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Mat. Sb., 2016, Volume 207, Number 11, Pages 127–152 (Mi msb8682)  

This article is cited in 2 scientific papers (total in 2 papers)

Projective toric polynomial generators in the unitary cobordism ring

G. D. Solomadina, Yu. M. Ustinovskiyb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Department of Mathematics, Princeton University, USA

Abstract: According to Milnor and Novikov's classical result, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: $\Omega^U_*\simeq \mathbb{Z}[a_1,a_2,…]$, $\deg(a_i)=2i$. In this paper we solve the well-known problem of constructing geometric representatives for the $a_i$ among smooth projective toric varieties, $a_n=[X^{n}]$, $\dim_\mathbb{C} X^{n}=n$. Our proof uses a family of equivariant modifications (birational isomorphisms) $B_k(X)\to X$ of an arbitrary complex manifold $X$ of complex dimension $n$ ($n\geq 2$, $k=0,…,n-2$). The key fact is that the change of the Milnor number under these modifications depends only on the dimension $n$ and the number $k$ and does not depend on the manifold $X$ itself.
Bibliography: 22 titles.

Keywords: unitary cobordism, toric varieties, blow-ups, convex polytopes.

Funding Agency Grant Number
Russian Science Foundation 14-11-00414
G. D. Solomadin's research was supported by a grant from the Russian Science Foundation (project no. 14-11-00414) in the Steklov Mathematical Institute of the Russian Academy of Sciences. Sections 1, 2.2, 3, 4.1, 5.2 and 6 are the work of Yu. M. Ustinovskiy. The other sections are due to G. D. Solomadin.


DOI: https://doi.org/10.4213/sm8682

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English version:
Sbornik: Mathematics, 2016, 207:11, 1601–1624

Bibliographic databases:

Document Type: Article
UDC: 515.165
MSC: Primary 14M25; Secondary 55N22, 57R77, 52B20
Received: 25.02.2016 and 01.07.2016

Citation: G. D. Solomadin, Yu. M. Ustinovskiy, “Projective toric polynomial generators in the unitary cobordism ring”, Mat. Sb., 207:11 (2016), 127–152; Sb. Math., 207:11 (2016), 1601–1624

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. G. D. Solomadin, “Kvazitoricheskiepolnostyu normalno rasschepimye predstaviteli v koltse kompleksnykh kobordizmov”, Matem. zametki, 105:5 (2019), 771–791  mathnet  crossref  elib
    2. I. Yu. Limonchenko, T. E. Panov, G. S. Chernykh, “$SU$-bordizmy: strukturnye rezultaty i geometricheskie predstaviteli”, UMN, 74:3(447) (2019), 95–166  mathnet  crossref
  • Математический сборник Sbornik: Mathematics (from 1967)
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