Geometric pushforward in Hodge filtered complex cobordism and secondary invariants

We construct a functorial pushforward homomorphism in geometric Hodge filtered complex cobordism along proper holomorphic maps between arbitrary complex manifolds. This significantly improves previous results on such transfer maps and is a much stronger result than the ones known for differential cobordism of smooth manifolds. This enables us to define and provide a concrete geometric description of Hodge filtered fundamental classes for all proper holomorphic maps. Moreover, we give a geometric description of a cobordism analog of the Abel-Jacobi invariant for nullbordant maps which is mapped to the classical invariant under the Hodge filtered Thom morphism. For the latter we provide a new construction in terms of geometric cycles.

The study of the analytic submanifolds of a given compact Kähler manifold is a central theme in complex geometry.Fundamental classes provide important invariants for this study.For a classical example, let X be a compact Kähler manifold and Z ⊂ X a submanifold of codimension p.The Poincaré dual of the pushforward of the fundamental class of Z along the inclusion defines a cohomology class [Z] in H 2p (X; Z).In fact, [Z] lies in the subgroup Hdg 2p (X) = H 2p (X; Z) ∩ H p,p (X; C) of integral classes of Hodge type (p, p).This induces a homomorphism from the free abelian group Z p (X) generated by submanifolds of codimension p of X to Hdg 2p (X).This map lifts to a homomorphism to Deligne cohomology 2020 Mathematics Subject Classification.55N22, 14F43, 58J28, 19E15, 32C35.The second-named author was partially supported by the RCN Project No. 313472 Equations in Motivic Homotopy.

H 2p
D (X; Z(p)).The latter group fits in the short exact sequence 0 → J 2p−1 (X) → H 2p D (X; Z(p)) → Hdg 2p (X) → 0 (1) where J 2p−1 (X) denotes Griffiths' intermediate Jacobian (see for example [29, §12]).On the subgroup Z p hom (X) of submanifolds whose fundamental class is homologically trivial sequence (1) induces the Abel-Jacobi map Z p hom (X) → J 2p−1 (X).As described in [29, §12.1] this map has a concrete geometric description via evaluating integrals over singular cycles in X, and one may consider it as a secondary cohomology invariant.In [22,23] Karoubi constructed an analog of Deligne cohomology for complex K-theory over complex manifolds in which secondary invariants for vector bundles can be defined (see also [10] for a study of induced secondary invariants).In [19] the authors show that there is a bigraded analog of Deligne cohomology E D for every rationally even cohomology theory E. If X is a compact Kähler manifold, there is a short exact sequence D (p)(X) → Hdg 2p E (X) → 0 generalizing sequence (1).Let X be a smooth projective complex algebraic variety and M p (X) be the free abelian group generated by isomorphism classes [f ] of projective smooth morphisms f : Y → X of codimension p between complex algebraic varieties.Based on the work of Levine and Morel [24] on algebraic cobordism, it is shown in [19] that for E = M U there is a natural homomorphism ϕ : M p (X) → M U 2p D (p)(X) where X also denotes the underlying complex manifold of complex points of X.On the subgroup M p (X) top of topologically cobordant maps this induces an Abel-Jacobi type homomorphism AJ : M p (X) top → J 2p−1 MU (X).This homomorphism has been studied in more detail in [26].However, both ϕ and AJ are only defined for complex algebraic varieties and are not induced by a geometric procedure as their classical analogs, but by a rather abstract machinery.
In [15] the authors define for every complex manifold X and integers n and p, geometric Hodge filtered complex cobordism groups M U n (p)(X) recalled below.The main result of [15] is that there is a natural isomorphism of Hodge filtered cohomology groups The aim of the present paper is to construct pushforward homomorphisms along proper holomorphic maps for geometric Hodge filtered cobordism.This will allow us to give a concrete description of the Hodge filtered fundamental classes of holomorphic maps f : Y → X for any complex manifold X and of the Abel-Jacobi invariant AJ for topologically trivial cobordism cycles on compact Kähler manifolds.We note that the results of the present paper are independent of the comparison isomorphism (2) of [15].The only results from [15] that we assume here are the verification of some natural properties of the groups M U n (p)(X).
We will now briefly describe the construction of the groups M U n (p)(X) of [15] which we also recall in more detail in section 2 and will then describe our main results in more detail.Consider the genus φ : M U * → V * := M U * ⊗ Z C given by multiplication by (2πi) n in degree 2n.By Thom's theorem, M U n is the bordism group of n-dimensional almost complex manifolds Z. Hirzebruch showed that any genus φ : M U * → V * is of the form for a multiplicative sequence K φ , where T Z denotes the tangent bundle of Z.This yields a V * -valued characteristic class of complex vector bundles.For p ∈ Z, we consider the characteristic class K p = (2πi) p • K φ .If ∇ is a connection on a complex vector bundle E, Chern-Weil theory gives a form K p (∇) representing K p (E).Given a form ω on Z and a proper oriented map f : Z → X, we consider the pushforward current f * ω, which acts on compactly supported forms on X by σ → Z ω ∧ f * σ.We define Hodge filtered cobordism cycles as triples (f, ∇, h) where f is a proper complex-oriented map f : Z → X, ∇ is a connection on the complex stable normal bundle of f and h is a current on X such that f * K p (∇) − dh is a smooth form in F p A n (X; V * ).
After defining a suitable Hodge filtered bordism relation we then obtain the group M U n (p)(X) of Hodge filtered cobordism classes.The main new technical contribution of the present paper is the construction of pushforwards for geometric Hodge filtered complex cobordism.
Theorem 1.1.Let g : X → Y be a proper holomorphic map between complex manifolds of complex codimension d = dim C Y − dim C X. Then there is a pushforward homomorphism of M U * ( * )(Y )-modules which is functorial for proper holomorphic maps and compatible with pullbacks.
In [19,Section 7], the authors show that there is an M U D -pushforward along projective morphisms between smooth projective complex varieties.In fact, they show that there are pushforward maps for a logarithmically refined version of M U D for quasi-projective smooth complex varieties, as a rather formal consequence of the projective bundle formula.This theory coincides with M U D for projective smooth complex varieties.Hence, assuming the comparison isomorphism (2) of [15], Theorem 1.1 extends the existence of pushforwards to a significantly larger class of maps than the one in [19].Since pushforwards in cohomology g * : M U n (X) → M U n+2d (Y ) only exist for proper and complex oriented continuous maps of (real) codimension d, the class of proper holomorphic maps is the largest possible subclass of holomorphic maps for which a pushforward with good properties may exist.
The construction of g * in Theorem 1.1 is similar to the one of pushforwards for differential cobordism for smooth manifolds in [6].However, the pushforward in differential cobordism exists only for proper submersions with a choice of a smooth M U -orientation.We will now explain why the pushforward for Hodge filtered cobordism exists for all proper holomorphic maps.In section 3 we first define the group of a Hodge filtered M U -orientation as a Grothendieck group of triples (E, ∇, σ) where E is a complex vector bundle with connection ∇ and σ is a form on X such that K(∇) − dσ ∈ F 0 A 0 (X; V * ).The relations involve a Chern-Simons transgression form associated to the multiplicative sequence K.A Hodge filtered M U -oriented map is then a holomorphic map with a lift of the stable normal bundle to the group of Hodge filtered M U -orientations.Then we show in section 4 that there is a pushforward along every proper Hodge filtered M U -oriented map.Finally, we show in section 5 that there is a canonical choice of a Hodge filtered M U -orientation for every proper holomorphic map.The key idea is a variation of a result of Karoubi's [22,Theorem 6.7] which establishes a mapping of virtual holomorphic vector bundles to the group of Hodge filtered M U -orientations by picking a Bott connection. 1 Applying this result to the virtual normal bundle of a holomorphic map defines a canonical orientation which we call the Bott orientation.
Another crucial point for the construction of pushforwards is that there is a currential version of Hodge filtered cobordism which we introduce in section 2.3.A key difference to differential cohomology theories on smooth manifolds, such as differential cobordism or differential K-theory, is that, for Hodge filtered cobordism, the currential description and the one using forms are canonically isomorphic.This is not the case for differential theories as explained for differential K-theory in [11], where particularly the exact sequences [11, (2.20)] and [11, (2.29)] make it clear that the smooth and currential differential K-theory groups are different in general.The main reason is that the space of closed currents D n (X) cl is strictly larger than the space of closed forms A n (X) cl .In the Hodge filtered context, however, we use H n (X; F p A * ) and H n (X; F p D * ), and H n (X; A * /F p ) and H n (X; D * /F p ) instead of A n (X)/Im (d) and D n (X)/Im (d).Since the Dolbeault-Grothendieck lemma holds both for currents and forms, in both cases the canonical map from the first to the second group is an isomorphism (see Lemma 2.2 and Theorem 2.16).
We will now describe the remaining content and results of the paper.In section 4 we show that the pushforward is functorial, compatible with pullbacks and satisfies a projection formula.In section 6 we introduce the Hodge filtered fundamental class [f ] = f * (1) ∈ M U 2p (p)(X) associated to a holomorphic map f : Y → X of codimension p as the pushforward of the unit element along f .If X is a compact Kähler manifold and f is a nullbordant proper holomorphic map, then [f ] has image in the subgroup J 2p−1 MU (X) which has the structure of a complex torus.In this case we also write AJ(f ) := [f ] for the class of f in J 2p−1 MU (X).Since the pushforward f * has a geometric construction, we are able to give a geometric description of the secondary invariant AJ(f ) in section 6.The main ingredient in the formulas are certain Chern-Simons transgression forms mediating between an arbitrary connection on the normal bundle N f and Bott connections on the corresponding tangent bundles.In section 7 we present a cycle model for Deligne cohomology inspired by but slightly simpler than the one of Gillet and Soulé in [12].The main difference is that we use currents of integration instead of integral currents in the sense of geometric measure theory (see also [14]).The new construction may be of independent interest and useful for other applications.This enables us to give a cycle description of the Hodge filtered Thom morphism M U n D (p)(X) → H n D (X; Z(p)) for every complex manifold X and integers n and p.In section 8 we report on our knowledge of the current status of examples and phenomena related to the kernel and image of the Hodge filtered Thom morphism for compact Kähler manifolds.
Parts of the present paper grew out of the work on the first-named author's doctoral thesis.Both authors would like to thank the Department of Mathematical Sciences at NTNU for the continuous support during the work on the thesis and this paper.We thank the anonymous referee for many comments and valuable suggestions which helped improving the paper.

