复射影5维空间上的复秩3向量丛分类
A classification of complex rank 3 vector bundles on complex projective 5-space
AI总结：
该研究明确了复射影5维空间上秩3拓扑丛的同构类数与陈类的关系，构造$\boldsymbol{\textit{Z}}/3$值不变量区分同陈类的不同丛，完善了陈类不完备性的相关结论。
摘要：
给定整数$a_1,a_2,a_3$，当且仅当$a_1,a_2,a_3$满足Schwarzenberger条件时，复射影5维空间$\boldsymbol{\textit{CP}}^5$上存在第$i$个陈类为$a_i$的复秩3拓扑丛。在满足Schwarzenberger条件的前提下，我们证明：若$a_1$和$a_2$均被3整除，则$\boldsymbol{\textit{CP}}^5$上满足$c_i(V)=a_i$的秩3丛$V$的同构类数为3；否则同构类数为1。这表明陈类是$\boldsymbol{\textit{CP}}^5$上拓扑秩3丛的不完备不变量。为解决该问题，我们在与$BU(3)$相关的Thom谱的$tmf$上同调中构造一个通用类，其中$tmf$表示在3处局部化的拓扑模形式。基于该类和定向数据，我们构造了所研究丛的一个$\boldsymbol{\textit{Z}}/3$值不变量，并证明该不变量可区分具有相同陈类的不同丛。
英文摘要：
Given integers $a_1,a_2,a_3$, there is a complex rank $3$ topological bundle on $\mathbb CP^5$ with $i$-th Chern class equal to $a_i$ if and only if $a_1,a_2,a_3$ satisfy the Schwarzenberger condition. Provided that the Schwarzenberger condition is satisfied, we prove that the number of isomorphism classes of rank $3$ bundles $V$ on $\mathbb C P^5$ with $c_i(V)=a_i$ is equal to $3$ if $a_1$ and $a_2$ are both divisible by $3$ and equal to $1$ otherwise. This shows that Chern classes are incomplete invariants of topological rank $3$ bundles on $\mathbb CP^5$. To address this problem, we produce a universal class in the $tmf$-cohomology of a Thom spectrum related to $BU(3)$, where $tmf$ denotes topological modular forms localized at $3$. From this class and orientation data, we construct a $\mathbb Z/3$-valued invariant of the bundles of interest and prove that our invariant separates distinct bundles with the same Chern classes.
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