The K\"unneth Formula for the Twisted de Rham and Higgs Cohomologies

We prove the K\"unneth formula for the irregular Hodge filtrations on the exponentially twisted de Rham and the Higgs cohomologies of smooth quasi-projective complex varieties. The method involves a careful comparison of the underlying chain complexes under a certain elimination of indeterminacy.


The main result
Let U be a smooth quasi-projective variety over the field C of complex numbers, and f ∈ Γ(U, O U ) a regular function. Attached to such a pair (U, f ) and a non-negative integer k, one has the k-th de Rham cohomology H k dR (U, f ) and Higgs cohomology H k Hig (U, f ), defined in Section 2. The two spaces H k (U, f ), ∈ {dR, Hig}, are of the same finite dimension over C (see [1,Remark 1.3.3]), and are equipped with the decreasing irregular Hodge filtrations indexed by the ordered set Q of rational numbers with finitely many jumps. In the following, we omit the adjective and just call them the Hodge filtrations. For the motivations and the basic properties of the Hodge filtration, including in particular the degeneration of the Hodge to de Rham spectral sequence, see [1,3] and the references therein. We recall the construction in Section 2. Now consider two such pairs (U i , f i ), i = 1, 2. On the product U := U 1 × U 2 , consider the regular function f defined by f : (x 1 , x 2 ) → f 1 (x 1 ) + f 2 (x 2 ). We call the pair (U, f ) the product of the two (U i , f i ). For ∈ {dR, Hig}, there is the canonical map induced by cup product. We equip the space on the left hand side with the product filtration, i.e., the λ-th filtration for λ ∈ Q is given by the subspace i+j=k a+b=λ where the inner sum is taken inside H i (U 1 , f 1 )⊗H j (U 2 , f 2 ). In this article, we prove the following Künneth formula.
Theorem 1. With notations as above, the map (1) is an isomorphism of filtered spaces.
In particular, denoting Gr λ F V the λ-th graded piece of a filtered space (V, F ), one has the identification Recently different proofs of the Künneth formula in a more general setting appear in [8,Theorem 3.39] where the involved coefficients in the cohomology are allowed to be exponential twists of complex Hodge modules. One of the main ideas in that work is to enrich the de Rham and Higgs cohomologies into the Brieskorn lattice (see the last section Section 4) or a twistor structure and treat the irregular Hodge filtration as a byproduct of the enrichment. The notion of a V -adapted trivializing lattice for a meromorphic connection on P 1 of special type is introduced [8, Section 3.2.b] in order to obtain the Künneth formula for irregular Hodge filtrations. The methods depend on the deep theory of twistor D-modules mainly developed by Sabbah and Mochizuki (see [6]). On the other hand, our approach is much more elementary. We believe that the concrete filtered de Rham complex used here would be suitable for computations in some interesting examples of irregular Hodge structures in the future work.
The rest of the article begins in Section 2 with a brief of the construction of the Hodge filtration. We follow the approach of [11] by putting a filtration on the de Rham complex or the Higgs complex via a certain compactification of the pair (U, f ). Here we introduce the notion of a non-degenerate compactification, which is weaker than that of a good compactification used in [11], but appears naturally in the later section (see also [7,Section 4], [9, Section 7.3], [5]). In order to compare the cohomologies with the filtrations of the summands (U i , f i ) and of their product (U, f ), we construct a particular compactification of (U, f ) from the fixed ones of (U i , f i ) in Section 3. The proof of the Künneth formula is obtained by a careful investigation of the relations between the involved complexes on the compactifications. In the last Section 4, we remark that one can interpolate the two spaces H k dR (U, f ) and H k Hig (U, f ) as the fibers of the Kontsevich bundle on the projective line P 1 over 1 and 0, respectively. In fact, the fiber over c ∈ C \ {0} of the bundle is equal to H k dR (U, f /c) and hence the Künneth formula also holds true. However at ∞ the situation is more complicated in this regard and the direct analogue of the Künneth formula does not hold in general.

