The K¨
unneth Formula for the Twisted de Rham
and Higgs Cohomologies
Kai-Chieh CHEN † and Jeng-Daw YU ‡
† Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA E-mail: kaichiehchen@berkeley.edu
URL: https://math.berkeley.edu/~kaichieh/
‡ Department of Mathematics, National Taiwan University, Taipei, Taiwan E-mail: jdyu@ntu.edu.tw
URL: http://homepage.ntu.edu.tw/~jdyu/
Received February 23, 2018, in final form June 02, 2018; Published online June 12, 2018
https://doi.org/10.3842/SIGMA.2018.055
Abstract. We prove the K¨unneth formula for the irregular Hodge filtrations on the expo-nentially 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.
Key words: de Rham complex; Hodge filtration; K¨unneth formula 2010 Mathematics Subject Classification: 14F40; 18F20; 14C30
1
The main result
Let U be a smooth quasi-projective variety over the field C of complex numbers, and f ∈ Γ(U,OU) a regular function. Attached to such a pair (U, f ) and a non-negative integer k, one has the k-th de Rham cohomology HkdR(U, f ) and Higgs cohomology HkHig(U, f ), defined in Section2. The two spaces Hk
(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
FλHk(U, f ), λ ∈ Q,
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 (Ui, fi), i = 1, 2. On the product U := U1× U2, consider the regular function f defined by f : (x1, x2) 7→ f1(x1) + f2(x2). We call the pair (U, f ) the product of the two (Ui, fi). For ∈ {dR, Hig}, there is the canonical map
M i+j=k
Hi(U1, f1) ⊗ Hj(U2, f2) → Hk(U, f ) (1)
This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The full collection is available athttp://www.emis.de/journals/SIGMA/modular-forms.html
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
M i+j=k X a+b=λ FaHi(U1, f1) ⊗ FbHj(U2, f2) ! ,
where the inner sum is taken inside Hi(U1, f1)⊗Hj(U2, f2). In this article, we prove the following K¨unneth formula.
Theorem 1. With notations as above, the map (1) is an isomorphism of filtered spaces. In particular, denoting GrλFV the λ-th graded piece of a filtered space (V, F ), one has the identification GrλF Hk(U, f ) =M r,s GrrFHs(U1, f1) ⊗ Grλ−rF H k−s (U2, f2).
Recently different proofs of the K¨unneth 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 P1 of special type is introduced [8, Section 3.2.b] in order to obtain the K¨unneth 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 (Ui, fi) and of their product (U, f ), we construct a particular compactification of (U, f ) from the fixed ones of (Ui, fi) in Section 3. The proof of the K¨unneth 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 HkdR(U, f ) and HkHig(U, f ) as the fibers of the Kontsevich bundle on the projective line P1 over 1 and 0, respectively. In fact, the fiber over c ∈ C \ {0} of the bundle is equal to HkdR(U, f /c) and hence the K¨unneth formula also holds true. However at ∞ the situation is more complicated in this regard and the direct analogue of the K¨unneth formula does not hold in general.
2
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,OU).
2.1 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,OX(∗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 Predbe 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 → P1.
If X is non-degenerate, analytically locally at a point of P , there exists a coordinate system {x1, . . . , xl, y1, . . . , ym, z1, . . . , zr} such that S = (xy) and f = 1 xe or f = z1 xe (2)
for some e ∈ Zl>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 (OX(∗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].
2.2 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, Ω1X(log S)(P ).
Definition.
