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∂ ¯
∂-lemma, double complex and L
cohomology
Sheng Rao, Yongpan Zou
To cite this version:
SHENG RAO AND YONGPAN ZOU One world, one fight
Abstract. Inspired by the recent works of J. Stelzig and M. Khovanov–Y. Qi on the structure of bounded double complex, we improve further this theory with more emphasis on the bounded double complexes possibly without real structures by calculating out six types of indecomposable double complexes in the complex de-composition, to characterize various (weak) ∂∂-lemmata by de Rham, Dolbeault, Bott–Chern, Appeli and Varouchas’ cohomology groups. In particular, we ob-tain new proofs of D. Angella–A. Tomassini’s characterization of ∂∂-lemma and Fr¨olicher-type inequalities, a weak ∂∂-lemma for logarithmic forms with a unitary local system to confirm a conjecture of X. Wan as a direct corollary, and also several new injectivity results on complex manifolds. From the double complex perspective, one can easily see the mechanism behind the validity or invalidity of the (weak) ∂∂-lemmata and injectivity.
Contents
1. Introduction 1
2. Decomposing double complexes 7
3. Morphism and tensor product of double complexes 21
4. ∂∂-lemma, Fr¨olicher spectral sequence and L2 cohomology 28
5. ∂∂-lemma under the holomorphic morphism 38
References 40
1. Introduction
One most important class of complex manifolds in complex differential and analytic geometry is compact K¨ahler manifolds, which admit many nice properties, such as Hodge decomposition on complex differential forms, K¨ahler identity and the induced ∂∂-lemma by them as follows:
Lemma 1.1 ([Dm12, Lemma 8.6 in Chapter VI] for example). On a compact K¨ahler manifold, the following exactness properties for every d-closed pure-type smooth com-plex form are equivalent
d-exactness ⇐⇒ ∂-exactness ⇐⇒ ∂-exactness ⇐⇒ ∂∂-exactness.
Date: July 22, 2020.
2010 Mathematics Subject Classification. Primary 13D02; Secondary 18G40, 32S35, 14F40, 14A21, 32G05.
Key words and phrases. Syzygies, resolutions, complexes and commutative rings; Spectral se-quences, hypercohomology, Mixed Hodge theory of singular varieties (complex-analytic aspects), de Rham cohomology and algebraic geometry, Logarithmic algebraic geometry, log schemes, Deforma-tions of complex structures.
Both authors are partially supported by NSFC (Grant No. 11671305, 11771339, 11922115) and the Fundamental Research Funds for the Central Universities (Grant No. 2042020kf1065).
The compactness and K¨ahlerness plays an essential role in the proof of the ∂∂-lemma for K¨ahler manifolds, such that the compactness yields that the space of smooth complex differential forms can be decomposed as the direct sum of image and kernel space of elliptic operator, while the K¨ahlerness gives rise to the K¨ahler identity on smooth complex differential forms. Conversely, the ∂∂-lemma is a powerful tool to study K¨ahler manifolds, such as the functional of solving the ∂-equation ∂x = ∂α with closed ∂α on a K¨ahler manifold and the degeneration of Fr¨olicher spectral sequence. As for both of these two aspects, we will try to relax various settings mentioned above, such as compactness generalized by completeness, smooth complex differential forms replaced by logarithmic differential forms and so on, by the structure theory of bounded double complex possibly without real structures and L2 Hodge theory on complete manifolds.
As we know, there are many interesting alternative characterizations of the ∂∂-lemma on compact complex manifolds, among which are P. Deligne–P. Griffiths–J. Morgan–D. Sullivan’s one by the decomposition type of Dolbeault complex and D. Angella–A. Tomassini’s one by Fr¨olicher-type equality in terms of Betti, Bott–Chern and Aeppli numbers, cf. Theorem 2.9.(1) and Theorem 2.9.(7) in the general bounded double complex setting, respectively.
More recently, a question proposed by L. Alessandrini in [Al17, Introduction] whether the modification of a ∂ ¯∂-manifold (i.e., a compact complex manifold sat-isfying the ∂∂-lemma) is still a ∂ ¯∂-manifold arouses intense interest, such as [YY17, RYY19, RYY20, ASTT, Mn18, Mn20b, St18a, St18b, CY19, Zo19, RYYY]. Notice that the converse of this question is confirmed by [Pa66] or [DGMS75, Theorem 5.22]. Up to now, the threefold case of this question can be answered positively by S. Yang– X. Yang–the first author [RYY19] by use of Dolbeault, de Rham cohomologies blow-up formulae and an equivalent characterization of ∂ ¯∂-manifold in Remark 2.14, and by S. Yang–X. Yang [YY17] via abstract Bott–Chern cohomology blow-up formula and Angella–Tomassini’s characterization of the ∂∂-manifold in Theorem 2.9.(7). See also several quick solutions to this question in Corollary 5.5.