The filtrations
In this section, we fix a pair (U, f ) consisting of a smooth quasi-projective variety U over C and a regular function f ∈ Γ(U, O U ).

The compactification
Let X be a projective variety over C containing U such that the reduced subvariety S := X \U is a normal crossing divisor. Regard f ∈ Γ(X, O X ( * S)) as a rational function on X. Let P and Z be the pole divisor and the zero divisor of f on X, respectively, and let P red be the support of P .

Definition.
(i) We call X a non-degenerate compactification of (U, f ) if there exists a neighborhood V ⊂ X of P such that Z ∩ V is smooth and Z + S forms a reduced normal crossing divisor on V .
(ii) We call X a good compactification of (U, f ) if f indeed defines a morphism f : X → P 1 .
If X is non-degenerate, analytically locally at a point of P , there exists a coordinate system {x 1 , . . . , x l , y 1 , . . . , y m , z 1 , . . . , z r } such that S = (xy) and for some e ∈ Z l >0 . (Here and afterwards, we use the standard multi-index convention.) If X is good, the second case of f in the above expression does not occur.
For example, consider the case where U is a complex torus and f a Laurent polynomial. Assume that f is non-degenerate with respect to its Newton polytope (for the definition, see [11,Section 4]). Then any toric smooth compactification compatible with the Newton polytope is a non-degenerate compactification of (U, f ) by [11,Proposition 4.3] or [5, Lemma 6.6]. The non-degenerate compactification also appears in the considerations of rescaling from a good compactification [9,Section 7.3], and of Fourier transform [7,Section 4]. It is discussed in [5] where in this situation, the author calls the meromorphic function f on X non-degenerate along S [5, Definition 2.6]. In this case, the author investigates the structures of the twistor D-module associated with the meromorphic connection (O X ( * S), d + df ); it is shown in particular that if S equals the pole divisor of f , the resulting twistor D-module is pure [5, Lemma 2.10 and Corollary 3.12].

The filtered complexes
Fix a non-degenerate compactification X of (U, f ) with boundary S. Regard f as a rational function on X and let P = f * (∞) be the pole divisor with multiplicities. We have df ∈ Γ X, Ω 1 X (log S)(P ) . Definition.
(i) The twisted de Rham complex and the Higgs complex are the complexes where Θ = d + df and df , respectively. Here df sends a local section ω to df ∧ ω.
(ii) We call the associated hypercohomology groups H k (X, (Ω • X (log S)( * P ), Θ)) the de Rham cohomology and the Higgs cohomology of (U, f ), and denote respectively by (iii) For an effective divisor D on X and µ ∈ Q, let otherwise.
The subindex X in F λ X (Θ) will be omitted if the base variety is clear. The Hodge filtrations of the de Rham and the Higgs cohomologies are induced by the inclusions of complexes.
(iv) For α ∈ Q, the Kontsevich sheaf of differential p-forms is the coherent subsheaf of Ω p X (log S)( * P ) Proof . Both statements are local properties for coherent sheaves on X and we can restrict to the coordinates such that (2) holds. Consider the second case f = z 1 x e so that In this chart, the O X -module Ω p X (log S) is freely generated by The O X -module Ω p f (α) is indeed freely generated by where ω 1 and ω 2 run through elements in (4) and (5), respectively.
(ii) Let D be a divisor supported on P red , and E an irreducible component of P red . Let To obtain the quasi-isomorphisms, it suffices to show that the inclusion of complexes is a quasi-isomorphism for any k, α and all choices of D (cf. [11,Proposition 1.2]). In fact, the first desired quasi-isomorphism then follows immediately. The last two quasi-isomorphisms can be derived by a decreasing induction (see the analogous statement [1,Proposition 1.4.2] in the case of good compactification and its proof, which works for both Θ = d + df, df ).
Remark. In a more functorial way, one can consider the D-module M on X attached to the meromorphic connection (O X ( * S), d + df ) and define the irregular Hodge filtration on M as given in [