(i) The twisted de Rham complex and the Higgs complex are the complexes (ΩX• (log S)(∗P ), Θ) =OX(∗P )−Θ→ · · ·−→ ΩΘ iX(log S)(∗P )
Θ
−→ Ωi+1X (log S)(∗P )−→ · · ·Θ , where Θ = d + df and df , respectively. Here df sends a local section ω to df ∧ ω. (ii) We call the associated hypercohomology groups Hk(X, (Ω•X(log S)(∗P ), Θ)) the de Rham
cohomology and the Higgs cohomology of (U, f ), and denote respectively by HkdR(U, f ) = Hk(U, d + df ) and HkHig(U, f ) = Hk(U, df ). (iii) For an effective divisor D on X and µ ∈ Q, let
ΩiX(log S)(bµDc)+= ( ΩiX(log S)(bµDc), if µ ≥ 0, 0, otherwise. For Θ ∈ {d + df, df }, λ ∈ Q, let FXλ(Θ) = Fλ(Θ) =OX(b−λP c)+−→ · · ·Θ −→ ΩΘ iX(log S)(b(i − λ)P c)+ −→ · · ·Θ .
The subindex X in FXλ(Θ) will be omitted if the base variety is clear. The Hodge filtrations of the de Rham and the Higgs cohomologies are
FλHk(U, Θ) = ImageHk(X, Fλ(Θ)) → Hk(U, Θ) (3)
induced by the inclusions of complexes.
(iv) For α ∈ Q, the Kontsevich sheaf of differential p-forms is the coherent subsheaf of ΩpX(log S)(∗P )
Ωpf(α) = kerΘ : ΩpX(log S)(bαP c) → Ωp+1X (∗S)/Ωp+1X (log S)(bαP c) ,
and it forms the Kontsevich complex (Ω•f(α), Θ) equipped with the filtration (Ω•f(α), Θ)•≥p by direct truncation. We simply write Ωpf for Ωpf(0).
Proposition 1.
(i) For α ∈ Q and p ∈ Z, the OX-module Ωpf(α) is locally free of rank dim Xp with Ωpf(α) = Ωpf ⊗OX OX(bαP c).
(ii) For Θ ∈ {d + df, df } the three inclusions F0(Θ) → F−α(Θ), 0 ≤ α, (Ω•f(α), Θ) → F−α(Θ), 0 ≤ α,
(Ω•f(α), Θ)•≥p→ F−α+p(Θ), 0 ≤ α < 1 are quasi-isomorphisms.
Proof . Both statements are local properties for coherent sheaves on X and we can restrict to the coordinates such that (2) holds.
(i) In case f = x1e, the local freeness and the quasi-isomorphisms are obtained in [1, equa-tion (1.3.1)], [3, Lemma 2.12(a)] and [11, Proposition 1.3.], [1, Proposition 1.4.2], respectively. In fact, the methods are similar to the arguments below.
Consider the second case f = z1xe so that
z1 df f = x edf = dz 1− z1 X ei dxi xi .
In this chart, the OX-module ΩpX(log S) is freely generated by
z1 df f ∧ p−1 ^ i=1 ξi, {ξi}p−1i=1 ⊂ dz2, . . . , dzr, dx1 x1 , . . . ,dxl xl ,dy1 y1 , . . . ,dym ym (4) and p ^ i=1 ηi, {ηi}pi=1⊂ dz2, . . . , dzr, dx1 x1 , . . . ,dxl xl ,dy1 y1 , . . . ,dym ym . (5)
The OX-module Ωpf(α) is indeed freely generated by
x−bαecω1 and xe−bαecω2, (6)
(ii) Let D be a divisor supported on Pred, and E an irreducible component of Pred. Let Apk(D) = Ωp(log S)(D + pP + kE), p, k ∈ Z, α ∈ Q.
To obtain the quasi-isomorphisms, it suffices to show that the inclusion of complexes
(A•k−1(D), Θ) → (A•k(D), Θ) (7)
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 ).
To check the quasi-isomorphism (7), we may assume that E = (x1) in the local model. Consider local sections h ∈ OX(D), and ω1 and ω2 in (4) and (5), respectively. Then h · x−pex−k1 ωi generates Apk(D) and
Θ : Apk(D)/Apk−1(D) → Ap+1k (D)/Ap+1k−1(D), h xpexk 1 · ( ω1, ω2 7→ ( 0, h x(p+1)exk 1 (xedf ) ∧ ω2.
