论文标题
几乎复杂的圆环歧管 - 图形，内脑属和Petrie类型的问题
Almost complex torus manifolds -- graphs, Hirzebruch genera, and problem of Petrie type
论文作者
论文摘要
令$ k $ - 维圆环$ t^k $在$ 2N $二维紧凑型连接的几乎复杂的歧管$ m $带有隔离固定点的情况下。至于圆圈动作，我们表明存在一个（有指示的标记）多编码，该多编码在$ m $的固定点上编码权重。这包括GKM图的概念，作为一个特殊情况，每个固定点处的权重是成对独立的。如果另外，$ k = n $，即$ m $是几乎复杂的圆环歧管，则多编码是图形；它没有多个边缘。我们表明，Hirzebruch $χ_y$ -genus $χ_y（m）= \ sum_ {i = 0}^n a_i（m）\ cdot（-y）\ cdot（-y）^i $ y几乎复杂的圆环$ m $ $ m $满足$ a_i（m）> 0 $ 0 $ 0 $ 0 $ for $ 0 \ leq i \ leq i \ leq leq n $。特别是，$ m $的TODD属为正，至少有$ n+1 $固定点。 Petrie的猜想断言，如果同型$ \ mathbb {cp}^n $接受了非平凡的圆圈动作，则其Pontryagin类与$ \ Mathbb {cp}^n $相符。 Petrie如果取而代之的是$ T^n $ Action，则证明了这一猜想。我们证明，如果$ 2N $ dimenional几乎复杂的圆环$ m $仅与复杂的投影空间$ \ mathbb {cp}^n $共享Euler号码，则相关的图与线性$ t^n $ -action的$ \ Mathbb {cp}^n $相符。因此，$ m $在固定点，Chern数字，Equivariant Coobordism类，Hirzebruch $χ_y$ -genus，Todd属和签名中具有相同的权重。如果此外，如果$ m $是正式的，那么$ m $和$ \ mathbb {cp}^n $的chern cromology和chern类也同意。
Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As for circle actions, we show that there exists a (directed labeled) multigraph that encodes weights at the fixed points of $M$. This includes the notion of a GKM graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, i.e., $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch $χ_y$-genus $χ_y(M)=\sum_{i=0}^n a_i(M) \cdot (-y)^i$ of an almost complex torus manifold $M$ satisfies $a_i(M) > 0$ for $0 \leq i \leq n$. In particular, the Todd genus of $M$ is positive and there are at least $n+1$ fixed points. Petrie's conjecture asserts that if a homotopy $\mathbb{CP}^n$ admits a non-trivial circle action, its Pontryagin class agrees with that of $\mathbb{CP}^n$. Petrie proved this conjecture if instead it admits a $T^n$-action. We prove that if a $2n$-dimensional almost complex torus manifold $M$ only shares the Euler number with the complex projective space $\mathbb{CP}^n$, an associated graph agrees with that of a linear $T^n$-action on $\mathbb{CP}^n$; consequently $M$ has the same weights at the fixed points, Chern numbers, equivariant cobordism class, Hirzebruch $χ_y$-genus, Todd genus, and signature as $\mathbb{CP}^n$. If furthermore $M$ is equivariantly formal, the equivariant cohomology and the Chern classes of $M$ and $\mathbb{CP}^n$ also agree.