The independence
Proposition. For ∈ {dR, Hig}, the space H k (U, f ) with the Hodge filtration is independent of the choice of the non-degenerate compactification of (U, f ). More precisely, if π : X → X is a morphism between non-degenerate compactifications extending the identity on U , then there is a natural quasi-isomorphism Proof . The assertion for good compactifications is proved in [11,Theorem 1.7], by comparing the degree p components F λ X (Θ) p of F λ X (Θ) on various X for a fixed p (and hence the proof works for both Θ = d + df and df ).
In the following we show that by performing successively certain blowups, one can replace a non-degenerate compactification X (thus in the second case of (2)) by a good one and compare the involved chain complexes (cf. [11, Section 4(b)]). Let : X → X be the blowup along the intersection Ξ of Z = (z 1 ) and the irreducible component (x 1 ) of P with multiplicity e 1 . Let E be the exceptional divisor, S = X \ U and P the pole divisor off := * f . We want to establish that R * F λ X (Θ) and F λ X (Θ) are canonically quasi-isomorphic. In case e 1 > 1, it can be proved similarly as [11,Proposition 4.4]. In more details, write λ = −α + p where 0 ≤ α < 1 and p ∈ Z. On X define the complex By [11,Proposition 4.4(i)], each component of the complex R λ (Θ)/ * F λ X (Θ) is a direct sum of copies of O E/Ξ (−1). Hence the adjunction F λ X (Θ) → R * R λ (Θ) is a quasi-isomorphism. On the other hand, there are increasing complexes R λ q (Θ) on X (those denoted by R λ (q) in [11, equation (27)], which is a complex under either d + df or df ) with is quasi-isomorphic to the complex consisting of a direct sum of copies of O E/Ξ (−1) concentrated at degree (p + q). In fact, in [11,Lemma 4.5], we further introduce the complexes (K • ρ,η,ξ , d+df ) and prove that the inclusion (K • ρ,η,ξ , d+df ) ⊂ (K • ρ,η+1,ξ , d + df ) is a quasi-isomorphism by showing the quotient complex is exact. Now one notices that K • ρ,η,ξ is indeed also stable under df ; on the quotient K • ρ,η+1,ξ /K • ρ,η,ξ , one has the equality d + df = df of the differential maps under the condition e 1 > 1. Therefore the proof of [11,Proposition 4.4(ii)] describing R λ q (Θ)/R λ q−1 (Θ) goes through in both cases Θ = d + df and Θ = df . Hence one concludes that the inclusion F λ We first compute Ω p f (α). Explicitly the blowup X is defined by the equation the O X -module Ω p f (α) is generated by the basis and On the chart u = 0 with local coordinates x 1 , . . . , x l , y 1 , . . . , y m ,v = v u , z 2 , . . . , z r , the sheaf Ω p f (α) is generated by the basis and On the intersection u, v = 0, one has dx 1 x 1 = dz 1 z 1 + dū u . Using the basis (6) of Ω p f (α), a direct computation reveals that * Ω p f (α) is contained in Ω p f (α). Moreover, the O E -module Ω p f (α)/ * Ω p f (α) equals either zero if α(e 1 − 1) = αe 1 or otherwise l+m+r p copies of O E/Ξ (−1) generated by (9), (10) and (11), (12) on the two charts, respectively. Therefore we obtain that F λ X (Θ) and R * F λ X (Θ) are naturally quasi-isomorphic. One then iteratively takes the blowups along the intersections of irreducible components of the zero and the pole divisors as in [11,Section 4(b)] (the diagram (26) therein) to obtain a good compactification X of (U, f ) from the non-degenerate X with the canonical quasi-isomorphism (8).

Remark.
(i) By the E 1 -degeneration [ (3) is injective for any non-degenerate X and indices k, λ.
(ii) For X non-degenerate, we have H k dR (U, f ) = H k (X, (Ω • X ( * S), d + df )) by the arguments of [11,Corollary 1.4]. On the other hand, H k Hig (U, f ) = H k (X, (Ω • X ( * S), df )) in general which can be seen by considering the case U affine and f = 0.