Currential geometric Hodge filtered cobordism
First we briefly recall some facts about currents and the construction of geometric Hodge filtered complex cobordism groups from [15].Then we introduce a currential version of Hodge filtered cobordism.
2.1.Currents.Let X be a smooth manifold and let Λ X denote the orientation bundle of X.Let A * c (X; Λ X ) be the space of compactly supported smooth forms on X with values in Λ X .Let D * (X) denote the space of currents on X, defined as the topological dual of A * c (X; Λ X ).Given a form ω ∈ A * (X) and a current T ∈ D * (X), their product acts by There is an injective map A * (X) ֒→ D * (X) given by We equip D * with a grading so that this injection preserves degrees.That is, D k (X) consists of the currents which vanish on a homogeneous Λ X -valued form σ, unless possibly if deg σ = dim R X − k.We will not always distinguish ω from T ω in our notation.If X is a manifold without boundary, Stokes' theorem implies for ω ∈ A k (X): Hence the exterior differential can be extended to a map d : For a vector space V we set D * (X; V ) = D * (X) ⊗ V , and for an evenly graded complex vector space V * = ⊕ j V 2j we set An orientation of a map f : If f is proper and oriented, we therefore get a map where d = codim f = dim X − dim Z.We also denote by f * the homomorphism D * (Z; V * ) → D * (X; V * ) induced by tensoring f * with the identity of the various V 2j .We then have the identity Remark 2.1.In the case of a submersion π : W → X the pushforward π * preserves smoothness.We thus obtain the integration over the fiber map defined by the equation T W/X ω = π * T ω .Now we assume that X is a complex manifold.Then the space of currents is bigraded as follows.We write D p,q (X) for the subgroup of those currents which vanish on compactly supported (p ′ , q ′ )-forms unless p ′ + p = dim C X = q ′ + q.Then A * , * (X) → D * , * (X), ω → T ω , is a morphism of double complexes.The Hodge filtration on currents is defined by For an evenly graded complex vector space V * we set With similar notation for F p A * (X; V * ) and A * F p (X; V * ) we get the following result which will be crucial for the proof of Theorem 2.16: Lemma 2.2.Let X be a complex manifold.For every p, the maps of complexes of sheaves Proof.It suffices to prove the assertions for V * = C.Let Ω p be the sheaf of holomorphic p forms on X.The maps of complexes are quasi-isomorphisms by the Dolbeault-Grothendieck Lemma as formulated and proven in [13, pages 382-385] (see also [8,Lemma 3.29 on page 28]), where we consider Ω p as a complex concentrated in a single degree.The sheaves A p,q and D p,q , being modules over A 0 , are fine.Therefore, it follows from [29,Lemma 8.5] that the solid inclusions are quasi-isomorphisms.This implies that the dotted arrow is a quasi-isomorphism and proves the first assertion.By the same argument we have de Rham's theorem and the inclusion A * → D * is a quasi-isomorphism.Hence the induced map between the cokernels of the maps A * → D * and F p A * → F p D * is a quasi-isomorphism as well.This proves the second assertion.
2.2.Geometric Hodge filtered cobordism groups.We briefly recall the construction of the geometric Hodge filtered cobordism groups of [15].For further details we refer to [15, §2].Let Man C denote the category of complex manifolds with holomorphic maps.For X ∈ Man C let f : Z → X be a proper complex-oriented map, and let N f be a complex vector bundle which represents the stable normal bundle of f .Let ∇ f be a connection on N f .We call the triple f = (f, N f , ∇ f ) a geometric cycle over X.We let ZM U n (X) denote the abelian group generated by isomorphism classes, in the obvious sense, of geometric cycles over X of codimension n with the relations Let M U * be the graded ring with M U n = M U −n (pt).A map of rings M U * → R for R an integral domain over Q is called a complex genus.Complex genera may be constructed in the following way.For each i ∈ N, let x i be an indeterminate of degree i.Let Q ∈ R[[y]] be a formal power series in the variable y of degree 2. Let σ i denote the i-th elementary symmetric function in x 1 , x 2 , . . . .We may then define a sequence of polynomials since the right-hand side is symmetric in the x i .Then we get a characteristic class K Q defined on a complex vector bundle E → X of dimension n by where c i (E) denotes the i-th Chern class of E. In fact, by [16, §1.8], all genera are of the form where N X denotes the complex vector bundle representing the stable normal bundle of X obtained from the complex orientation of X → pt.From now on we set V * := M U * ⊗ Z C. We assume that the power series Q(y) = 1 + r 1 y + r 2 y 2 + • • • has total degree 0. This is equivalent to assuming φ Q to be a degree-preserving genus.Then K Q (E) has total degree 0. By [6,Lemma 3.26], φ Q extends to a morphism of multiplicative cohomology theories Here H n (X; V * ) ∼ = j H n+2j (X; V 2j ), so that in particular H −2j (pt; V * ) ∼ = V 2j .Now we fix the multiplicative natural transformation Let f : Z → X be a proper complex-oriented map, and let ∇ f be a connection on N f .By Chern-Weil theory there is a well-defined form c(∇ f ) ∈ A * (Z) representing the total Chern class c(N f ).In fact, with respect to local coordinates, we have where F ∇ f denotes the curvature of ∇ f .Then the form represents the cohomology class K(N f ).
Definition 2.3.For a geometric cycle f ∈ ZM U n (X) we define, using the orientation of f induced by its complex orientation, the current . By de Rham's work [9,Theorem 14] we can always find a current h Definition 2.4.Let X be a complex manifold and n, p integers.The group of Hodge filtered cycles of degree (n, p) on X is defined as the subgroup Remark 2.5.To simplify the notation, we will often write φ and K instead of φ p and K p , respectively.We may sometimes consider a Hodge filtered cobordism cycle as a triple Next we introduce the cobordism relation.The group of geometric bordism data over X is the subgroup of elements b ∈ ZM U n (R × X), with underlying maps b = (c b , f ) : W → R × X such that 0 and 1 are regular values for c b .Then W t = c −1 b (t) is a closed manifold for t = 0, 1, and We will often write ψ instead of ψ p to simplify the notation.By [15, Proposition 2.17], a geometric bordism datum b over X satisfies Hence we consider (∂ b, ψ p ( b)) as a Hodge filtered cycle of degree (codim b, p).We call such cycles nullbordant and let BM U n geo (p)(X) ⊂ ZM U n (p)(X) denote the subgroup they generate.We follow Karoubi in [22, §4.1] and denote We define the map n−1 denotes the subset of elements in A n−1 (X; V * ) which are sent to the subgroup F p A n (X; V * ) under d : A n−1 (X; V * ) → A n (X; V * ).The group of Hodge filtered cobordism relations is defined as Definition 2.6.Let X ∈ Man C and let n and p be integers.The geometric Hodge filtered cobordism group of X of degree (n, p) is defined as the quotient We denote the Hodge filtered cobordism class of the cycle γ We define maps R and I on the level of cycles as follows: Note that the maps R, I, and a above induce well-defined homomorphisms on cohomology by [15,Proposition 2.19].
Remark 2.7.Note that [R(γ)] = φ(I(γ)).In that sense, R refines the topological information of I with Hodge filtered differential geometric content.It is shown in [15] that R and I fit in a homotopy pullback in a suitable model category which can be used to construct Hodge filtered cobordism.
For the following theorem we let φ denote the composition of φ with the homomorphism induced by reducing the coefficients modulo F p .Theorem 2.8.For every p ∈ Z, the assignment X → M U * (p)(X) has the following properties: • For every X ∈ Man C there is the following long exact sequence: • For every holomorphic map g : Y → X and every n there is a homomorphism Hence M U n (p) is a contravariant functor on Man C .
• For every X ∈ Man C , there is a structure of a bigraded ring on Proof.The first assertion is proven in [15, §2.6] and follows from a direct verification of the exactness.The second and third assertions are proven in [15, §2.7] and [15, §2.8], respectively.We will, however, recall the construction of the pullback and of the ring structure in section 4.
For later purposes we now show how the Hodge filtered cobordism class depends on the connection on the representative of the normal bundle.Definition 2.9.Let X be a smooth manifold, and let be a short exact sequence of complex vector bundles over X with connections ∇ Ei on E i .Let π : [0, 1] × X → X denote the projection.Let ∇ π * E2 be a connection on π * E 2 which equals π * ∇ E2 near {1} × X and equals π * (∇ E1 ⊕ ∇ E3 ) near {0} × X.
The Chern-Simons transgression form of the short exact sequence E associated to the multiplicative sequence K is given by Remark 2.10.The construction of CS K (E) requires choosing a section s : E 3 → E 2 as well as a connection ∇ π * E2 .However, the form CS K (E) is independent of these choices in the quotient A −1 (X; V * )/Im (d).By Stokes' theorem, the derivative of the Chern-Simons form CS K (∇ E1 , ∇ E2 , ∇ E3 ) satisfies Remark 2.11.We will often consider the following special case.Let E be a complex vector bundle over the smooth manifold X.Let ∇ 0 and ∇ 1 be two connections on E. We can form the short exact sequence and define CS K (∇ 0 , ∇ 1 ) := CS K (E).This Chern-Simons transgression form can be expressed as and its derivative satisfies Lemma 2.12.Let f 0 = (f, N, ∇ 0 ), and f 1 = (f, N, ∇ 1 ) ∈ ZM U n (X) be two geometric cycles over X with the same underlying complex-oriented map f : Z → X.Then there is a geometric bordism b with ∂ b = f 1 − f 0 and and let π Z : R × Z → Z denote the projection.With the product complex orientation, using N id R = 0, we have 2.3.Currential Hodge filtered complex cobordism.Now we introduce a new and alternative description of Hodge filtered cobordism groups by considering the Hodge filtration on currents instead of forms.The difference to the previous definition may seem negligible but turns out to be crucial for the construction of a general pushforward later.Definition 2.13.Let X be a complex manifold and n, p integers.We define the group of currential Hodge filtered cycles ZM U n δ (p)(X) as the subgroup We will sometimes write a currential Hodge filtered cycle ( f , h) as a triple ( f , T, h) with T = φ( f ) − dh.Let a δ denote the map Definition 2.14.For X ∈ Man C and integers n, p, we define the currential Hodge filtered cobordism groups by 6), we define maps on the level of currential cycles as follows: By slight abuse of notation, we also denote by the symbols R δ , I δ and a δ the corresponding induced homomorphisms on cohomology groups.When the context is clear, we will often drop the subscript δ from the notation.For the next statement let φ δ denote the composition of φ with the homomorphism induced by reducing the coefficients modulo F p .
Proposition 2.15.Let X be a complex manifold.There is a long exact sequence Proof.The proof follows that of [15,Theorem 2.21] closely.We provide the details of the proof for the convenience of the reader.We start with exactness at M U n δ (p)(X).By definition of a δ and I δ we have To show the converse we work at the level of cycles.Let γ = ( f , h) ∈ ZM U n δ (p)(X) and suppose I δ (γ) = 0.That means f = ∂b for some bordism datum b.We may extend the geometric structure of f over b and obtain a geometric bordism datum b such that ∂ b = f .We then have The last equality follows from the observation that, since (0, Next we show exactness at M U n (X).The vanishing φ δ • I δ = 0 follows from the following commutative diagram, where the bottom row is exact: ' ' P P P P P P P P P P P P Let ∇ f be a connection on N f so that we get a geometric cycle f with We may build from f a geometric bordism datum b with underlying map where 1 2 denotes the constant map with value 1 2 .Clearly ∂ b = 0.Moreover, we have and we conclude that Since ∂ b = 0, we have