Therefore the quotient Apk(D)/Apk−1(D), Θp∈Z of (7) is exact. Remark. In a more functorial way, one can consider the D-module M on X attached to the meromorphic connection (OX(∗S), d + df ) and define the irregular Hodge filtration on M as given in [9, Definition 5.1]. By [9, Lemma 9.17], the filtered complex Fλ(d + df ) defined above is quasi-isomorphic to the filtered de Rham complex associated with the filteredD-module M . When X is a good compactification, this is obtained in [1, Proposition 1.7.4]. Forgetting the filtrations, the quasi-isomorphisms of various de Rham complexes of M are also derived in [5, Lemmas 2.13 and 2.15] in the case where X is non-degenerate, including the compactly supported counterpart.
2.3 The independence
Proposition. For ∈ {dR, Hig}, the space Hk(U, f ) with the Hodge filtration is independent of the choice of the non-degenerate compactification of (U, f ). More precisely, if π : X0 → X is a morphism between non-degenerate compactifications extending the identity on U , then there is a natural quasi-isomorphism
Rπ∗FXλ0(Θ) ' FXλ(Θ), λ ∈ Q. (8)
Proof . The assertion for good compactifications is proved in [11, Theorem 1.7], by comparing the degree p components FXλ(Θ)p of FXλ(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 $ : eX → X be the blowup along the intersection Ξ of Z = (z1) and the irreducible component (x1) of P with multiplicity e1. Let E be the exceptional divisor, eS = eX \ U and eP the pole divisor of ˜f := $∗f . We want to establish that R$∗Fλ
e
X(Θ) and F λ
X(Θ) are canonically quasi-isomorphic.
In case e1 > 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 eX define the complex
Rλ(Θ) = Ω• e
X log eS(b(α − p + •) P + E)ce
We have
$∗FXλ(Θ), Fλ e
X(Θ) ⊂ R λ(Θ).
By [11, Proposition 4.4(i)], each component of the complex Rλ(Θ)/$∗FXλ(Θ) is a direct sum of copies of OE/Ξ(−1). Hence the adjunction FXλ(Θ) → R$∗Rλ(Θ) is a quasi-isomorphism. On the other hand, there are increasing complexes Rλq(Θ) on eX (those denoted by Rλ(q) in [11, equation (27)], which is a complex under either d + df or df ) with Rλ−1(Θ) = Fλ
e
X(Θ) and Rλ
dim eX−p(Θ) = R
λ(Θ) such that Rλ
q(Θ)/Rq−1λ (Θ) is quasi-isomorphic to the complex consisting of a direct sum of copies ofOE/Ξ(−1) concentrated at degree (p + q). In fact, in [11, Lemma 4.5], we further introduce the complexes (Kρ,η,ξ• , d+df ) and prove that the inclusion (Kρ,η,ξ• , d+df ) ⊂ (Kρ,η+1,ξ• , d + df ) is a quasi-isomorphism by showing the quotient complex is exact. Now one notices that Kρ,η,ξ• is indeed also stable under df ; on the quotient Kρ,η+1,ξ• /Kρ,η,ξ• , one has the equality d + df = df of the differential maps under the condition e1 > 1. Therefore the proof of [11, Proposition 4.4(ii)] describing Rλq(Θ)/Rλq−1(Θ) goes through in both cases Θ = d + df and Θ = df . Hence one concludes that the inclusion Fλ
e X(Θ) ⊂ R λ(Θ) induces a quasi-isomorphism R$∗Fλ e X(Θ) → R$∗R
λ(Θ) under the assumption e 1> 1.