The proof of the main result
We begin with two pairs (U i , f i ), i = 1, 2, and their product (U, f ). Fix good compactifications X i of (U i , f i ) such that S i := X i \ U i are strict normal crossing divisors. The proof of Theorem 1 consists of two steps. In the first step Section 3.1, we construct explicitly a non-degenerate compactification X of (U, f ) from X 1 × X 2 by successive blowups. In step two Section 3.2, we compare the filtered de Rham or the Higgs complex on X, which gives the Hodge filtration on H k (U, f ), with a certain filtered complex on X 1 × X 2 that gives the product filtration using the explicit construction of X.

An explicit elimination
For each i = 1, 2, take an open covering of X i with a system of local coordinates As for the initial data in the inductive construction, we consider the compactification X 1 × X 2 of U with the systems of local coordinates {x i,j }. Suppose we have constructed a compactification Y of U and its systems of local coordinates {y 1 , . . . , y l , y l+1 , . . . , y l+m , y l+m+1 , . . . , y l+m+r }, together with a birational map π : Y → X 1 × X 2 such that π * f 1 = 1 y a 1 1 · · · y a l l , π * f 2 = 1 Notice that if ∆ Y (D 1 , D 2 ) ≥ 0 for any pair (D 1 , D 2 ), i.e., a ≥ b or b ≥ a in terms of the systems of local coordinates as above, then the zero divisor of π * f is smooth in a neighborhood of T and intersects T transversally. That is, Y is a non-degenerate compactification of (U, f ) in this case.
Otherwise, pick a pair (D 1 , D 2 ) such that ∆ Y (D 1 , D 2 ) < 0 and is the smallest among all possible values of ∆ Y . Let Y be the blowup of Y along D 1 ∩ D 2 andπ : Y → X 1 × X 2 the induced map. Then T := Y \ U consists of the exceptional divisor E and { D} where D denotes the proper transform of an irreducible component D of T . To construct the explicit systems of local coordinates of Y , we may assume that D i = (y i ) for i = 1, 2 with a 1 > b 1 and a 2 < b 2 after rearrangement. The blowup is defined by the equation y 1 v = y 2 u where [u : v] is the homogeneous coordinate of P 1 . Over this chart of coordinates of Y , we add two charts to Y (and away from the blowup center, we pass the charts of Y to Y ). In the chart v = 0 of P 1 , we consider the local coordinates ū := u v , y 2 , y 3 , . . . , y l+m+r .
One has D 1 ∩ D 2 = ∅ and Notice that We then replace Y with its systems of local coordinates by Y with the coordinates constructed above.
Observe that after a finite number of blowups in this procedure, the smallest possible value of the function ∆, if it is negative on Y , strictly increases. Hence repeating this construction, it produces a non-degenerate compactification X of (U, f ) obtained by a sequence X → · · · → X 1 × X 2 of explicit blowups.