S. Yang–X. Yang–the first author also propose the Bott–Chern cohomology blow-up formula in [RYY19, Conjecture 1.9], which is proved by J. Stelzig in [St18a] via the structure Theorem 1.21of bounded double complexes and the morphism between them [St18b]. Actually, based on the Dolbeault cohomology blow-up formula, Stelzig even can uniformly prove the blow-up formulae for all cohomology groups induced by the Dolbeault complex, such as de Rham, Bott–Chern and Aeppli cohomology groups, etc, by introducing the E1-isomorphism of double complexes with real structures,
while a direct proof of Bott–Chern cohomology blow-up formula seems still desirable. Theorem 1.2. For any bounded double complex K over a field K there exist unique cardinal numbers multS(K) and a (non-functorial) isomorphism K ∼=LSS⊕multS(K),
where S runs over squares and zigzags.
Much motivated by these, we try to improve this powerful new structure theory of double complex to study the ∂∂-lemma, its weak versions and their connection to other topics in complex geometry, such as ∂-equation, Fr¨olicher spectral sequence and L2 cohomology.
We first list all kinds of indecomposable double complexes, i.e., squares and zigzags, and find that in each indecomposable double complex, there are at most four different
1As mentioned in [St18b], Theorem 1.2 usually quotes an unpublished note of M. Khovanov, now
types of vector spaces, up to their cohomological properties. So we calculate each case and make up tables to compare their dimensional relationships in Subsection 2.2. Theorem 1.3. For any bounded double complex K over a field K there exist unique cardinal numbers multS(K) and a (non-functorial) isomorphism K ∼=LSS⊕multS(K),
where S runs over the six types of indecomposable double complexes: Type 1 (square). K1 K2 K3 K4. ∂1 ∂2 ∂1 ∂2
Type 2 (upper-right even zigzags).
Type 2.1. upper-right even zigzags with lengths ≥ 4 K1 K2 K5 K3 K4. ∂1 ∂2 ∂1 ··· ∂2 ∂1
Type 2.2. right length 2 zigzag
K1 ∂1 K2.
Type 3 (lower-left even zigzags).
Type 3.1. lower-left even zigzags with lengths ≥ 4 K1 K2 K3 K5 K4. ∂1 ∂2 ∂2 ∂1 ··· ∂2
Type 3.2. upper length 2 zigzag
K1
K2. ∂2
Type 4 (upper-right odd zigzags).
Type 4.1. upper-right odd zigzags with lengths ≥ 5 K1 K2 K3 K4. ∂1 ∂2 ∂1 ··· ∂2
Type 4.2. upper-right length 3 zigzag
K1 K2
K3. ∂1
∂2
Type 5.1. lower-left odd zigzags with lengths ≥ 5 K1 K2 K3 K5 K6 K4. ∂1 ∂2 ··· ∂2 ∂1
Type 5.2. lower-left length 3 zigzag
K1 K2 K3. ∂1 ∂2 Type 6 (dot). K1•
One classical result in [DGMS75] is that any double complex with a real structure satisfies the ∂1∂2-lemma if and only if it is decomposed as a sum of indecomposable
double complexes of single dots and squares. Since dots and squares are the simplest double complexes, we first investigate what properties are equivalent to it. By explicit cohomological dimensional calculations from Theorem 1.3, we can present a uniform proof of all equivalent characterizations for the ∂1∂2-lemma in [DGMS75, ATo13,
ATa17, CS15, CHT15] in Theorem 2.9, and several Fr¨olicher-type inequalities with Angella-Tomassini’s [ATo13] and four new ones in terms of the cohomology groups of (2.1) in Corollary 2.12.
