In previous work, the authors have constructed their (geometric) Hodge filtered complex cobordism theory. This theory provides groups \(MU^n(p)(X)\) for every complex manifold \(X\) and integers \(n\) and \(p\). The present paper aims to present pushforward homomorphisms (for this theory) along proper holomorphic maps \(f : X \rightarrow Y\) between complex manifolds \(X\) and \(Y\). These pushforward maps are homomorphisms of \(MU^*_*(Y)\)-modules \(f_* : MU^n(p)(X) \rightarrow MU^{n+2d}(p+d)(Y)\), where \(d\) is the difference of complex dimensions between \(Y\) and \(X\), and their existence is guaranteed by Theorem 1.1, the main result in this work, appearing in a more extensive form as Theorem 5.12 and stemming from Theorems 4.8 and the projection formula of Theorem 4.14. Theorem 4.8 actually provides pushforwards for proper holomorphic maps that are \(MU_{\mathcal{D}}\)-oriented, a notion studied in Section 3 and appearing as Definition 3.1. As seen in Section 5, every homolorphic map has a natural choice for an \(MU_{\mathcal{D}}\)-orientation, the Bott orientation of Definition 5.9, which is functorial and compatible with pullbacks (Lemma 5.10.)
The above pushforward homomorphisms have the expected properties: like the Bott orientation, they are functorial and compatible with pullbacks, and moreover satisfy the projection formula of Theorem 4.14. Besides that, Proposition 4.10 shows how they relate to the pushforwards of complex cobordism and sheaf cohomology.
In order to introduce the pushforward morphisms and to prove their properties, the authors start by considering a different definition of Hodge filtered complex cobordism, which they call currential Hodge filtered complex cobordism, and which depends on the currents of Section 2.1 and not on forms. This program is carried out in Section 2, where these new rings are denoted by \(MU^n_\delta(p)(X)\). The main point here is that there is an isomorphism between the \(MU^n_\delta(p)(X)\) and the original geometric Hodge filtered complex cobordism groups \(MU^n(p)(X)\) (Theorem 2.16); since the currential version of the theory is easier to handle, it is the preferred tool in the presentation of pushforward maps in Sections 3 and 4.
Section 6 introduces the Hodge filtered fundamental class \([f]\) of a holomorphic map \(f : Y \rightarrow X\), an element in \(MU^{2p}(p)(X)\) (where \(p\) is the codimension of \(f\)) that is given by applying the pushforward \(f_*\) from before to the unit element. If \(X\) is a compact K\(\mathrm{\ddot{a}}\)hler manifold and \(f\) is nullbordant and proper, \([f]\) belongs to the subgroup \(J^{2p-1}_{MU}(X)\), a version of Griffiths’s intermediate Jacobian, and can then be considered an Abel-Jacobi-type secondary invariant. Since \(f_*\) has been constructed geometrically, the authors can present a geometric description of these secondary invariants (Theorem 6.10). Section 7 defines a Thom morphism from Hodge filtered complex cobordism to Deligne cohomology, using a new cycle model for the latter that depends on currents and not on geometric measure theory (as in H. Gillet and C. Soulé [in: Algebraic \(K\)-theory: Connections with geometry and topology, Proc. Meet., Lake Louise/Can. 1987, NATO ASI Ser., Ser. C 279, 29–68 (1989; Zbl 0719.14003)].) Section 8 then reports on recent progress on the kernel and image of this Thom morphism applied to compact K\(\mathrm{\ddot{a}}\)hler manifolds.