Toric genera. (English) Zbl 1206.57039

Buchstaber, Victor; Panov, Taras; Ray, Nigel

Int. Math. Res. Not. 2010, No. 16, 3207-3262 (2010).

Authors’ abstract: Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus \(T^k\). In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus \(\Phi \), which was introduced independently by Krichever and Löffler in 1974, albeit from radically different viewpoints. In fact, \(\Phi \) is a version of tom Dieck’s bundling transformation of 1970, defined on \(T^k\)-equivariant complex cobordism classes and taking values in the complex cobordism algebra \(\Omega^*_U (BT^k_+)\) of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera and refer to the index-theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework and to introduce parametrized versions that apply to bundles equipped with a stably complex structure on the tangents along their fibers. In the presence of isolated fixed points, we obtain universal localization formulae, whose applications include the identification of Krichever’s generalized elliptic genus as universal among the genera that are rigid on \(SU\)-manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites, we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s.

Reviewer: Karl Heinz Dovermann (Honolulu)

MSC:

57R77	Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism)
32M99	Complex spaces with a group of automorphisms

Keywords:

toric genera; complex cobordism; torus actions