We say that a general bounded double complex is a ∂1∂2-complex as it is isomorphic
to a sum of double complexes of Types of single dots and squares; in particular, if the Dolbeault complex on a complex manifold satisfies this, it will be called a ∂∂-complex and the ∂∂-complex manifold is called a ∂∂-manifold or satisfies the ∂∂-lemma. Besides the characterizations in Theorem 2.9, we find more new characterizations of ∂1∂2-complex:
Theorem 1.4 (= Theorem 2.13). For any general bounded double complex K = (K•,•, ∂1, ∂2), the following numerical properties for the associated Varouchas’
coho-mologies to K in (2.1) are equivalent: for any p, q, k, (1) K is a ∂1∂2-complex; (2) ap,q= bp,q, cp,q = fp,q; (3) ap,q= dp,q, ep,q = fp,q; (4) ap,q= bp,q, bk= P p+q=kh p,q ∂1, h p,q ∂2 = h p,q ∂1; (5) cp,q= fp,q, bk =Pp+q=kh∂p,q2, hp,q∂2 = hp,q∂1 ; (6) ep,q= fp,q, b k=Pp+q=kh p,q ∂1 , h p,q ∂2 = h p,q ∂1; (7) ap,q= dp,q, bk=Pp+q=kh p,q ∂2, h p,q ∂2 = h p,q ∂1; (8) cp,q= fp,q, bk = P p+q=kh p,q ∂2, h p,q ∂2 = h p,q ∂1 .
Moreover, the following numerical properties are also equivalent to either of the above properties when the double complex K has a real structure: for any p, q,
(9) ap,q= bp,q, ep,q= fp,q; (10) ap,q= dp,q, cp,q= fp,q; (11) ap,q= bp,q, hp,q∂ 2 = h q,p ∂2; (12) cp,q= fp,q, hp,q∂ 2 = h q,p ∂2; (13) ep,q= fp,q, hp,q∂ 2 = h q,p ∂2; (14) ap,q= dp,q, hp,q∂ 2 = h q,p ∂2; (15) cp,q= fp,q, hp,q∂ 2 = h q,p ∂2.
Proposition 1.5 (= Proposition 2.16). For any bounded double complex K = (K•,•, ∂1, ∂2),
the following properties are equivalent.
(1) K is isomorphic to a sum of double complexes of Types: square and zigzags with lengths ≤ 2;
(2) the associated cohomologies to K satisfy hp,qBC+ hp,qA = hp,q∂
2 + h
p,q
∂1 for any p, q;
(3) the associated cohomologies to K satisfy Ap,q = 0 and Fp,q= 0 for any p, q. We call a bounded double complex a weak ∂1∂2-complex if it is isomorphic to a sum
of double complexes of Types: square and zigzags with lengths ≤ 2; in particular, if the Dolbeault complex on a complex manifold satisfies this, it will be called a weak ∂∂-complex.
In Section 3, we first give the definition of morphism between double complexes, originally from [St18b], and also the equivalent characterization of (weak) E1-isomorphism
by types of indecomposable double complexes in complex decompositions. Then fol-lowing [Mn20a], we define the tensor product of double complexes. In [St18b], Stelzig deals with it completely by computing the Grothendieck rings of several categories of double complexes. We obtain:
Proposition 1.6 (= Propositions 3.10 + 3.11). If both (K•,•, ∂K
1 , ∂K2 ) and (L•,•, ∂1L, ∂2L)
are (weak) ∂1∂2-complexes, then so is the tensor product of (K•,•, ∂1K, ∂2K) and (L•,•, ∂1L, ∂2L).
Proposition 1.6 yields a result originally by L. Meng in [Mn19b] that if X and Y are ∂∂-manifolds, then so is X × Y .
Section 4 is much motivated by a recent work [LRW19] of K. Liu-X. Wan-the first author on the solution to a ∂-equation of logarithmic forms on a compact K¨ahler manifold and we will apply the results developed in Sections 2 and 3 to this topic. Notice that the logarithmic Dolbeault complex
ADX = (A0,•(X, Ω•X(log D)), ∂, ∂)
of logarithmic (p, q)-forms for a complex manifold X with a simple normal crossing divisor D has not a canonical real structure. There is a natural question about this work, whether the logarithmic Dolbeault complex AD
X is a ∂∂-complex on a more
general class of complex manifolds, such as the complete K¨ahler manifolds. In general, this answer to this question is negative due to [KT99, KT00], where H. Kazama–S. Takayama gave the counter-examples, toroidal groups of non-Hausdorff type and a simply connected 1-convex K¨ahler manifold, to the rather special Kodaira’s ∂∂-lemma on weakly pseudoconvex K¨ahler manifolds. Although of this, we apply the structure theory of double complex and L2 cohomology, to obtain a positive answer to this question on a strongly pseudoconvex K¨ahler manifold, a solution of ∂-equation for logarithmic forms with values in unitary local system and an injectivity result for the Dolbeault cohomology of logarithmic forms on K¨ahler manifolds.