The following arguments work for any e1 ≥ 1 and simplify those in [11, Section 4(b)]. (Cf., [3, Lemma 2.12(b)].) By Proposition1, it suffices to show that for any 0 ≤ α < 1, p ∈ Z, we have the inclusion relation $∗Ωpf(α) ⊂ Ωp˜
f(α) on eX and it induces an isomorphism Ω p f(α) → R$∗Ω p ˜ f(α) on X. We first compute Ωp˜
f(α). Explicitly the blowup eX is defined by the equation detx1 z1
u v
= 0
in X × P1 where [u : v] is the homogeneous coordinate on P1. On the chart v 6= 0 with local coordinates n ¯ u = u v, x2, . . . , xl, y1, . . . , ym, z1, . . . , zr o , theO e X-module Ω p ˜
f(α) is generated by the basis
δd ˜f ˜ f ∧ p−1 ^ i=1 ξi, δ = z−bα(e1−1)c 1 u¯−bαe1cx −bαe2c 2 · · · x −bαelc l , {ξi}p−1i=1 ⊂ dz1 z1 + d¯u ¯ u , dx2 x2 , . . . ,dxl xl ,dy1 y1 , . . . ,dym ym , dz2, . . . , dzr , (9) and δ1 ˜ f p ^ i=1 ηi, {ηi}pi=1⊂ dz1 z1 +d¯u ¯ u , dx2 x2 , . . . ,dxl xl ,dy1 y1 , . . . ,dym ym , dz2, . . . , dzr . (10)
On the chart u 6= 0 with local coordinates n x1, . . . , xl, y1, . . . , ym, ¯v = v u, z2, . . . , zr o , the sheaf Ωp˜
f(α) is generated by the basis
ε¯vd ˜f ˜ f ∧ p−1 ^ i=1 ξi, ε = x−bα(e1−1)c 1 x −bαe2c 2 · · · x −bαelc l , {ξi}p−1i=1 ⊂ dx1 x1 , . . . ,dxl xl ,dy1 y1 , . . . ,dym ym , dz2, . . . , dzr , (11)
and εv¯ ˜ f p ^ i=1 ηi, {ηi}pi=1⊂ dx1 x1 , . . . ,dxl xl ,dy1 y1 , . . . ,dym ym , dz2, . . . , dzr . (12)
On the intersection u, v 6= 0, one has dx1x1 = dz1z1 +d¯u¯u.
Using the basis (6) of Ωpf(α), a direct computation reveals that $∗Ωpf(α) is contained in Ωp˜ f(α). Moreover, theOE-module Ωpf˜(α)/$∗Ω
p
f(α) equals either zero if bα(e1− 1)c 6= bαe1c or otherwise l+m+r
p copies ofOE/Ξ(−1) generated by (9), (10) and (11), (12) on the two charts, respectively. Therefore we obtain that Fλ
X(Θ) and R$∗FXλe(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 X0of (U, f ) from the non-degenerate X with the canonical
quasi-isomor-phism (8).
Remark.
(i) By the E1-degeneration [1, Theorem 1.2.2], [4, Theorem 1.1] on a good compactification (see also [3, Theorem 2.18]), the above proposition implies that the arrow in (3) is injective for any non-degenerate X and indices k, λ.
(ii) For X non-degenerate, we have HkdR(U, f ) = Hk(X, (Ω•X(∗S), d + df )) by the arguments of [11, Corollary 1.4]. On the other hand, HkHig(U, f ) 6= Hk(X, (Ω•X(∗S), df )) in general which can be seen by considering the case U affine and f = 0.
3
The proof of the main result
We begin with two pairs (Ui, fi), i = 1, 2, and their product (U, f ). Fix good compactifications Xi of (Ui, fi) such that Si := Xi\ Ui 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 X1× X2 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 Hk(U, f ), with a certain filtered complex on X1× X2 that gives the product filtration using the explicit construction of X.