This finishes the proof.
There is a natural homomorphism , it follows that that there is an induced natural homomorphism 16.For every X ∈ Man C and all integers n and p, the natural homomorphism τ : The long exact sequences of Theorem 2.8 and Proposition 2.15 fit into the commutative diagram That the left-most vertical arrow is an isomorphism, is a corollary of the fact that the Dolbeault-Grothendieck lemma holds for currents as well as forms, see Lemma 2.2 for the full argument.The right-most arrow is the identity.The assertion now follows from the five-lemma.
Remark 2.17.In [15], M U n (p)(−) is defined on the larger category Man F of manifolds with a filtration of A * .We note that Theorem 2.16 does not extend to that context, since its proof uses that there is a Hodge filtration for currents which extends that of forms and that the inclusion Remark 2.18.As we discussed in the introduction, Theorem 2.16 reflects an important difference between Hodge filtered cohomology and differential cohomology.

Hodge filtered M U -orientations
We will now define the notion of a Hodge filtered M U -orientation of a holomorphic map in two steps: First as a type of Hodge filtered K-theory class with V * -coefficients.Then we apply this to the normal bundle of a holomorphic map.Recall from (4) the notation Definition 3.1.Let X be a complex manifold.We define the group K MUD (X) of Hodge filtered M U -orientations, M U D -orientations for short, to be the quotient of the free abelian group generated by triples ǫ := (E, ∇, σ) where E is a complex vector bundle on X, ∇ is a connection on E and is a form in filtration step F 0 modulo the subgroup generated by (C X , d, 0) for the trivial bundle C X on X with the canonical connection and by whenever there is a short exact sequence of the form with the identity We denote the image of (E, ∇, σ) in the quotient by [E, ∇, σ].Remark 3.2.For q ∈ Z we could modify the above definition and define an M U D -orientation of filtration q to be a triple (E, ∇, σ) as above such that K(ǫ) = K(∇)−dσ ∈ F q A 0 (X; V * ).Using appropriate relations, the addition on K MUD (X) extends to the direct limit of pointed sets over all q ∈ Z.The group K MUD (X) of Definition 3.1 is the subgroup of orientations of filtration 0. Since we do not know of applications to support the additional generality and complexity, we only consider orientations of filtration 0 in this paper.
We will now discuss the group K MUD (X) in more detail.
Lemma 3.3.The addition in K MUD (X) is given by where Proof.We consider the sequence Hence condition (11) for to be a relation reduces to Remark 3.6.There is a hidden symmetry in Equation ( 11) of Definition 3.1: Since σ 1 and σ 3 are of odd degree, we have Hence, modulo Im (d), we have Using the map K we can rewrite this relation as We now discuss further properties of the assignment (10).We note that there is a certain similarity in the behaviors and roles of the forms R( f , h) (respectively current R δ ( f , h)) and K(ǫ).Both R δ ( f , h) and K(ǫ) contribute to the construction of the pushforward along holomorphic maps in section 4 (see also Lemma 4.2, Remark 4.5, Proposition 4.10 and Remark 4.15).Moreover, while the current f * K(∇ f ) is not a cobordism invariant, the class of the difference R( f , h) = f * K(∇ f ) − dh is indeed invariant.Similarly, we will show in Proposition 3.8 that K(ǫ) = K(∇) − dσ is an invariant of the equivalence class [ǫ] in K MUD (X) while K(∇) is not.In fact, we will show that K respects the group structure on K MUD (X).This result will be used in several of our main results and their proofs.In particular, K plays a key role in the proof of the functoriality of the pushforward in Theorem 4.11.
Proof.To prove the assertion we use Lemma 3.3.Both dσ i and K(∇ i ) are closed and lie in A 0 (X i ; V * ) cl .In particular, this means that they are in the center of the ring A * (X i ; V * ).Using this fact we get Proposition 3.8.The map K descends to a morphism of monoids Proof.Since the triple (C N X , d, 0) represents the identity element in K MUD (X), we see that K sends the identity element to the identity.The fact that K descends to a map on K MUD (X) and respects the monoid structure then follows from Lemma 3.7 and the defining relations of K MUD (X).Remark 3.9.Let ǫ = (E, ∇, σ) in K MUD (X) be a representative of a generator in K MUD (X).We may consider K(ǫ) = K(∇) − dσ as a power series over the commutative ring A 2 * (X) in the generators of V * .Having leading term 1, K(ǫ) is an invertible power series.
Remark 3.10.The triple (C N X , d, 0) represents the identity element in K MUD (X).Given a generator ǫ = (E, ∇, σ) for K MUD (X), we can construct a class [ǫ ′ ] such that [ǫ] + [ǫ ′ ] = 0 in K MUD (X) as follows: Since X is a finite-dimensional manifold, we can find a complex vector bundle E ′ and an isomorphism E ⊕ E ′ ∼ = C N X for some N .We equip E ′ with the connection ∇ ′ induced from d by the direct sum decomposition To check that ǫ which proves the claim.
Remark 3.11.Assume we have a short exact sequence of complex vector bundles and that we have orientations o i = [E i , ∇ i , σ i ] involving two of the three bundles.
Then it follows from the defining relations in K MUD (X) and Remark 3.9 that for any connection ∇ j on the remaining bundle E j we can find a form σ j such that Definition 3.12.Let f : X → Y be a holomorphic map.Since the defining relations are compatible with pullbacks of bundles and connections, there is a welldefined pullback of orientations Next we define the notion of a Hodge filtered orientation of a holomorphic map.We will use this notion in the following section to define the pushforward along a holomorphic map.In section 5 we show that every holomorphic map has a canonical choice of a Hodge filtered orientation.Definition 3.13.Let g : X → Y be a holomorphic map.We define a Hodge filtered M U -orientation of g, or an M U D -orientation of g for short, to be a class where N g represents the complex stable normal bundle associated with g as a complex-oriented map.
With the notation of Definition 3.14, we recall that the stable normal bundle of g 2 • g 1 is isomorphic to the sum of the stable normal bundle of g 1 and the pullback of the stable normal bundle of g 2 along g 1 .Hence o 1 + g * 1 o 2 is, in fact, a Hodge filtered M U -orientation of g 2 • g 1 in the sense of Definition 3.13.

Pushforward along proper Hodge filtered M U -oriented maps
We will now define a pushforward homomorphism for proper M U D -oriented maps and show that it is functorial.Then we show that the pushforward is compatible with pullback and cup product.
Let g : X → Y be a holomorphic map and let o be an orientation of g.We write g o for g together with the orientation o and refer to g o as an M U D -oriented holomorphic map.If we want to specify the representative ǫ = (N g , ∇ g , σ g ) of o, we write g ǫ = (g, N g , ∇ g , σ g ), and we write g ǫ for the underlying geometric cycle g ǫ = (g, N g , ∇ g ).We will now define the pushforward of a Hodge filtered cycle along an oriented proper holomorphic map.Definition 4.1.Let g : X → Y be a proper holomorphic map of complex codimension d.Let ǫ = (N g , ∇ g , σ g ) be a representative of an orientation class of g in for the composed geometric cycle on Y .We define the pushforward homomorphism on currential Hodge filtered cycles by where g * denotes the pushforward of currents along g and R δ ( f , h) = f * K(∇ f )−dh is defined as in (8).
We will explain the choices made in Definition 4.1 further in Remarks 4.4 and 4.5 below.But first we need to check that the construction is well-defined, i.e., we have to show that g ǫ * ( f , h) actually is a currential Hodge filtered cycle.We will achieve this in two steps as follows: Proof.We check this claim by applying the definition of R δ and then rewrite the current as follows: We can now use this observation to show that g ǫ * ( f , h) is a currential Hodge filtered cycle: Proof.It follows from the definition that g ǫ • f is a geometric cycle.Hence, by definition of currential Hodge filtered cycles in 2.13, it remains to check that the current R δ (g ǫ * ( f , h)) = φ( g ǫ • f ) − dh satisfies condition (7) on the filtration step of a current in a Hodge filtered cycle, i.e., we have to show that R δ (g ǫ * ( f , h)) Remark 4.4.One might arrive at the formula for the current in the definition of g ǫ * ( f , h) as follows: If σ g = 0, then g * (K(∇ g ) ∧ h) is the only natural candidate, and it does satisfy the desirable formulas.Having made that choice, consider next the case o = [N g , ∇ g , σ g ] such that there is a connection ∇ ′ g with o = [N g , ∇ ′ g , 0] in K MUD (X).Then the rest of the formula can be derived using Lemma 2.12 and the relations in K MUD (X).
Hence, modulo Im (d), we have σ g ∧ dh = dσ g ∧ h, and it follows that modulo Im (d) we have Using the maps K and R δ we can rewrite this relation as We will now show that the map g ǫ * of Definition 4.1 induces a well-defined pushforward homomorphism on Hodge filtered cobordism.We first show that g ǫ * sends Hodge filtered bordism data to Hodge filtered bordisms in Lemma 4.6.
Lemma 4.6.We have Proof.Let h ∈ F p D n−1 (X; V * ).By definition of the map a in (5), we have a(h) = (0, h).Using relation (13) we get It remains to show  3), the fact that g is of even real codimension implies This shows that we have By definition of g ǫ * , we have .
Next we show that the equivalence class of g ǫ * ( f , h) does not depend on the choice of a representative of the M U D -orientation on g.
Proof.Let γ = ( f , h) be a currential cycle.By the definition of Similarly, for the representative ǫ ′ , we get We need to show that the two cycles in ( 14) and (15), respectively, are connected by a Hodge filtered bordism.By Proposition 3.8, we know By Remark 3.5 we can assume σ − σ ′ = CS K (∇, ∇ ′ ).Hence we get Since f * K(∇ f ) is of degree n and CS K (∇, ∇ ′ ) is of degree −1, switching factor on the right-hand side yields The projection formula The connections of g ǫ • f and The Chern-Simons form for these two connections satisfies Together with (18) this implies Hence, identities ( 16), ( 17) and (19) Since ∇ f ⊕ f * ∇ and ∇ f ⊕ f * ∇ ′ are the connections of g ǫ • f and g ǫ ′ • f , respectively, Lemma 2.12 and (20) imply that the difference of the cycles g ǫ * ( f , h) and g ǫ ′ * ( f , h) lies in BM U n+2d δ (p + d)(Y ).This proves the assertion of the lemma.
From now on we will use the canonical isomorphism τ : Putting the previous results together we have shown the following result: We refer to g o * as the pushforward along g o .Remark 4.9.Following Remark 3.2 we could consider an orientation o q of filtration q for q ∈ Z. Then we would get a pushforward homomorphism with an additional shift by q.Since we are mainly interested in the orientation of Definition 5.9 which is of filtration 0 in this terminology, we decided to skip the additional level of generality.We note, however, that all the computations in this section could be modified accordingly.
The following result shows how the pushforward of Theorem 4.8 relates to the pushforwards of complex cobordism and sheaf cohomology.Proposition 4.10.Let g o : X → Y be a proper M U D -oriented holomorphic map of complex codimension d, with o = [N g , ∇ g , σ g ] ∈ K MUD (X).Recall that we write K(o) = K(∇ g ) − dσ g .Then the following diagrams commute: Proof.
which proves that the left-hand square in (21) commutes.That the right-hand square in (21) commutes follows from the observation that the underlying complexoriented map of a composition of geometric cycles, is the composition of the underlying complex-oriented maps.Hence we have ) which shows that square (22) commutes as well.
We will now show that the pushforward is functorial: Theorem 4.11.Let the composition of proper holomorphic maps with Observe that the underlying geometric cycle of g ǫ12 12 , which we denote by g 12 , is the composed geometric cycle g 12 = g 2 • g 1 .Therefore, we know that the underlying geometric cycles of . By definition of the pushforward and Remark 4.5 we have On the other hand, applying Remark 4.5 to h • yields Hence we can rewrite h • as We apply again the projection formula to the pushforward along g 1 , once with T = K(o 1 )∧h+σ 1 ∧f * K(∇ f ) and ω = K(o 2 ), and once with T = f * K(∇ f )∧K(∇ 1 ) and ω = σ 2 .Since g * 1 K(o 2 ) and K(∇ 1 ) lie in A 0 (X 1 ; V * ), and hence in the center of the ring A * (X 1 ; V * ), we then get Next we collect the terms that are wedged with f * K(∇ f ) and obtain: By Proposition 3.8 we have Finally, by formula (23) for σ 12 , we get This shows h • = h 12 and finishes the proof.
Remark 4.12.Let g : X → Y and q : W → Y be transverse proper holomorphic maps of codimensions d and d ′ , respectively.Let π : W × Y X → Y be the map

