

Iskovskikh Seminar
April 14, 2016 18:00, Moscow, Steklov Mathematical Institute, room 540






Higher Chern numbers of toric varieties
G. D. Solomadin^{} ^{} Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

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Abstract:
It is a wellknown fact that the unitary bordism ring is isomorphic to a
polynomial ring with countably many multiplicative generators:
$\Omega^{U}_{*}\simeq \mathbb{Z}[a_{1},a_{2}…]$, $deg(a_{i})=2i$.
Subject of my talk is a proof of the following fact: there exists a sequence
of smooth projective toric varieties giving polynomial generators of the
ring $\Omega^{U}_{*}$, $a_{n}=[X^{n}]$, $\dim_{\mathbb{C}} X^{n}=n$. Method
of the proof is based on considering a family of equivariant modifications
(birational isomorphisms) $B_{k}(X)\to X$ of an arbtrary smooth complex
manifold $X$ of complex dimension $n$ ($n\geq 2$, $k=0,…,n2$). These
modifications change the Chern number $s_{n}$ in a way depending only on the
dimension $n$ and a value of the parameter $k$. In particular, the change
does not depend on the manifold $X$. The result is complementary to the
results of A. Wilfong '13. The talk is based on a joint work "Projective
toric generators in the unitary cobordism ring", '16, with Y. Ustinovsky.