Much based on J. Kohn’s remarkable solution of ∂-Neumann problem [Ko63], we obtain the first result:
Proposition 1.7 (= Proposition 4.4). If X ⊂ X0 is a strongly pseudoconvex K¨ahler manifold and moreover the harmonic spacesHp,q
∂ =H p,q
∂ for each p, q in the respective
domains of the Laplacians of ∂ and ∂, then the ∂∂-lemma holds on X.
canonical extension of V . We study the double complex ADX,V= (A0,•(X, Ω•X(log D) ⊗V), ∇, ∂). The following theorem shows that for any p, q, it holds that
im ∇p−1,q∩ ker ∂p,q= im ∇p−1,q∩ im ∂p,q, i.e., Ap,q= Bp,q.
Theorem 1.8 (= Theorem 4.7). On a compact K¨ahler manifold X, let D be a normal crossing divisor, V a locally free sheaf and
∇ :V → Ω1X(log D) ⊗V
an integrable logarithmic connection. Assume that for all component Di of D, the
real part of all eigenvalues of ResDi(∇) lies in [0, 1). Assume moreover that the local
constant system
V = ker(∇ :V|U−→ Ω1U⊗V|U)
is unitary for U = X − D. Then for any α ∈ A0,q(X, ΩpX(log D) ⊗V) with ∂∇α = 0, there exists a solution x ∈ A0,q−1(X, Ωp+1X (log D) ⊗V) such that ∂x = ∇α.
The case of logarithmic forms with trivial holomorphic vector bundle was origi-nally obtained by Liu–Wan–the first author in [LRW19] with quite different methods by combining de Rham–Kodaira’s current theory [deK50], Hodge theory on K¨ahler manifolds and a trick of Noguchi [No95].
Although ADX,V has no canonical real structure, Corollary 2.19 shows that A = B still implies that the Fr¨olicher (or Hodge to de Rham) spectral sequence degenerates at E1-page, a very important result in algebraic geometry, thanks to [Ti86, Theorem
D.2] and also [De71].
Corollary 1.9 (= Corollary 4.8). With the above notations, the spectral sequence associated to the Hodge filtration
E1p,q= Hq(X, ΩpX(log D) ⊗V) ⇒ Hp+q(U, V ) degenerates at E1-page.
Finally, we also study the pull-back induced by a holomorphic proper surjective map between two complex manifolds, which can be dated back to [Ap56, De68, GR70, We74]. We prove that the induced pull-backs between respective Varouchas’ cohomol-ogy groups by a surjective holomorphic map between connected complex manifolds with the same dimension are split injections in Corollary 5.3.
Notations and conventions 1.10. Unless specially mentioned, we will obey the fol-lowing notations and conventions. All double complexes are assumed to be bounded and a ‘general’ bounded double complex means that it possibly has no real structure. We identify two double complexes as they are E0-isomorphic. All complex
mani-folds are connected and n-dimensional and the upper-indices p, q on these complex manifolds will take values in the integers {0, · · · , n}.
author’s visit to Institut Fourier, Universit´e Grenoble Alpes from March 2019 and he would like to express his gratitude to the institute for their hospitality and the wonderful work environment during his visit.
2. Decomposing double complexes
We will list all kinds of indecomposable double complexes, i.e., squares and zigzags, and calculate each case and make up tables to compare their dimensional relationships, to characterize the (weak) ∂1∂2-complex and the degeneration of Fr¨olicher spectral
sequences.
2.1. Double complexes and cohomology groups. We will study the ∂∂-lemma in the framework of the general double complex and adopt the following notations throughout the whole paper.
By a general double complex (sometimes also called bicomplex ) over the field K, we mean a bigraded K-vector space
K = M
p,q∈Z
Kp,q
with two endomorphisms ∂1, ∂2 of bidegrees (1, 0) and (0, 1) that satisfy
∂1◦ ∂1 = 0, ∂2◦ ∂2 = 0
and anticommute, i.e.,
∂1◦ ∂2+ ∂2◦ ∂1 = 0.
