connected. To circumvent this problem, Johnston constructed in 2004 generators that are smooth and algebraic varieties; this was done by taking sequences of blow-ups of hypersurfaces and complete intersections in smooth projective algebraic varieties, starting with complex projective space. The author proposes here an approach that is based on Johnston’s construction, but obtains smooth projective toric varieties, which are also connected and algebraic.

The advantage in working with toric varieties is their intrinsic combinatorial properties, which result in their computational suitability. In fact, as Theorem 2.2 states, isomorphism classes of complex \(n\)-dimensional toric varieties are in bijective correspondence with equivalence classes of fans in \(\mathbb{R}^n\) (under unimodular transformations). These fans, described in section 2, are special sets of cones in \(\mathbb{R}^n\) (these are special convex sets, which are subsets of those generated by finite sets of points in \(\mathbb{Z}^n\)). By their construction, fans have convenient combinatorial properties, which reflect similar properties for toric varieties (Proposition 2.4).

The method the author uses to construct his generators for \(\Omega^U_*\) depends on a criterion by Milnor and Novikov involving the Milnor genus of a complex manifold \(M\) of real dimension \(2n\). Given \(n\) one less than a power of a prime \(p\), \(M\) will be a generator only if its Milnor genus is \(p\) or \(-p\); if \(n\) is not one less than a power of a prime, then the Milnor genus of \(M\) has to be \(1\) or \(-1\) if \(M\) is such a generator.

The Milnor genus of toric varieties is easily describable (Corollary 2.7), but not always easy to calculate. To obtain workable objects, the author considers blow-ups of some smooth projective toric varieties along a subvariety that is an orbit of the torus action (the resulting varieties are still smooth projective toric.) The basic toric varieties used are called \(Y_n^\epsilon(a,b)\), for \(n \geq 3\), \(\epsilon \in \{ 2, \cdots , n-1\}\) and \(a\) and \(b\) integers, and are stacks of two projectivized bundles: each \(Y_n^\epsilon(a,b)\) is a \(\mathbb{C}P^{n-\epsilon}\)-bundle over a variety which is a \(\mathbb{C}P^{\epsilon - 1}\)-bundle over \(\mathbb{C}P^{1}\). Proposition 2.12 computes their Milnor genera, and the crucial Proposition 2.13 shows the effect a blow-up has on the Milnor genus of a complex manifold.

Sections 3 and 4 construct then the desired toric polynomial generators in some dimensions, using carefully chosen instances of \(Y_n^\epsilon(a,b)\), their blow-ups, and the Milnor genus criterion above described. Theorem 3.1 does this whenever \(n = p-1\) for some prime \(p \geq 5\), obtaining \(Y_n^{n-2}(1,1)\) as a choice for the generator \(\alpha_n\); Theorem 3.4 tackles the case of \(n = p^m - 1\) for some odd prime \(p\) and some integer \(m \geq 2\), obtaining as chosen generator a variety that comes from blowing-up \(Y_n^{p^{m-1}}(1,b)\) enough times (for a determined \(b\)); Theorem 4.1 deals with \(n = 2^m - 1\) for some integer \(m \geq 2\), and here \(Y_n^{n-1}(1,1)\) can be chosen; finally, Theorem 4.4 solves the remaining odd cases, where one does blow-ups of \(Y_n^{2^m}(a,b)\) (for specially determined \(m\), \(a\) and \(b\)).

The dimensions for which no smooth projective toric generators are presented are the even values of \(n\) that are not one less than a prime power. The author discusses these cases in section 5, and provides convincing evidence for the case of the existence of such additional generators (related Conjecture 5.1 is true for \(n \leq 100 000\)). Finally, the author states that even more convenient smooth projective toric variety generators might be found; the main problem with some of those obtained in this work is, in his view, the lack of control one has on the number of blow-ups needed to obtain the generators for some values of \(n\). The amount of work already done here is, nonetheless, impressive in its own right, and is worth attention.

The advantage in working with toric varieties is their intrinsic combinatorial properties, which result in their computational suitability. In fact, as Theorem 2.2 states, isomorphism classes of complex \(n\)-dimensional toric varieties are in bijective correspondence with equivalence classes of fans in \(\mathbb{R}^n\) (under unimodular transformations). These fans, described in section 2, are special sets of cones in \(\mathbb{R}^n\) (these are special convex sets, which are subsets of those generated by finite sets of points in \(\mathbb{Z}^n\)). By their construction, fans have convenient combinatorial properties, which reflect similar properties for toric varieties (Proposition 2.4).

The method the author uses to construct his generators for \(\Omega^U_*\) depends on a criterion by Milnor and Novikov involving the Milnor genus of a complex manifold \(M\) of real dimension \(2n\). Given \(n\) one less than a power of a prime \(p\), \(M\) will be a generator only if its Milnor genus is \(p\) or \(-p\); if \(n\) is not one less than a power of a prime, then the Milnor genus of \(M\) has to be \(1\) or \(-1\) if \(M\) is such a generator.

The Milnor genus of toric varieties is easily describable (Corollary 2.7), but not always easy to calculate. To obtain workable objects, the author considers blow-ups of some smooth projective toric varieties along a subvariety that is an orbit of the torus action (the resulting varieties are still smooth projective toric.) The basic toric varieties used are called \(Y_n^\epsilon(a,b)\), for \(n \geq 3\), \(\epsilon \in \{ 2, \cdots , n-1\}\) and \(a\) and \(b\) integers, and are stacks of two projectivized bundles: each \(Y_n^\epsilon(a,b)\) is a \(\mathbb{C}P^{n-\epsilon}\)-bundle over a variety which is a \(\mathbb{C}P^{\epsilon - 1}\)-bundle over \(\mathbb{C}P^{1}\). Proposition 2.12 computes their Milnor genera, and the crucial Proposition 2.13 shows the effect a blow-up has on the Milnor genus of a complex manifold.

Sections 3 and 4 construct then the desired toric polynomial generators in some dimensions, using carefully chosen instances of \(Y_n^\epsilon(a,b)\), their blow-ups, and the Milnor genus criterion above described. Theorem 3.1 does this whenever \(n = p-1\) for some prime \(p \geq 5\), obtaining \(Y_n^{n-2}(1,1)\) as a choice for the generator \(\alpha_n\); Theorem 3.4 tackles the case of \(n = p^m - 1\) for some odd prime \(p\) and some integer \(m \geq 2\), obtaining as chosen generator a variety that comes from blowing-up \(Y_n^{p^{m-1}}(1,b)\) enough times (for a determined \(b\)); Theorem 4.1 deals with \(n = 2^m - 1\) for some integer \(m \geq 2\), and here \(Y_n^{n-1}(1,1)\) can be chosen; finally, Theorem 4.4 solves the remaining odd cases, where one does blow-ups of \(Y_n^{2^m}(a,b)\) (for specially determined \(m\), \(a\) and \(b\)).

The dimensions for which no smooth projective toric generators are presented are the even values of \(n\) that are not one less than a prime power. The author discusses these cases in section 5, and provides convincing evidence for the case of the existence of such additional generators (related Conjecture 5.1 is true for \(n \leq 100 000\)). Finally, the author states that even more convenient smooth projective toric variety generators might be found; the main problem with some of those obtained in this work is, in his view, the lack of control one has on the number of blow-ups needed to obtain the generators for some values of \(n\). The amount of work already done here is, nonetheless, impressive in its own right, and is worth attention.

Reviewer: Rui Miguel Saramago (Porto Salvo)