3.1 An explicit elimination
For each i = 1, 2, take an open covering of Xi with a system of local coordinates {xi,1, . . . , xi,li, xi,li+1, . . . , xi,li+mi, xi,li+mi+1, . . . , xi,li+mi+ri}
such that fi =
1 xei,1i,1 · · · xei,li
i,li
, ei,j > 0, Si = (xi,1· · · xi,li+mi).
As for the initial data in the inductive construction, we consider the compactification X1× X2 of U with the systems of local coordinates {xi,j}.
Suppose we have constructed a compactification Y of U and its systems of local coordinates {y1, . . . , yl, yl+1, . . . , yl+m, yl+m+1, . . . , yl+m+r}, (13)
together with a birational map π : Y → X1× X2 such that π∗f1 = 1 y1a1· · · yall , π ∗ f2 = 1 yb11 · · · ylbl, Y \ U = (y1· · · yl+m) (14) for some ai, bi ≥ 0 with bi > 0 if ai = 0. Let T be the boundary divisor Y \ U . For a pair of irreducible components D1, D2 of T , set
∆Y(D1, D2) = 2 Y i=1 ordDi(π∗f1) − ordDi(π∗f2), if D1 6= D2, D1∩ D2 6= ∅, 0, otherwise.
Notice that if ∆Y(D1, D2) ≥ 0 for any pair (D1, D2), 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 (D1, D2) such that ∆Y(D1, D2) < 0 and is the smallest among all possible values of ∆Y. Let eY be the blowup of Y along D1∩ D2 and ˜π : eY → X1 × X2 the induced map. Then eT := eY \ U consists of the exceptional divisor E and { eD} where eD denotes the proper transform of an irreducible component D of T . To construct the explicit systems of local coordinates of eY , we may assume that Di = (yi) for i = 1, 2 with a1 > b1 and a2 < b2 after rearrangement. The blowup is defined by the equation y1v = y2u where [u : v] is the homogeneous coordinate of P1. Over this chart of coordinates of Y , we add two charts to eY (and away from the blowup center, we pass the charts of Y to eY ). In the chart v 6= 0 of P1, we consider the local coordinates
n ¯ u := u v, y2, y3, . . . , yl+m+r o . (15) Then ˜ π∗f1 = 1 ¯ ua1ya1+a2 2 y a3 3 · · · y al l , π˜∗f2 = 1 ¯ ub1yb1+b2 2 y b3 3 · · · y bl l , (16) e T = (¯uy2· · · yl+m), De1 = (¯u), E = (y2). In the chart u 6= 0 of P1, we consider the local coordinates
n ¯ v := v u, y1, y3, . . . , yl+m+r o . (17) Then ˜ π∗f1 = 1 ¯ va2ya1+a2 1 y a3 3 · · · y al l , π˜∗f2 = 1 ¯ vb2yb1+b2 1 y b3 3 · · · y bl l , (18) e T = (¯vy1y3· · · yl+m), De2 = (¯v), E = (y1). One has eD1∩ eD2 = ∅ and
∆ e Y( eD, eD 0) = ∆ Y(D, D0) for (D, D0) 6= (D1, D2), (D2, D1), ∆ e Y( eDi, E) = ∆Y(D1, D2) + ordDi(π ∗f 1) − ordDi(π∗f2) 2 > ∆Y(D1, D2), ∆ e Y( eD, E) = ∆Y(D, D1) + ∆Y(D, D2). Notice that
and consequently
∆ e Y( eD, E)
(
> min{∆Y(D, D1), ∆Y(D, D2)} if ordD(π∗f1) 6= ordD(π∗f2),
= 0 if ordD(π∗f1) = ordD(π∗f2).
We then replace Y with its systems of local coordinates by eY 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 → · · · → X1× X2 of explicit blowups.
3.2 Relations between complexes
We shall put a filtered complex on each step of the sequence of blowups constructed in Section3.1 and compare them under push-forwards.
Consider a birational map π : Y → X1× X2 and let PY,fj be the pole divisor of the pullback of fj on Y . For any λ ∈ Q, consider the subsheaf OY(λ) of OY(∗(PY,f1+ PY,f2)) given by
O(λ)
Y =
X 0≤θ≤1
OY (bλ((1 − θ)PY,f1 + θPY,f2)c) ,
which is coherent but not locally free in general. Let T = Y \ π−1(U ) and Ωp,(λ)Y =OY(λ)⊗OY ΩpY(log T ).
For Θ ∈ {d + df, df }, consider the complex on Y
FY(λ)(Θ) =OY(−λ)+ −→ ΩΘ 1,(1−λ)+Y −→ ΩΘ 2,(2−λ)+Y −→ · · ·Θ , where Ωp,(p−λ)+Y = ( Ωp,(p−λ)Y , if p − λ ≥ 0, 0, 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 FY(λ)(Θ) is indeed the filtration defining the desired Hodge filtration. The following two lemmas describe the situations in the initial step Y = X1×X2 and in each blowup $ : eY → Y appeared in the sequence occurred in Section3.1, respectively. Lemma. Consider Y = X1× X2 and let Fλ(Θ) be the product filtration of FXiλ (Θi) whose p-th component is
Fλ(Θ)p:= X a∈Q,q∈Z
FX1a (Θ1)q FX2λ−a(Θ2)p−q
inside ΩpY(∗T ). (Here Θi = d + dfi and dfi if Θ = d + df and df , respectively.) (i) The filtration FY(λ)(Θ) coincides with Fλ(Θ).
(ii) For all λ ≥ ρ, the induced map Hk Y, Fλ(Θ) → Hk Y, F ρ (Θ)
is injective; it is an isomorphism if λ ≤ 0. One has
Hk Y, Fλ(Θ) = M i+j=k X a+b=λ Hi X1, FX1a (Θ1) ⊗ Hj X2, FX2b (Θ2) ! (19)
for any λ where the inner sum is taken inside the vector space Hi(U1, Θ1) ⊗ Hj(U2, Θ2). Proof . (i) Indeed the inclusion Fλ(Θ) ⊂ FY(λ)(Θ) is clear. Conversely suppose that µ := p − λ ≥ 0. The p-th degree component Ωp,(µ)Y of FY(λ)(Θ) is generated by products of elements in
ΩqX1(log S1)(b(1 − θ)µPX1c) and Ωp−qX2 (log S2)(bθµPX2c),
which are the q-th and the (p − q)-th components of FX1q−(1−θ)µ(Θ1) and FX2p−q−θµ(Θ2), respec-tively. The latter two contribute to Fλ(Θ).
(ii) We have the natural external products FX1a (Θ1) FX2b (Θ2) → Fλ(Θ)
for all a + b = λ. Using ˇCech resolution or representatives in smooth forms, one obtains the cup product
Hi X1, FX1a (Θ1) ⊗ Hj X2, FX2b (Θ2) → Hi+j Y, FX1a (Θ1) FX2b (Θ2). On the other hand, one has from the definition that
GrλF (Θ)= M a+b=λ GraF X1(Θ1) Gr b FX2(Θ2).
Again there is the cup product M i+j=k Hi X1, GraFX1(Θ1) ⊗ H j X 2, GrbFX2(Θ2) → H k Y, Gra FX1(Θ1) Gr b FX2(Θ2). (20)
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 E1 -degeneration of the spectral sequence attached to each filtered complex FXi(Θi) implies that there is the natural exact sequence
0 → [ µ>λ Φ(i, j, µ) → Φ(i, j, λ) → M a+b=λ Hi X1, GraF X1(Θ1) ⊗ H j X 2, GrbF X2(Θ2) → 0.
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 K¨unneth formula for coherent sheaves [10,
Theorem 1].
Lemma. Let $ : eY → Y be the blowup constructed in Section3.1. Then the pullback $∗FY(λ)(Θ) is a sub-complex of F(λ)
e
(i) Each component F(λ) e Y (Θ)
p/$∗F(λ)
Y (Θ)p of the quotient is supported on the exceptional di-visor E and is a direct sum of copies of relativeO(−1) of the P1-bundle E over the center of the blowup $.
(ii) We have the canonical quasi-isomorphism FY(λ)(Θ) → R$∗F(λ) e Y (Θ).
Proof . (i) Let eT = eY \ U . As in the proof of the previous lemma, assume that µ := p − λ ≥ 0. Since $∗ΩpY(log T ) = Ωp
e
Y(log eT ), we only need to consider the difference between O (µ) e Y and $∗OY(µ). We use the local coordinates (13) with the properties (14). Then OY(µ)is generated by various
g := 1
ybcc, (21)
where c = (1 − θ)µa + θµb for 0 ≤ θ ≤ 1. On the other hand, O(µ) e Y is generated by various h¯u:= 1 ¯ ubc1cybc1+c2c 2 y bc3c 3 · · · y bclc l and hv¯:= 1 ¯ vbc2cybc1+c2c 1 y bc3c 3 · · · y bclc l on the two charts (15) and (17) satisfying (16) and (18), respectively. We have
hhu¯iO e Y /$ ∗hgi OY = ( hhu¯iOE, if bc1+ c2c = bc1c + bc2c + 1, 0, if bc1+ c2c = bc1c + bc2c,
and similarly on the chart u 6= 0. Observe on the intersection that hu¯ = ¯v−1h¯v if bc1+ c2c 6= bc1c + bc2c. Thus O(µ)
e Y /$
∗O(µ)
Y is a direct sum of copies of the relativeO(−1).
(ii) There are only finitely many y−bcc occurred in the generators in (21); call the appeared (distinct) monomials ξ1, . . . , ξk ordered by the increment of the corresponding parameter θ. Consider locally the filtration Mi = hξ1, . . . , ξii of OY-submodules of OY(µ). On this chart one has the short exact sequence
0 → Ni+1→ Mi⊕OY · ξi+1→ Mi+1→ 0, (ω, η) 7→ ω − η. Here if ξi =Q y
−ci,j
j , then ci,j is monotone as a function of i and Ni+1 = Mi ∩OYξi+1 is the invertible sheaf generated by
Y
y− min{ci,j,ci+1,j} j
(with diagonal embedding to Mi⊕OYξi+1). Applying inductively the projection formula (e.g., [2, Exercise III.8.3]) to the locally freeOYξi+1 and Ni, one obtains that
R$∗$∗Mi= R$∗ OYe ⊗ $∗Mi = R$∗OYe ⊗ Mi= Mi,
since R$∗OYe =OY for the birational morphism $. Together with the computation in (i), the
assertion follows.
4
The Brieskorn lattice
We indicate that the K¨unneth formula for Hk
dR(U, f ) and HkHig(U, f ) can be put together into a family version (cf. [9, Sections 1.3 ans 6.2], [3, Section 3.2] and [4]).
Consider the affine line A1u with a fixed coordinate u.
Fix a pair (U, f ) and a non-degenerate compactification X. Let π : X × A1
u → A1u be the projection. The k-th Brieskorn lattice of (U, f ) (cf. [9, Section 6.1]) is the coherent sheaf on A1u
Gk(U, f ) := Rkπ
∗ Ωqf(α) OA1
u, u · dX + df
q≥0.