induced by the following cartesian diagram in Man
Let o g ∈ K MUD (X) and o q ∈ K MUD (W ) be M U D -orientations of g and q, respectively.We then have natural isomorphisms of stable normal bundles (g ′ ) * N q = N q ′ , (q ′ ) * N g = N g ′ , and Hence (g ′ ) * o q + (q ′ ) * o g is an orientation of π, and Theorem 4.11 implies that we have the following identity Next we will show that the pushforward is compatible with pullbacks.First we briefly recall the construction of pullback homomorphisms in M U * (p)(−) from [15,Theorem 2.22].For further details we refer to [15, §2.7] and the references therein.Let k : Y ′ → Y be a holomorphic map.We consider the following cartesian diagram of manifolds where k and f are transverse, and f = (f, N f , ∇ f ) is a geometric cycle on Y .By transversality we get that k * f is complex-oriented with N kZ = k Z * N f .We define , it remains to define the pullback of the current h.Since the pullback of an arbitrary current is not defined, this requires to restrict to the subgroup where WF(h) denotes the wave-front set of h and N (k) is the normal set of f as defined in [17, 8.1].For γ = ( f , h) ∈ ZM U n k (p)(Y ), we then have a well-defined pullback where k * h is well-defined by [17,Theorem 8.2.4].By [15,Theorem 2.25], this induces a pullback homomorphism

Theorem 4.13. Suppose we have a cartesian diagram in Man
with k transverse to g, and g proper of codimension d.Let o be an M U D -orientation of g.We equip g ′ with the pullback orientation o ′ := k ′ * o.Then we have Since transversality is generic we can assume f to be transverse with k ′ .Let k ′ Z : Z ′ → Z be the induced map in the top cartesian rectangle in Since both rectangles are cartesian, the outer rectangle is cartesian as well.Hence the map k Z : Z ′ → Z induced by the outer cartesian diagram agrees with k ′ Z .We write k * g o * ( f , h) =: (f , h ) and (g ′ ) o ′ * k ′ * ( f , h) =: (f , h ).Let ǫ = (N g , ∇ g , σ g ) be a representative of the orientation o of g.Then we have (24), is cartesian we have Now we check the effect on the current h using that we have k * g * = g ′ * k ′ * by [15, Theorem 2.27] whenever the involved maps are defined: We recall from [15, §2.8] that there is a natural product of the form and is induced by the following construction: We consider the operation We then define the symbol ×, and refer to it as the external product of Hodge filtered cycles by potential slight abuse of terminology, by The product in ( 25) is then defined as the pullback along the diagonal map ∆ X : X → X × X: ).The following theorem shows that g * is a homomorphism of M U * ( * )(Y )-modules.
Theorem 4.14.Let g : X → Y be a proper holomorphic map of codimension d and let o be an M U D -orientation of g.Then, for all integers n, p, m, q, and all elements x ∈ M U n (p)(X) and y ∈ M U m (q)(Y ), we have the following projection formula Proof.Since the product is defined by pulling back an exterior product along the diagonal, we consider the following commutative diagram We denote by π Y : Y × X → Y and π X : Y × X → X, and by pr 1 : Y × Y → Y and pr 2 : Y × Y → Y the projections onto the first and second factors, respectively.We endow the map G := id Y × g with the pullback M U D -orientation o ′ := π * X o.We claim that in order to prove the assertion of the theorem it suffices to show the identity To prove that it suffices to show (29), we observe that (29) by definition of the cup product on M U * ( * )(Y ).Hence it remains to show that To do so we consider the following diagram where the last equality uses the definition of the cup product on M U * ( * )(X).This proves the claim.
We will now show that identity (29) holds by proving the corresponding formula on the level of cycles.Let ǫ = (N g , ∇ g , σ g ) be a representative of o.
) and ( f y , h y ) be cycles such that x = [ f x , h x ] and y = [ f y , h y ].We write h y×x for the current defined by ( 27) such that y × x = [ f y × f x , h y×x ].The theorem will then follow once we have proven the identity of cycles 31) Formula ( 31) can be checked separately on the level of geometric cycles and on the level of currents.To simplify the notation we denote the cycle We write ( f g * (x) , h g * (x) ) for the cycle g ǫ * ( f x , h x ), and ( f y×g * (x) , h y×g * (x) ) for the cycle ( f y , h y )×g ǫ * ( f x , h x ).For the geometric cycles the formula f G = f y×g * (x) follows directly from the definition of the pushforward and the definition of the map G = id Y × g.Now we show that (31) holds for the corresponding currents.Recall that we use the notation φ(γ) = (f γ ) * K(∇ fγ ) and R(γ) = φ(γ) − dh γ for a cycle γ = (f γ , h γ ).We then have by definition of the exterior product × By definition of the pushforward we have Using formula (32) and the formula R(y By definition of ⊗ in ( 26) and the fact that R(y) is of degree m we then get Now we use the definition of G as G = id Y × g to get: On the other hand we compute Comparing the expressions for h G and h y×g ǫ * (x) it remains to show modulo Im (d).Since, by definition of R in (6), R(x) is a closed form, we have Since h y is of degree m, we therefore get Hence, modulo image of d, we get the following identity Since R(y) = φ(y) − dh y by definition, we can thus conclude modulo Im (d).This shows (31) and finishes the proof.
We end this section with a further observation on the relationship of the maps R and K.

