On singular generalizations of the Singer–Hopf conjecture
On singular generalizations of the Singer–Hopf conjecture
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关于辛格-霍普夫猜想的奇异推广
DOI：
10.1002/mana.202200322
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发表时间：
2023
影响因子：
1
通讯作者：
L. Maxim
中科院分区：
文献类型：
--
作者：
L. Maxim
Singer-Hopf猜想预言了闭合非球面流形的拓扑欧拉特征的符号。本文根据非球面复射影流形的闭不可约子簇的Euler-Mather特征、交同调Euler特征和虚Euler特征，提出Singer-Hopf猜想的奇异推广。我们在环境簇的余切丛是数值有效的(NEF)的假设下，或者更一般地，当环境流形允许具有新余切丛的复射影流形的有限态射时，证明了这些新的猜想。证明的主要内容是nef向量丛的半正性性质以及Kashiwara证明的Riemann-Roch定理的拓扑版。
The Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara.