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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. |
Author to whom correspondence should be addressed
DOI:
https://doi.org/10.4213/sm8682
Full text:
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English version:
Sbornik: Mathematics, 2016, 207:11, 1601–1624
Bibliographic databases:
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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http://mi.mathnet.ru/eng/msb8682https://doi.org/10.4213/sm8682 http://mi.mathnet.ru/eng/msb/v207/i11/p127
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This publication is cited in the following articles:
-
G. D. Solomadin, “Quasitoric Totally Normally Split Representatives in the Unitary Cobordism Ring”, Math. Notes, 105:5 (2019), 763–780
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I. Yu. Limonchenko, T. E. Panov, G. Chernykh, “$SU$-bordism: structure results and geometric representatives”, Russian Math. Surveys, 74:3 (2019), 461–524
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