A canonical Hodge filtered M U -orientation for holomorphic maps
We will now show that for every holomorphic map there is a natural choice for an M U D -orientation.The key result is Theorem 5.6 which provides us with a canonical choice of a class of connections.The existence of a canonical choice of a class of orientation and Theorem 5.12 may be seen as justification for defining M U D -orientations as a K-group and not just as a set.We recall from [22, §6.3] the following terminology.Definition 5.1.Let X be a complex manifold and let D be a smooth connection on a holomorphic vector bundle E over X.Then with respect to local coordinates (U i , g i ), D acts as d + Γ i , where Γ i = (Γ jk i ) is a matrix of 1-forms.Recall that we have Γ i = g −1 ji dg ji + g −1 ji Γ j g ji where the g ij denote the transition functions.Conversely any such cocycle {Γ i } defines a connection.Then D is called a Bott connection if for each i, j, k we have As noted in the introduction, Bott connections are more commonly referred to as connections compatible with the holomorphic structure.Here we follow Karoubi, who uses the terminology in [22] in a context where Bott connections generalize both connections compatible with a holomorphic structure and Bott connections of foliation theory.Since Bott connections are frequently used in what follows, we adopt Bott connection as a convenient terminology.
Remark 5.3.Every holomorphic vector bundle on a complex manifold admits a Bott connection.In fact, the Chern connection on a holomorphic bundle with a hermitian metric is defined as the unique Bott connection which is compatible with the hermitian structure.By [21, Proposition 4.1.4]every complex vector bundle admits a hermitian metric.By [21, Proposition 4.2.14]every holomorphic bundle with a hermitian structure has a Chern connection.Alternatively, one can show the existence of Bott connections as in [22, §6] using a local trivialization of the bundle and a partition of unity.
belongs to F 1 A2 (X; End(E)).This implies the following key fact about Bott connections: Remark 5.5.Let D be a Bott connection on E. Then (33) implies that the triple (E, D, 0) defines an element in K MUD (X).The following result, inspired by [22,Theorem 6.7], shows that the associated orientation class [E, D, 0] is independent of the choice of Bott connection D.
We will now prove the key technical result of this section.
Theorem 5.6.For every X ∈ Man C , there is a natural homomorphism for each holomorphic vector bundle E where D is any Bott connection on E.
Proof.The existence of a Bott connection was pointed out in Remark 5.3.The assertion of the theorem then follows from the following two lemmas.
As a first step we analyze the Chern-Simons form of two Bott connections on a given holomorphic vector bundle and show that they lead to the same orientation class: Lemma 5.7.Let D and D ′ be two Bott connections for a holomorphic vector bundle E → X.
Proof.Let Γ i and Γ ′ i be the connection matrices of D and D ′ , respectively, with respect to local holomorphic coordinates z 1 , . . ., z l on U i .Let I = [0, 1] be the unit interval and let π : I × X → X denote the projection.Then consider the connection Each term is of filtration 1 in the sense that at least one of the dz j s appears in each term of each entry.Hence the Chern form c k (D ′′ ) has at least k many dz j s appearing in its local expression, and in that sense belongs to Integrating out dt maps this filtration step F k A * ([0, 1] × X) to the Hodge filtration F k A * (X).This implies Next, we show that all the defining relations of K 0 hol (X) and K MUD (X) are respected by B: Lemma 5.8.Let E be a holomorphic bundle over X and D be a Bott connection on E. The assignment E → (E, D, 0) induces a map B : be a short exact sequence of holomorphic vector bundles, and let D i be a Bott connection on E i .By the defining relations for K MUD (X) we need to establish where we recall that the notation F has been introduced in (4).Let γ : E 3 → E 2 be a smooth splitting.This yields a smooth isomorphism of bundles u = (α, γ) : where σ is a left-inverse of α.We choose holomorphic coordinates for each E j over an open U i ⊂ X.We then get the following equations of matrix valued forms: and Since D 2 is a Bott connection, it is represented by a matrix Γ 2 i with coefficients in F 1 .Note that, since γ and σ may not be holomorphic, ∆ 2 := u * D 2 may not be a Bott connection.However, locally on U i , ∆ 2 takes the form Since and let θ i be the connection matrix of ∇ with respect to local coordinates on U i ⊂ X.We continue to use the notion of filtration on is in F 1 , and we have just shown that t • π * ∆ 2 is upper triangular modulo F 1 .Thus, θ i is upper triangular modulo F 1 as well.Hence the local curvature form of ∇, i.e., Ω i = dθ i + θ i ∧ θ i , is upper triangular modulo F 1 as well.This implies that c i (∇) ∈ F i A 2i ([0, 1] × X) and hence K(∇) ∈ F 0 A 0 ([0, 1] × X; V * ).Now we note that we defined the Chern-Simons form CS K (D 1 , D 2 , D 3 ) as the integral of K(∇ ′ ), and not K(∇), for the connection Locally we can express the curvature of ∇ ′ as which finishes the proof of the lemma and of Theorem 5.6.
A key application of Theorem 5.6 is that it allows us to make a canonical choice of a Hodge filtered M U -orientation for each holomorphic map: Definition 5.9.Let g : X → Y be a holomorphic map, and let hol (X) denote the virtual holomorphic normal bundle of g.We define the Bott M U Dorientation of g, or Bott orientation of g for short, to be B(N g ) ∈ K MUD (X), i.e., the image of N g under B : K 0 hol (X) → K MUD (X).
The next lemma shows that the Bott orientation is functorial, i.e., it is compatible with pullbacks in the following way: with f transverse to g.Let N g and N g ′ be the virtual holomorphic normal bundles of g and g ′ , respectively.Then we have Proof.Since f is transverse to g, we have f ′ * N g = N g ′ in K 0 hol (X ).Since the choice of Bott connection does not matter for B by Theorem 5.6, this induces the identity ). Hence the Bott orientation of g 2 • g 1 is the composed M U D -orientation of the Bott orientations of g 1 and g 2 , respectively.Together with Lemma 5.10 this may justify to call the Bott orientation a canonical Hodge filtered M U -orientation for a holomorphic map.
Applying Theorems 4.8, 4.11, 4.13, and 4.14 with the Bott orientation together with Remark 5.11 yields the following result: Theorem 5.12.Let X and Y be complex manifolds, and let g : X → Y be a proper holomorphic map of codimension d.We equip g with its Bott orientation o := B(N g ).Then g * := g o * defines a functorial pushforward map This is a homomorphism of M U * ( * )(Y )-modules in the sense that, for all integers n, p, m, q, and all elements x ∈ M U n (p)(X) and y ∈ M U m (q)(Y ), we have Furthermore, if f : Y ′ → Y is holomorphic and transversal to g, letting f ′ and g ′ denote the induced maps as in (34), the following formula holds In the remainder of this section we further reflect on the Bott orientation class B(N g ).We note that [N g ] = [g * T Y ] − [T X] merely is a virtual bundle and, in general, there may not be a holomorphic bundle N g over X which represents [g * T Y ] − [T X] in K 0 hol (X).We can, however, obtain a representative of the orientation class B(N g ) in K MUD (X) as follows: Let g : X → Y be a holomorphic map and i : X → C k a smooth embedding.We then get a short exact sequence of complex vector bundles of the form Proposition 5.13.Let X be a Stein manifold, Y any complex manifold and g : X → Y a holomorphic map.Then we can represent the virtual normal bundle of g, [g * T Y ] − [T X] ∈ K 0 (X) by a holomorphic vector bundle on X.
Proof.Since X is Stein, we can assume i in (35) to be holomorphic.Hence N (g,i) admits a holomorphic structure.
For general X, however, we cannot expect N (g,i) to be holomorphic.In par- ).Yet we have the following result which follows from the defining relations in K MUD (X) (see also Remark 3.11): Proposition 5.14.With the above notation, let D X be a Bott connection for T X, and D Y a Bott connection for T Y .Let ∇ (g,i) be a connection on N (g,i) .We set For a projective complex manifold we can represent the canonical M U D -orientation in the following way: Proposition 5.15.Let g : X → Y be a proper holomorphic map.Assume that X is a projective complex manifold.Then there is a holomorphic vector bundle N on X and a Bott connection Proof.Recall the Euler sequence where γ 1 → CP n is the tautological line bundle.There is a canonical inclusion γ 1 → C n+1 , and we denote the quotient by γ ]. Thus we obtain the identity in K 0 hol (CP n ).Now let X be a projective manifold and let ι : X ֒→ CP n denote a holomorphic embedding.We have a short exact sequence of holomorphic vector bundles over X In K 0 hol (X) this implies the identities and hence We define the holomorphic bundle N := g * T Y ⊕ N X ⊕ ι * (γ ⊥ 1 ) ⊕(n+1) .Since B(C n 2 +2n ) = 0, we then get the identity B(N g ) = B(N ) in K MUD (X).Thus we have B(N g ) = [N, D, 0] for any Bott connection D on N .

