论文标题
在实际枚举几何形状中减半空间和下限
Halving spaces and lower bounds in real enumerative geometry
论文作者
论文摘要
我们发展了将空间减半的理论，以在实际枚举几何形状中获得下限。一半的空间是拓扑空间，其作用是Lie组$γ$，具有附加的共同体特性。对于$γ= \ Mathbb {z} _2 $，我们恢复了Hausmann，Holm和Puppe的共轭空间。对于$γ= \ Mathrm {u}（1）$，我们获得了圆形空间。我们表明，实际的均匀和Quaternionic部分旗帜歧管是圆空间，导致无处不在的下限，即使是真实和Quaternionic Schubert问题。为了证明给定的空间是一个减半的空间，我们将Borel和Haefliger的结果推广到真实次要体及其复杂性的同类阶级。新颖性是，我们能够以合理的共同点而不是模型2的形式获得结果。圆形空间理论的模棱两可的扩展导致Borel和Haefliger在Thom多项式上的结果概括。
We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $Γ$ with additional cohomological properties. For $Γ=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm and Puppe. For $Γ=\mathrm{U}(1)$ we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haefliger on the cohomology classes of real subvarieties and their complexifications. The novelty is that we are able to obtain results in rational cohomology instead of modulo 2. The equivariant extension of the theory of circle spaces leads to generalizations of the results of Borel and Haefliger on Thom polynomials.
