论文标题
拓扑彼得森产品
The topological Petersson product
论文作者
论文摘要
Petersson内部产品在尖峰形式上的非平稳性，以及Hecke操作员在Petersson产品方面是自我参与的事实，这意味着尖峰形式具有由Hecke特征形式组成的基础。在有关拓扑模块化形式的文献中，找不到彼得森产品的拓扑类似物，并且尚不清楚哪些拓扑空间具有其拓扑缘形式的特性，该特性允许由贝克拓扑Hecke运营商的作用的特征形式组成。在本说明中，我们在复杂的拓扑尖缘形式上定义和研究一种自然拓扑彼得森产品，其在单点空间上的价值恢复了古典彼得森产品。我们发现拓扑彼得森产品通常是退化的：特别是，如果$ x $在任何积极的程度上都是非平凡合理同源性的空间，那么彼得森的产品复杂化的$ tcf^*（x）$是堕落的。尽管$ TCF $ -ROMOMOLOGY是一种稳定的不变性，但拓扑彼得森的产品还是不稳定的，在所有悬架上都消失了。然而，我们通过在复杂的$ TCF $ TCF $ -THOPOLY对复杂的投射平面的拓扑彼得森产品进行明确计算来证明了拓扑彼得森产品的非平地。我们表明，对于紧凑型卡勒歧管$ x $，彼得森的产品在一系列Atiyah-hirzebruch滤清室中的复杂化$ tcf $ tcf $ - $ x $的浓度（本质上是$ tcf $ tcf $ xcolomology of $ x $ x $ x $的三分之一。
The nondegeneracy of the Petersson inner product on cusp forms, and the fact that Hecke operators are self-adjoint with respect to the Petersson product, together imply that the cusp forms have a basis consisting of Hecke eigenforms. In the literature on topological modular forms, no topological analogue of the Petersson product is to be found, and it is not known which topological spaces have the property that their topological cusp forms admit a basis consisting of eigenforms for the action of Baker's topological Hecke operators. In this note we define and study a natural topological Petersson product on complexified topological cusp forms, whose value on a one-point space recovers the classical Petersson product. We find that the topological Petersson product is usually degenerate: in particular, if $X$ is a space with nontrivial rational homology in any positive degree, then the Petersson product on complexified $tcf^*(X)$ is degenerate. Despite $tcf$-cohomology being a stable invariant, the topological Petersson product is an unstable invariant, vanishing on all suspensions. Nevertheless, we demonstrate nontriviality of the topological Petersson product by giving an explicit calculation of the topological Petersson product on complexified $tcf$-cohomology of the complex projective plane. We show that, for a compact Kahler manifold $X$, the Petersson product is nondegenerate on complexified $tcf$-cohomology of $X$ in a range of Atiyah-Hirzebruch filtrations (essentially one-third of the possibly-nonzero Atiyah-Hirzebruch filtrations in the $tcf$-cohomology of $X$).