Fundamental classes and secondary cobordism invariants
The existence of pushforwards along proper holomorphic maps allows us to define special types of Hodge filtered cobordism classes.In particular, we can define fundamental classes as follows: Definition 6.1.Let f : Y → X be a proper holomorphic map of codimension d.Let 1 Y ∈ M U 0 (0)(Y ) be the identity element of the graded commutative ring M U * ( * )(Y ).We endow f with its Bott orientation.We then refer to the element Let f : Y → X be a proper holomorphic map of codimension d.Let i : Y → C k be a smooth embedding.We then get a short exact sequence of the form With this notation, we have the following result: Assume that f and g are transverse.Then we have Proof.Since f and g are transverse, we can apply Theorem 4.13 to get f * g * = g ′ * f ′ * .Since π = g ′ • f by definition, Theorem 4.11 implies Now we apply Theorem 4.14 to y = [g] and x = 1 Y to conclude Finally, we note that the product in the subring of even cohomological degrees M U 2 * ( * )(X) is commutative to conclude the proof.Remark 6.4.If f : Y ֒→ X is the embedding of a complex submanifold of codimension d, then the normal bundle N f is a holomorphic bundle.Hence, in this case, the Bott orientation of f is given by B(N f ) = (N f , D f , 0) with a Bott connection D f on N f , and we have In particular, two homotopic maps f 0 and f 1 do not define the same class in Hodge filtered cobordism in general (see also Lemma 2.12 and [15,Lemma 5.9]).This shows that the current ψ( b) contains information that is not detected by M U 2d (X).Following Remark 6.5 we will now study the case of a topologically cobordant fundamental class in more detail.For the rest of this section we assume that X is a compact Kähler manifold.Then we can split the long exact sequence of Proposition 2.15 into a short exact sequence as follows.Let Hdg 2p MU (X) = I(M U 2p (p)(X)).We write .
Remark 6.7.As noted in [19,Remark 4.12], it follows from the Hodge decomposition that J 2p−1 MU (X) is isomorphic to the group M U 2p−1 (X) ⊗ R/Z.This implies that, as a real Lie group, J 2p−1 MU (X) is a homotopy invariant of X, while as a complex Lie group J 2p−1 MU (X) depends on the complex structure of X. Definition 6.8.Assume we have an element [γ] in M U 2p (p)(X) such that I([γ]) vanishes in M U 2p (X).Then sequence (37) shows that we may use J 2p−1 MU (X) as the target for secondary cobordism invariants.For example, let f : Y → X be a proper holomorphic map of codimension p. Assume that the fundamental class of f in M U 2p (X), given as the pushforward of 1 Y ∈ M U 0 (Y ) along f , vanishes.Then the fundamental class of f in M U 2p (p)(X) has image in the subgroup J 2p−1 MU (X).Because of the similarity to the Abel-Jacobi map of Deligne-Griffiths (see e.g.[29, §12]) we will denote the image of f in the subgroup J 2p−1 MU (X) by AJ(f ) and will refer to AJ(f ) as the Abel-Jacobi invariant of f .Let f : Y → X be a proper holomorphic map of codimension p.We will now describe AJ(f ) in more detail.Let [γ f ] := [f, N (f,i) , ∇ (f,i) , f * σ (f,i) ] be as in Proposition 6.2.We assume that f * (1 Y ) = 0 in M U 2p (X).Then there is a topological bordism datum b : W → R × X such that ∂b = f .Let N b be the associated normal bundle.We can extend the connection ∇ (f,i) on N (f,i) to get a connection ∇ b on N b , and obtain a geometric cobordism datum b.Then we have by definition of ψ( b) in (3).Hence we get which is induced by the map defined in (5).The class Thus, after taking the quotient, we get a well-defined class.We summarise these observations in the following theorem.Theorem 6.9.With the above assumptions on f and X, the fundamental class of Now we give an alternative description of AJ(f ).Let V ′ * be the C-dual graded algebra with homogeneous components Then the canonical pairing given by evaluation ev : V ′ * ⊗ V * → C has degree 0, if C is interpreted as a graded vector space concentrated in degree 0. Let n = dim C X. Poincaré duality and the fact that all vector spaces involved are finite-dimensional imply that the pairing 40) vanishes.For the other integral we note that by Stokes' theorem we have Since D X and D Y are Bott connections, we know that K(f * D X ) and K(D Y ) are in F 0 A 0 (Y ; V * ).This implies again for reasons of type that the integrals both vanish.The remaining term to analyse is the integral Y K(∇ f ) ∧ f * ψ which we already have shown to vanish.Thus integral (41) vanishes and the functional is well-defined.Finally, we note that integral (41) is independent of the chosen bordism datum, while the difference between the integrals (40) corresponding to two different bordism data is an element in φ ′ (M U 2p−1 (X)).Remark 6.11.The formula in Theorem 6.10 simplifies if the orientation o f admits a representative of the form (N, ∇, 0).If f is projective, we obtain such a representative from Proposition 5.15, and if f is a holomorphic embedding, B(f * T X/T Y ) will do.We do not know if such representatives exist for the canonical orientations of general holomorphic maps.

Hodge filtered Thom morphism
We will now define a Thom morphism from Hodge filtered cobordism to Deligne cohomology.In order to define a map on the level of cycles we will first construct a new cycle model for Deligne cohomology.Our construction is similar to that of Gillet-Soulé in [12] (see also [14]).However, our construction is more elementary than the one in [12] in the sense that it avoids the use of geometric measure theory.
Let X be a complex manifold and U ⊆ X an open subset.For an integer p ≥ 0, let Z(p) denote (2πi) p • Z and let Z D (p) be the complex of sheaves where Z(p) is placed in degree 0. Then the Deligne cohomology group H q D (X; Z(p)) may be defined as the q-th hypercohomology of the complex Z D (p).We recall the group of smooth relative chains defined as the quotient The restriction maps of C k are induced by quotienting out the appropriate additional chains.The presheaf C k is very close to being a sheaf since it satisfies the sheaf condition for coverings of X.However, it does not satisfy the sheaf condition for general collections of open subsets of X. Hence let C k be the sheafification of C k .The sheaf C k is not fine, but it is homotopically fine, meaning that its endomorphism sheaf admits a homotopy partition of unity.We refer to [5, page 172], from which we also recall the implication that H * (H j (C * (U ))) = 0 for j > 0.
Hence the hypercohomology spectral sequence degenerates on the E 2 -page, past which only the row H 0 (C * (U )) survives.On stalks the sheaf C k coincides with the presheaf C k .Let U be a small contractible open subset of X.By excision we have for D the closed unit disc.Hence we get This proves the following result: In other words, the sheaf cohomology H k (X; Z(p)) can be computed as the cohomology of the complex C * (X).Now we consider the map of complexes induced by integration.Let D * Z (X) be the image of T in D * (X).Since T is a map of chain complexes, it follows that D * Z (X) is a complex as well.Proposition 7.3.The map T : C * (X) → D * Z (X) induces an isomorphism on cohomology.