Set d = ∂1+∂2. One writes ∂1p,qfor the map from Kp,qto Kp+1,qinduced by restriction
and similarly for ∂2p,q, and always assumes the double complexes to be bounded, i.e., Kp,q = 0 for almost all (p, q) ∈ Z2. As for the double complex of (p, q)-forms on complex manifolds, we often consider the differentials
∂ and ∂,
as the endomorphisms ∂1 and ∂2, respectively.
Definition 2.1. We say that a (nonzero) double complex K•,• has a real structure if there is a conjugation-antilinear involution σ on K•,•, satisfying
σKp,q= Kq,p and σ∂1σ = ∂2.
In particular, as for the double complex on a complex manifold, one just takes σ as the complex conjugation and calls the real structure canonical if it exists. By a morphism of double complexes, we mean a K-linear morphism of the underlying vector spaces, compatible with the bigrading and two endomorphisms ∂1, ∂2.
Example 2.2 (Dolbeault complex). Consider a complex manifold X and the space Ap,qX of the complex smooth differential (p, q)-forms, associated to which is a natural bounded double complex with a real structure
AX = (A•,•X , ∂, ∂).
We call AX as the Dolbeault complex of X.
By a morphism of double complexes, we mean a K-linear morphism of the under-lying vector spaces, compatible with the bigrading and two endomorphisms ∂1, ∂2.
are defined by H∂•,• 2 (K) := ker ∂2 im ∂2 and HdR•,•(K) := ker d im d, and similarly for H∂•,•
1 (K), while the Bott–Chern and Aeppli cohomology groups are
defined as HBC•,•(K) := ker ∂1∩ ker ∂2 im ∂1∂2 and HA•,•(K) := ker ∂1∂2 im ∂1+ im ∂2 , respectively.
For every p, q, we will also consider the following magical quotient spaces:
(2.1) Ap,q= Ap,q(K) := im ∂ p−1,q 1 ∩ im ∂ p,q−1 2 im ∂1∂2p−1,q−1 , Bp,q= Bp,q(K) := im ∂ p−1,q 1 ∩ ker ∂ p,q 2 im ∂1∂2p−1,q−1 , Cp,q= Cp,q(K) := ker ∂1∂ p,q 2 im ∂1p−1,q+ ker ∂2p,q, Dp,q= Dp,q(K) := ker ∂ p,q 1 ∩ im ∂ p,q−1 2 im ∂1∂2p−1,q−1 , Ep,q= Ep,q(K) := ker ∂1∂ p,q 2 ker ∂1p,q+ im ∂2p,q−1, Fp,q= Fp,q(K) := ker ∂1∂ p,q 2 ker ∂1p,q+ ker ∂p,q2 ,
which we call Varouchas’ cohomology groups, and whose dimensions ap,q, bp,q, cp,q, dp,q, ep,q, fp,q are called Varouchas’ numbers, cf. [Va86]. It is easy to see that Varouchas’
cohomol-ogy groups are closely related to the solvability to various kinds of ∂-equations. The equivalence in the following two examples is often used later.
Example 2.3 (A=B). The following two statements are equivalent: For any p, q, (1) Ap,q= Bp,q;
(2) for any α ∈ Kp,q with ∂2∂1α = 0, there exists a solution x ∈ Kp+1,q−1 for
∂2x = ∂1α.
Example 2.4 (B=0). The following two statements are equivalent: For any p, q, (1) Bp,q= 0;
(2) for any α ∈ Kp−1,q with ∂2∂1α = 0, there exists a solution x ∈ Kp−1,q−1 for
∂1∂2x = ∂1α.
These spaces are related by the following exact sequences, where all maps are induced by the identity (for more details, one may refer to [ATo13, Va86, Al19]):
0 → Ap,q→ Bp,q → H∂p,q 2 → H p,q A → C p,q→ 0, (2.2) 0 → Dp,q→ HBCp,q→H∂p,q 2 → E p,q → Fp,q → 0, 0 → Ap,q→ Dp,q→ H∂p,q 1 →H p,q A → E p,q→ 0, 0 → Bp,q→ HBCp,q→H∂p,q 1 → C p,q→ Fp,q→ 0.
Definition 2.5. A (nonzero) double complex K is called indecomposable if there is no nontrivial decomposition K = K1⊕ K2.
Example 2.6. The following double complexes over a field K are indecomposable: square Kp−1,q Kp,q Kp−1,q−1 Kp,q−1 ∼ = ∼ = ∼ = ∼ = and zigzags Kp,q , Kp,q+1 Kp,q ∼ = , K p,q ∼= Kp+1,q , Kp,q+1 Kp,q Kp,q+1 ∼ = ∼ = , Kp−1,q Kp,q Kp,q−1 ∼ = ∼ = , Kp,q Kp+1,q Kp+1,q−1 Kp+2,q−1 ∼ = ∼ = ∼= , · · · .
The following theorem plays an essential role in this article. As mentioned in [St18b], Theorem 2.7 usually quotes an unpublished note of M. Khovanov, now written as [KQ20, Section 2], cf. also [Ag15, Section 1], [St18b, Theorem A].
Theorem 2.7. For any bounded double complex K over a field K there exist unique cardinal numbers multS(K) and a (non-functorial) isomorphism K ∼=LSS⊕multS(K),
where S runs over squares and zigzags.
For any double complex K = (K•,•, ∂1, ∂2) and Kd =
L
p+q=dKp,q, we set the
dimensional notations as the dimensions are finite: bd:= dim HdRd (K), h p,q ∂2 := dim H p,q ∂2 (K), h d ∂2 := Σp+q=ddim H p,q ∂2 (K); hp,q∂ 1 := dim H p,q ∂1 (K), h d ∂1 := Σp+q=ddim H p,q ∂1 (K), h p,q BC := dim H p,q BC(K);
hdBC := Σp+q=ddim HBCp,q(K), hp,qA := dim HAp,q(K), hdA:= Σp+q=ddim HAp,q(K),
while ap,q, bp,q, cp,q, dp,q, ep,q, fp,qrepresent the dimensions of Ap,q, Bp,q, Cp,q, Dp,q, Ep,q, Fp,q, respectively.
Table 1. The cohomology dimensions of Type 1 h∂2 h∂1 hBC hA a b c d e f K1 0 0 0 0 0 0 0 0 0 0 K2 0 0 0 0 0 0 0 0 0 0 K3 0 0 0 0 0 0 0 0 0 0 K4 0 0 0 0 0 0 0 0 0 0
Type 2 (upper-right even zigzags).
Type 2.1. upper-right even zigzags with lengths ≥ 4
K1 K2 K5 K3 K4. ∂1 ∂2 ∂1 ··· ∂2 ∂1
Here, as in all the following examples, the length of a zigzag is the number of its vertices and we assume that K1 ∈ Kd and K1, K2, K3, K4 represent all the different
types of spaces.
Table 2. The cohomology dimensions of Type 2.1 h∂2 h∂1 hBC hA a b c d e f
K1 1 0 0 1 0 0 0 0 1 0 hBC+ hA= h∂1 + h∂2
K2 0 0 1 0 1 1 0 1 0 0 hBC+ hA> h∂1 + h∂2
K3 0 0 0 1 0 0 1 0 1 1 hBC+ hA> h∂1 + h∂2
K4 1 0 1 0 0 1 0 0 0 0 hBC+ hA= h∂1 + h∂2
Table 3. The dimensional relationship of Type 2.1
d d + 1 bd= 0 bd+1= 0 bd<Pp+q=dh p,q ∂2 bd+1< P p+q=d+1h p,q ∂2 bd= P p+q=dh p,q ∂1 bd+1= P p+q=d+1h p,q ∂1 hdBC+ hdA> 2bd hd+1BC + hd+1A > 2bd+1 hdBC < hdA hd+1BC > hd+1A hdBC = bd hd+1BC > bd+1 hdA> bd hd+1A = bd+1
Type 2.2. right length 2 zigzag
K1 K2.
∂1
Table 4. The cohomology dimensions of Type 2.2 h∂2 h∂1 hBC hA a b c d e f
K1 1 0 0 1 0 0 0 0 1 0 hBC + hA= h∂1+ h∂2
K2 1 0 1 0 0 1 0 0 0 0 hBC + hA= h∂1+ h∂2
Table 5. The dimensional relationship of Type 2.2
d d + 1 bd= 0 bd+1= 0 bd<P p+q=dh p,q ∂2 bd+1< P p+q=d+1h p,q ∂2 bd=P p+q=dh p,q ∂1 bd+1= P p+q=d+1h p,q ∂1 hd BC+ hdA> 2bd h d+1 BC + h d+1 A > 2bd+1 hd BC < hdA h d+1 BC > h d+1 A hd BC= bd hd+1BC > bd+1 hdA> bd hd+1A = bd+1
Type 3 (lower-left even zigzags).