Here dX is the derivative with respective to the component X only. Let
F−α+pGk(U, f ) = Rkπ∗ Ωqf(α) OA1
u, u · dX + df
q≥p, 0 ≤ α < 1. According to the quasi-isomorphisms in Proposition 1(ii) and the E1-degeneration
dim Hk(U, Θ) =X q dim Hk−q X, Ωqf(α), Θ ∈ {d + df, df }, we have that • theOA1 u-moduleG
k(U, f ) is free and independent of the choice of α, and
• the canonical maps F−α+pGk(U, f ) →Gk(U, f ) indeed define a filtration by free subsheaves of Gk(U, f ) with free quotients whose fibers are
F−α+pGk(U, f )|u=c = (
F−α+pHkHig(U, f ), c = 0,
F−α+pHkdR(U, f /c), c 6= 0 (22)
under the base-change map (cf. the arguments in the proof of [1, Proposition 1.5.1]). Now consider as in Section3 two pairs (Ui, fi) and their product (U, f ). We again have the natural map M 0≤i≤k Gi(U 1, f1) ⊗O A1u G k−i(U 2, f2) →Gk(U, f )
obtained by cup product. Similar to the proof of [1, Proposition 1.5.1], the fiber-wise K¨unneth 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 Gk(U, f ), F−α+pp∈Z into a filtered bundle on P1 by adding the filtered cohomology space
FpHkd,α(U, f ) := ImageHk X, (Ω•f(α), d)•≥p → Hk X, (Ω • f(α), d)
as the fiber over u = ∞. The resulting sheaf on P1 is called the Kontsevich bundle and denoted by Kαk(U, f ). The filtered space Hkd,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 bundleKαk(U, f ) can be obtained (as a Deligne extension) by using a natural algebraic connection onGk(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 Hkd,α(U, f ), F• = Kk
α(U, f ), HN •
|u=∞, (23)
where HNpKαk(U, f ) is the Harder–Narasimhan filtration on the locally free sheafKαk(U, f ) normalized with GrpHN isomorphic to a direct sum of copies ofO(p) on P1.
In fact, we also have
F−α+•Gk(U, f ) = Kαk(U, f ), HN•
|u6=∞. (24)
However, one does not have the direct analogue of the K¨unneth formula for H•d,α in general. For example, in the case (Ui, fi) = (A1, x2i) where xi is a global coordinate on the affine line A1, one has
dim Hkd,0 A1, x2i = (
1, if k = 1, 0, otherwise, and the cup product
H1d,0 A1, x21 ⊗ H1d,0 A1, x22 → H2d,0 A2, x21+ x22
(25) is zero. The vanishing can be obtained by a careful cohomology calculation or manipulating Kontsevich bundles as follows.
Indeed, consider the complexΩ0fi −→ Ωd 1
fi on P1, whose hypercohomology gives Hkd,0 A1, x2i. Let π : eX → P1× P1 be the blowup at (∞, ∞). Then eX is a non-degenerate compactification of
A2, f = f1+ f2. OnX we have the inclusione A := π∗ 2i=1(Ω•fi, d) → B := (Ω
• f, d)
and the arrow (25) factors through the induced map H2 X, Ae → H2 X, B. One checks thate the last map is zero.
On the other hand, for Θ = d + dx2i or dx2i, we have H1 A1, Θ = Gr1/2F H1 A1, Θ = H0 P1, Ω1(2[∞]),
which is generated by the class dxi. By (22), (23), (24), one obtains that K1 α A1, x2i = (OP1 · dxi, 0 ≤ α < 1 2, OP1([∞]) · dxi, 12 ≤ α < 1,
and in particular, H1d,0 A1, x2i is generated by dxi as a fiber of K01 A1, x2i. Similar argument shows that
K2
α A2, x21+ x22 =OP1([∞]) · dx1dx2, 0 ≤ α < 1.
We conclude that the map (25) sends the generator dx1⊗ dx2 to dx1dx2, which represents zero in H2d,0 A2, x21+ x22 as a fiber ofK02 A2, x21+ x22.
Acknowledgements
The second author thanks Noriko Yui for providing him the wonderful opportunity to work with her during the years 2006 to 2008 and the happy lunch time at Queen’s University. We thank the referee for the careful reading, helpful comments and in particular pointing out the insufficiency of the proof of Proposition 1 in an earlier version. This work was partially supported by the MoST and the NCTS, Taiwan.
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