Proof.By Whitehead's triangulation theorem, we may pick a smooth triangulation of X, i.e., a set S = {f i : ∆ ki → X} such that each f i is a continuous embedding which extends to a smooth mapping of a neighborhood of ∆ k ⊂ R k , and each x ∈ X is in the interior of a unique cell S i = Im (f i ).It is well-known that the inclusion of cellular chains C * (S; Z(p)) → C * (X; Z(p)) is a quasi-isomorphism.Hence it suffices to show that T restricts to a quasi-isomorphism on the cellular chains of S. Since each point x ∈ X is contained in the interior of a unique cell of S, we can show that T is injective on cellular chains as follows.We can construct for each i a form ω i ∈ A ki (X) such that ∆ k i f * i ω i = 0, and such that the only k icell intersecting the support of ω i is S i .Suppose T (c) = 0 for c = a i f i .Then T (c)(ω i ) = a i T (f i )(ω i ) is a nonzero multiple of a i , and we get a i = 0 for all i.To see that the map induced by T from cellular homology is injective, we first note that the inclusion of cellular chains into singular chains is a deformation retract since it is a quasi-isomorphism between complexes of projective modules.Let r be a retraction onto the cellular chains.Now let c be a cellular cycle with T (c) = dT (α) for α an arbitrary integral chain α ∈ C * (X).Then we have T (c) = dT (α) = T (∂α) and thus T (c) = T (r(c)) = T (r(∂α)) = T (∂r(α)).Since T is injective on cellular chains, we get c = ∂r(α).Hence c represents 0 in cellular homology, and the map induced by T on cellular homology is injective.It remains to see that T restricted to cellular chains is surjective on homology.
By definition of D * Z (X) as the image of T , every element of D * Z (X) is of the form i T (a i • g i ) where g i are smooth maps ∆ k → X. Assume that i T (a i • g i ) is a cycle and hence represents a class in H k (X; D Z ).To simplify the notation, we write g := i a i • g i .By assumption, we have dT (g) = 0. Since r is a deformation retraction, there is a homotopy h of the cellular chains such that ∂h + h∂ = 1 − r.
By applying r, we define a cellular chain f := r(g).Omitting the inclusion from cellular chains into chains from the notation we then have the identity of chains Applying r again defines a cellular chain r(g ′ ) such that where we use the assumption dT (g) = 0. Hence we get T (∂(r(g ′ ))) = dT (r(g ′ )) = 0. Since T is injective on cellular chains, this implies ∂r(g ′ ) = 0, i.e., that f ) is an exact current, we have found a cellular cycle f ′ whose homology class is mapped to the homology class of g under T .This completes the proof.
We are now ready to give our presentation of Deligne cohomology.Let i F : F p A * → D * be the map of sheaves induced by T , and let i c : D * Z (X) → D * (X) be the inclusion.We will show that the following cochain complex computes the Deligne cohomology of X.In degree k we have the group Theorem 7.4.The cohomology of the cochain complex C * D (p)(X) is naturally isomorphic to Deligne cohomology.
To prove the theorem we will use multicomplexes, which are more flexible than bicomplexes.We recall from [3] that a multicomplex of abelian groups consists of the data of a bigraded abelian group, E s,t , and differentials d s,t r : One can consider multicomplexes of objects in any abelian category.We are considering here multicomplexes of abelian sheaves.
Proof of Theorem 7.4.We will construct a series of quasi-isomorphisms of complexes of sheaves Z D (p) ≃ C ′ * D (p) ≃ Tot(M ) and a quasi-isomorphism of complexes of abelian groups Tot(M )(X) → C * D (X), where M is the following multicomplex of sheaves on X: To define the differentials let Π s,k−s : D k → D s,k−s be the projection.For s > 0, there is only d 0 and d 1 .The differentials of M are 0 < s < p (∂, i F − ∂) : F p A s,t ⊕ D s−1,t → F p A s+1,t ⊕ D s,t s ≥ p d 0,t r = Π r,t−r • i c : C t → D r,t−r .The total complex of M is given by There is therefore a natural map Tot * (M (X)) → C * D (p)(X) defined by Tot * (M (X)) ∋ (c, ω, h) → (aT (c), ω, h) ∈ C * D (p)(X) where we write aT for the sheafified map induced by T .This map of complexes induces an isomorphism on cohomology since each of the maps id : F p A * (X) → F p A * (X), T : C * (X) → D * Z (X) and id : D * (X) → D * (X) is a quasi-isomorphism.We define yet another complex of sheaves with α(ω) = (dω, ω).We claim that this is a quasi-isomorphism of complexes of sheaves.This is clear in degrees < p, and in degrees > p it follows from the fact that C ′ * D (p) is exact in that range.In degree p we need to show that f induces an isomorphism on cohomology of stalks.Let U be a polydisc.Then Proof.This follows from the definition of τ 0 and the fact K 0 = 1 since K is a multiplicative sequence.Theorem 7.7.For every X ∈ Man C , the map τ Z induces a natural homomorphism τ Z : M U n (p)(X) → H n D (X; Z(p)) which fits into a morphism of long exact sequences Proof.It is clear that τ Z is a group homomorphism.We need to prove that, for a cycle γ = ( f , h) ∈ ZM U n (p)(X), we have dτ Z (γ) = 0 and τ Z (BM U n (p)(X)) ⊂ dC n−1 D (p)(X).
Let X be a compact Kähler manifold.Let f : Y → X be the inclusion of a complex submanifold of codimension p such that its fundamental class in M U 2p (X) vanishes.The latter condition implies that the fundamental class of f in H 2p (X; Z) vanishes as well.Hence both the classical Abel-Jacobi invariant AJ H (f ) of Deligne-Griffiths (see e.g.[29, §12]) and the invariant AJ(f ) of Theorems 6.9 and 6.10 are defined.
Proof.By Theorem 6.10 the invariant AJ(f ) may be represented by the functional The image of the Chern-Simons form σ f under τZ and τ 0 is zero since σ f is a form in degree −1.By Lemma 7.6, K(∇ b ) is mapped to 1. Thus, τ 0 maps AJ(f ) to the class of the functional in F n−p+1 H 2n−2p+1 (X; C ′ ) ′ defined by  Let M p (X) be the free abelian group generated by isomorphism classes [f ] of proper holomorphic maps f : Y → X of codimension p.For a proper holomorphic map f : Y → X of codimension p we denote its fundamental class in M U 2p (p)(X) by ϕ(f ) and its fundamental class in M U 2p (X) by ϕ(f ).This defines homomorphisms of abelian groups ϕ : M p (X) → M U 2p (p)(X) and ϕ : M p (X) → M U 2p (X).
We denote the kernel of ϕ by M p (X) top .Then the Abel-Jacobi invariant of Definition 6.8 defines a homomorphism AJ : M p (X) top → J 2p−1 MU (X).
Note that every element in M p (X) top is homologically equivalent to zero and therefore has a well-defined image in J 2p−1 (X).By Theorems 7.7 and 7.8 composition with the respective maps of diagram (43) produces the classical invariants.Diagram (43) shows that studying the kernel and image of τ Z is equivalent to analysing the kernel and image of τ J and τ , respectively.We expect the maps ϕ and AJ to be useful to discover new phenomena and examples that the classical invariants with values in Deligne cohomology are not able to detect.We will now briefly report on some results in this direction.
First we look at the image of τ Z .Let X be a smooth projective complex algebraic variety.In [28], Totaro showed that an element in H 2 * (X(C); Z) which is not in the image of τ : M U 2 * (X(C)) → H 2 * (X(C); Z) cannot be algebraic.This is a refinement of the obstruction induced by the Atiyah-Hirzebruch spectral sequence (see also [1]).It follows from [19,Corollary 7.12] that an algebraic class in H 2 * (X(C); Z) has to be in the subgroup τ Hdg 2 * MU (X(C) .In [2, §3.4], Benoist shows that this obstruction to algebraicity of cohomology classes is in fact finer than the one of [28].Now we consider the kernel of τ J .Since τ 0 is an epimorphism of vector spaces, the map τ J is surjective, and the snake lemma implies that there is a short exact sequence 0 → ker τ J → ker τ Z → ker τ → 0.
Hence ker τ Z contains information on the failure of the Thom morphism τ to be injective on Hodge classes, and on the failure of τ J to be injective.We have a filtered M U -orientations 14 4. Pushforward along proper Hodge filtered M U -oriented maps 17 5.A canonical Hodge filtered M U -orientation for holomorphic maps 27 6.Fundamental classes and secondary cobordism invariants 33 7. Hodge filtered Thom morphism 37 8. Image and kernel for compact Kähler manifolds 43