Type 3.1. lower-left even zigzags with lengths ≥ 4
K1 K2 K3 K5 K4. ∂1 ∂2 ∂2 ∂1 ··· ∂2
Suppose that K2 ∈ Kd and K1, K2, K3, K4 represent all the different types of
spaces.
Table 6. The cohomology dimensions of Type 3.1 h∂2 h∂1 hBC hA a b c d e f
K1 0 1 1 0 0 0 0 1 0 0 hBC + hA= h∂1+ h∂2
K2 0 0 0 1 0 0 1 0 1 1 hBC + hA> h∂1+ h∂2
K3 0 0 1 0 1 1 0 1 0 0 hBC + hA> h∂1+ h∂2
Table 7. The dimensional relationships of Type 3.1 d d + 1 bd= 0 bd+1= 0 bd=Pp+q=dh p,q ∂2 bd+1= P p+q=d+1h p,q ∂2 bd<Pp+q=dh p,q ∂1 bd+1< P p+q=d+1h p,q ∂1 hdBC+ hdA> 2bd hd+1BC + hd+1A > 2bd+1 hd BC < hdA h d+1 BC > h d+1 A hd BC= bd h d+1 BC > bd+1 hd A> bd hd+1A = bd+1
Type 3.2. upper length 2 zigzag
K1
K2. ∂2
Assume that K2 ∈ Kd and K1, K2 represent all the different types of spaces.
Table 8. The cohomology dimensions of Type 3.2 h∂2 h∂1 hBC hA a b c d e f
K1 0 1 1 0 0 0 0 1 0 0 hBC+ hA= h∂1 + h∂2
K2 0 1 0 1 0 0 1 0 0 0 hBC+ hA= h∂1 + h∂2
Table 9. The dimensional relationship of Type 3.2
d d + 1 bd= 0 bd+1= 0 bd=Pp+q=dh p,q ∂2 bd+1= P p+q=d+1h p,q ∂2 bd< P p+q=dh p,q ∂1 bd+1< P p+q=d+1h p,q ∂1 hdBC+ hdA> 2bd hd+1BC + hd+1A > 2bd+1 hdBC < hdA hd+1BC > hd+1A hdBC = bd hd+1BC > bd+1 hdA> bd hd+1A = bd+1
Type 4 (upper-right odd zigzags).
K1 K2 K3 K4. ∂1 ∂2 ∂1 ··· ∂2
One supposes that K1 ∈ Kd and K1, K2, K3, K4 represent all the different types of
spaces.
Table 10. The cohomology dimensions of Type 4.1 h∂2 h∂1 hBC hA a b c d e f
K1 1 0 0 1 0 0 0 0 1 0 hBC+ hA= h∂1 + h∂2
K2 0 0 1 0 1 1 0 1 0 0 hBC+ hA> h∂1 + h∂2
K3 0 0 0 1 0 0 1 0 1 1 hBC+ hA> h∂1 + h∂2
K4 0 1 0 1 0 0 1 0 0 0 hBC+ hA= h∂1 + h∂2
Table 11. The dimensional relationship of Type 4.1
d d + 1 bd= 1 bd+1= 0 bd=Pp+q=dh p,q ∂2 bd+1= P p+q=d+1h p,q ∂2 bd=Pp+q=dh p,q ∂1 bd+1= P p+q=d+1h p,q ∂1 hdBC+ hdA> 2bd hd+1BC + h d+1 A > 2bd+1 hdBC < hdA hd+1BC > hd+1A hdBC < bd hd+1BC > bd+1 hdA> bd hd+1A = bd+1
Type 4.2. upper-right length 3 zigzag
K1 K2
K3. ∂1
∂2
One assumes that K1 ∈ Kdand K1, K2, K3represent all the different types of spaces.
Table 12. The cohomology dimensions of Type 4.2 h∂2 h∂1 hBC hA a b c d e f
K1 1 0 0 1 0 0 0 0 1 0 hBC+ hA= h∂1 + h∂2
K2 0 0 1 0 1 1 0 1 0 0 hBC+ hA> h∂1 + h∂2
Table 13. The dimensional relationship of Type 4.2 d d + 1 bd= 1 bd+1= 0 bd= P p+q=dh p,q ∂2 bd+1= P p+q=d+1h p,q ∂2 bd=Pp+q=dhp,q∂1 bd+1=Pp+q=d+1hp,q∂1 hdBC+ hdA= 2bd hd+1BC + h d+1 A > 2bd+1 hdBC < hdA hd+1BC > hd+1A hdBC < bd hd+1BC > bd+1 hdA> bd hd+1A = bd+1
Type 5 (lower-left odd zigzags).