Relations between complexes
We shall put a filtered complex on each step of the sequence of blowups constructed in Section 3.1 and compare them under push-forwards.
Consider a birational map π : Y → X 1 × X 2 and let P Y,f j be the pole divisor of the pullback which is coherent but not locally free in general. Let T = Y \ π −1 (U ) and otherwise.
(It is indeed a sub-complex of (Ω • Y ( * T ), Θ).) If Y is a non-degenerate compactification of (U, f ), e.g., Y equals the iterated blowup X constructed in the previous subsection, then F (λ) Y (Θ) is indeed the filtration defining the desired Hodge filtration. The following two lemmas describe the situations in the initial step Y = X 1 ×X 2 and in each blowup : Y → Y appeared in the sequence occurred in Section 3.1, respectively.
Lemma. Consider Y = X 1 × X 2 and let F λ (Θ) be the product filtration of F λ X i (Θ i ) whose p-th component is inside Ω p Y ( * T ). (Here Θ i = d + df i and df i if Θ = d + df and df , respectively.) (ii) For all λ ≥ ρ, the induced map for any λ where the inner sum is taken inside the vector space and which are the q-th and the (p − q)-th components of F , respectively. The latter two contribute to F λ (Θ).
(ii) We have the natural external products for all a + b = λ. UsingČech resolution or representatives in smooth forms, one obtains the cup product On the other hand, one has from the definition that Again there is the cup product To complete the assertions, it suffices to show that the arrow above is an isomorphism. Indeed, denote the sum inside the big round brackets in the right side of (19) by Φ(i, j, λ). The E 1degeneration of the spectral sequence attached to each filtered complex F X i (Θ i ) implies that there is the natural exact sequence Together with the isomorphism of (20), a decreasing induction on the index λ in (19) then gives the desired statements. Finally the isomorphism of (20) can be obtained by directly truncating the involved complexes and inductively using the Künneth formula for coherent sheaves [10, Theorem 1].
Consider the affine line A 1 u with a fixed coordinate u. Fix a pair (U, f ) and a non-degenerate compactification X. Let π : X × A 1 u → A 1 u be the projection. The k-th Brieskorn lattice of (U, f ) (cf. [9, Section 6.1]) is the coherent sheaf on A 1 Here d X is the derivative with respective to the component X only. Let According to the quasi-isomorphisms in Proposition 1(ii) and the E 1 -degeneration we have that is free and independent of the choice of α, and • the canonical maps F −α+p G k (U, f ) → G k (U, f ) indeed define a filtration by free subsheaves of G k (U, f ) with free quotients whose fibers are Now consider as in Section 3 two pairs (U i , f i ) and their product (U, f ). We again have the natural map obtained by cup product. Similar to the proof of [1, Proposition 1.5.1], the fiber-wise Künneth formula Theorem 1 shows that the above map is an isomorphism strictly compatible with the filtrations.
On the other hand, fix 0 ≤ α < 1. One can naturally complete G k (U, f ), F −α+p p∈Z into a filtered bundle on P 1 by adding the filtered cohomology space as the fiber over u = ∞. The resulting sheaf on P 1 is called the Kontsevich bundle and denoted by K k α (U, f ). The filtered space H k d,α (U, f ) depends on α but does not depend on the choice of the non-degenerate compactification X since in fact ([9, Theorem 1.11(a), Section 1.3], [4, Theorem 1.2(i,ii)]) (i) the bundle K k α (U, f ) can be obtained (as a Deligne extension) by using a natural algebraic connection on G k (U, f ) (see [9,Lemma 6.2] with the aid of Proposition 1(ii) in the case of non-degenerate compactification X) with regular singularity at u = ∞, and (ii) under the base-change, one has where HN p K k α (U, f ) is the Harder-Narasimhan filtration on the locally free sheaf K k α (U, f ) normalized with Gr p HN isomorphic to a direct sum of copies of O(p) on P 1 . In fact, we also have → Ω 1 f i on P 1 , whose hypercohomology gives H k d,0 A 1 , x 2 i . Let π : X → P 1 × P 1 be the blowup at (∞, ∞). Then X is a non-degenerate compactification of A 2 , f = f 1 + f 2 . On X we have the inclusion and the arrow (25) factors through the induced map H 2 X, A → H 2 X, B . One checks that the last map is zero. On the other hand, for Θ = d + dx 2 i or dx 2 i , we have H 1 A 1 , Θ = Gr 1/2 F H 1 A 1 , Θ = H 0 P 1 , Ω 1 (2[∞]) , which is generated by the class dx i . By (22), (23), (24), one obtains that and in particular, H 1 d,0 A 1 , x 2 i is generated by dx i as a fiber of K 1 0 A 1 , x 2 i . Similar argument shows that We conclude that the map (25) sends the generator dx 1 ⊗ dx 2 to dx 1 dx 2 , which represents zero in H 2 d,0 A 2 , x 2 1 + x 2 2 as a fiber of K 2 0 A 2 , x 2 1 + x 2 2 .