An Abel-Jacobi invariant for cobordant cycles (Q504306)

This article builds on [\textit{M. J. Hopkins} and \textit{G. Quick}, J. Topol. 8, No. 1, 147--183 (2015; Zbl 1349.32009)]. Recall the following classical construction: for a complex projective variety \(X\), there is the \textit{cycle class map} \(\mathrm{cl}: \mathrm{CH}^p(X) \to H^{2p}(X, \mathbb{Z})\) where on the right hand side we mean ordinary singular cohomology. The kernel is denoted \(\mathrm{CH}^p_{\mathrm{hom}}(X)\), the group of cycles homologically equivalent to zero. Recall also that \(H^*(X, \mathbb{C})\) has a \textit{Hodge structure}, i.e. a decomposition into certain subspaces \(H^{p,q}(X) \subset H^{p+q}(X, \mathbb{C})\). A class \(x \in H^{2p}(X, \mathbb{Z})\) is called \textit{Hodge} if its image in \(H^{2p}(X, \mathbb{C})\) lands in \(H^{p,p}\), write \(\mathrm{Hdg}^{2p}(X)\) for the subgroup of Hodge classes. It is an important fact that \(cl\) takes values in \(\mathrm{Hdg}^{2p}(X) \subset H^{2p}(X, \mathbb{C})\). In fact this story can be extended. There exist so-called \textit{Deligne cohomology groups} \(H^{2p}_\mathcal{D}(X, \mathbb{Z}(p))\) which fit into exact sequences \[ 0 \to J^{2p-1}(X) \to H^{2p}_\mathcal{D}(X, \mathbb{Z}(p)) \to \mathrm{Hdg}^{2p}(X) \to 0. \] Here \(J^{2p-1}(X)\) is a certain complex torus called an intermediate Jacobian. The cycle class map \(cl\) lifts canonically to \(cl_\mathcal{D}: \mathrm{CH}^p(X) \to H^{2p}_\mathcal{D}(X, \mathbb{Z}(p))\). From this it follows that the composite \(\mathrm{CH}^p_{\mathrm{hom}}(X) \to \mathrm{CH}^p(X) \rightarrow{cl_\mathcal{D}} H^{2p}_\mathcal{D}(X, \mathbb{Z}(p))\) lifts uniquely to \(J^{2p-1}(X)\), defining the \textit{Abel-Jacobi map} \(\Phi: \mathrm{CH}^p_{\mathrm{hom}}(X) \to J^{2p-1}(X)\). The article under review is concerned with an extension of this construction where ordinary cohomology is replaced by a richer cohomology theory, namely cobordism. In the reference cited above, it is shown that there exists a cohomology theory \(MU^{2p}_{\log}(p)(X)\) fitting into an exact sequence \[ 0 \to J^{2p-1}_{MU}(X) \to MU^{2p}_{\log}(p)(X) \to \mathrm{Hdg}^{2p}_{MU}(X) \to 0, \] where \(J^{2p-1}_{MU}(X)\) is again a complex torus and \(\mathrm{Hdg}^{2p}_{MU}(X)\) is the subgroup of \(MU^{2p}(X)\) taken to hodge classes in the cohomology. Moreover there exists a cycle map \(\phi_{MU}: \Omega^p(X) \to \mathrm{Hdg}^{2p}(X)\), where \(\Omega^p(X)\) is the \(p\)-th \textit{algebraic cobordism group} as defined by Levine-Morel. Defining \(\Omega^p_{\mathrm{top}}(X)\) to be the kernel of \(\phi_{MU}\), there exists then an analog of the Abel-Jacobi map \[ \Phi_{MU}: \Omega^p_{top}(X) \to J^{2p-1}_{MU}(X). \] This map is the main object of study of the article under review. The author gives an explicit description of elements of \(MU^{2p}_{\log}(p)(X)\) in terms of triples \((\alpha, x, h)\) of holomorphic forms \(\alpha\) and cobordism elements \(x\), connected by an appropriate type of homotopy \(h\). Then the author explains how, given a class \(r \in \Omega^p_{\mathrm{top}}(X)\), to construct a triple \((\alpha, x, h)(r)\) such that \(\Phi_{MU}(r) = [(\alpha, x, h)(r)]\).