分野別セミナー

To find nice geometric representatives of bordism classes and bordism ring generators for various bordism theories has been a classical problem in algebraic and differential topology since 1960s. In 1958 F.Hirzebruch stated a problem, which remains open until now: to find a nonsingular (connected) complex algebraic variety in a given unitary bordism class. It was proved in 1960s that Milnor hypersurfaces generate the unitary bordism ring over integers, which is a polynomial ring due to a classical result of J.Milnor and S.P.Novikov, and similar generators also exist for unoriented and oriented bordism rings.

In 1962 S.P.Novikov proved that the special unitary bordism ring over integers with 2 reversed is isomorphic to a polynomial ring with one generator in each even real dimension greater than two. Z.Lu and T.E.Panov (2014) constructed a quasitoric representative for each multiplicative generator of this ring, starting with real dimension 10; quasitoric manifolds represent zero in dimensions 4, 6, and 8.

In this talk we are going to discuss the analogue of the Hirzebruch's problem for SU-bordism. J.Mosley (2016) proved that a nonsingular complex algebraic variety may not exist in a given SU-bordism class already in dimension 4. However, we show that for each multiplicative generator in the SU-bordism ring such a representative (disconnected in general) can be found using V.V.Batyrev's construction (1993) of Calabi--Yau hypersurfaces in toric Fano varieties over reflexive polyhedra.

The talk is based on a joint work with Zhi Lu (Fudan University) and Taras E. Panov (Moscow State University).