Remark 3 . 4 .
It follows from Lemma 3.3 that the identity element of K MUD (X) is represented by the triple (0, d, 0) where the first 0 denotes the zero-dimensional trivial bundle.Remark 3.5.Suppose we have two triples ǫ 1 = (E, ∇ 1 , σ 1 ) and ǫ 2 = (E, ∇ 2 , σ 2 ) with the same underlying bundle E. By Remark 3.4, the triple 0 = (0, d, 0) represents the identity element in K MUD (X).Since K(d) = 1, considering E id − → E as a short exact sequence as in Remark 2.11, we get a relation [ǫ 2 ] − [ǫ 1 ] − [0] for K MUD , i.e., we have [ǫ 1 ] = [ǫ 2 ] in K MUD (X), if and only if Y ).This follows from[6, Lemma 4.35].We provide a proof for the reader's convenience.Let b ∈ ZM U n (R × X)be a geometric bordism datum on X.Let e denote the geometric cycle id R × g : R × X → R × Y with the obvious geometric structure.Then e • b is a geometric bordism datum over Y .By definition of ψ( b) in ( diagram in (30) is cartesian and since G and ∆ Y are transverse, we can apply Theorem 4.13 to get

Remark 4 . 15 .
As in the proof of Proposition 4.10, we can express the identity shown in Lemma 4.2 asR(g o * (γ)) = g * (K(o) ∧ R(γ)) for every element [γ] and proper holomorphic map g : X → Y with M U D -orientation o.For the special case that γ is the identity element 1 X of the ring M U

Remark 5 . 4 .
If D is a Bott connection, then the curvature of D, which in local coordinates is represented by the matrix as the fundamental class of f .If the context of f and X is clear, we may also write [Y ] for [f ] and call it the fundamental class of Y .

Lemma 6 . 3 .
and D Y on T Y .Proof.This follows directly from the description of the Bott orientation in Proposition 5.14 and the definition of the pushforward map using 1 Y = [id Y , d, 0].Next we show that the fundamental class is compatible with products in the following sense: Let f : Y → X and g : Z → X be proper holomorphic maps of codimension d and d ′ , respectively.Let π denote the map induced by the following cartesian diagram in Man C

Remark 6 . 5 .
Let f 0 : Y 0 → X and f 1 : Y 1 → X be two embeddings of complex submanifolds of codimension d.By Remark 6.4 we can write the associated fundamental classes as[f 0 ] = [f 0 , N f0 , D f0 , 0] and [f 1 ] = [f 1 , N f1 , D f1 , 0].Now assume that f 0 and f 1 are cobordant, i.e., they represent the same element in M U 2d (X).Then we can find a geometric bordism b with ∂b = f 1 − f 0 .The bordism b is, in general, not sufficient to show [f 0 ] = [f 1 ] in M U 2d (d)(X), since the associated current ψ( b) defined in (3) may not vanish.In fact, b defines a Hodge filtered bordism datum between f 0 and f 1 if and only if and D Y on T Y , and an arbitrary connection ∇ f on the normal bundle.The derivative of σ f satisfies

8 . 1 MU
Image and kernel for compact Kähler manifoldsWe assume again that X is a compact Kähler manifold.Then the morphism of long exact sequences (42) induces a map of short exact sequences 0 / / J 2p−