Type 5.1. lower-left odd zigzags with lengths ≥ 5
K1 K2 K3 K5 K6 K4. ∂1 ∂2 ··· ∂2 ∂1
Here we assume that K2∈ Kdand K1, K2, K3, K4 represent all the different types of
spaces.
Table 14. The cohomology dimensions of Type 5.1
h∂2 h∂1 hBC hA a b c d e f
K1 0 1 1 0 0 0 0 1 0 0 hBC+ hA= h∂1+ h∂2
K2 0 0 0 1 0 0 1 0 1 1 hBC+ hA> h∂1+ h∂2
K3 0 0 1 0 1 1 0 1 0 0 hBC+ hA> h∂1+ h∂2
K4 1 0 1 0 0 1 0 0 0 0 hBC+ hA= h∂1+ h∂2
Table 15. The dimensional relationship of Type 5.1
Type 5.2. lower-left length 3 zigzag
K1
K2 K3.
∂1
∂2
Here we assume that K2 ∈ Kd and K1, K2, K3 represent all the different types of
spaces.
Table 16. The cohomology dimensions of Type 5.2 h∂2 h∂1 hBC hA a b c d e f
K1 0 1 1 0 0 0 0 1 0 0 hBC + hA= h∂1+ h∂2
K2 0 0 0 1 0 0 1 0 1 1 hBC + hA> h∂1+ h∂2
K3 1 0 1 0 0 1 0 0 0 0 hBC + hA= h∂1+ h∂2
Table 17. The dimensional relationship of Type 5.2
d d + 1 bd= 0 bd+1= 1 bd= P p+q=dh p,q ∂2 bd+1= P p+q=d+1h p,q ∂2 bd=Pp+q=dh p,q ∂1 bd+1= P p+q=d+1h p,q ∂1 hdBC+ hdA> 2bd hd+1BC + h d+1 A > 2bd+1 hdBC < hdA hd+1BC > hd+1A hd BC = bd hd+1BC > bd+1 hdA> bd hd+1A < bd+1 Type 6 (dot). K1• Set K1 ∈ Kd.
Table 18. The cohomology dimensions of Type 6 h∂2 h∂1 hBC hA a b c d e f
K1 1 1 1 1 0 0 0 0 0 0 hBC + hA= h∂1+ h∂2
From the point of decomposing double complexes, we can easily reprove some classical results.
where bk(·) denotes the k-th Betti number.
Proof. One just needs to check all the cases of indecomposable double complexes.
One fundamental example is the Dolbeault complex (Ap,qX , ∂, ∂) of (p, q)-forms on the complex manifold X. If f : X → Y is a holomorphic morphism between X and Y , then the pull-back f∗ : (A•,•Y , ∂, ∂) → (A•,•X , ∂, ∂) induced by f is the corresponding morphism of Dolbeault complexes. As a double complex has a real structure, it has some kind of symmetry. Crucially, if S•,• is an indecomposable complex of Type
2.1, 2.2, 3.1 or 3.2, then σS•,• becomes the indecomposable complex of Type
3.1, 3.2, 2.1 or 2.2,
respectively, while σ maps the indecomposable complexes of Types 4.1, 4.2, 5.1, 5.2,
to their respective types.
2.3. Fr¨olicher spectral sequence and ∂1∂2-complex. A general double complex
has two natural filtrations by columns and rows F1• :=M
p≥•
Kp,q and F2• :=M
q≥•
Kp,q,
on the total complex, which induce a filtration on the total cohomology HdRp+q(K) by
FipHdRp+q(K) = im ker d ∩ F p i Kp+q im d ∩ FipKp+q −→ H p+q dR (K) .
We will denote by Fi these filtrations and call them Hodge filtrations. The
filtra-tions by columns and rows also induce the converging Fr¨olicher spectral sequences, which compute the Hodge filtrations on the total cohomology from the column or row cohomology of the double complex:
(S1) Ep,q1 :=1E1p,q= H p,q ∂2 (K) =⇒ HdRp+q(K), F1 ; (S2) 2E1p,q = H p,q ∂1 (K) =⇒ HdRp+q(K), F2 .
If the double complex has a real structure, we can investigate the Hodge structure on the total cohomology. Let Kp,q := σKp,q and we may identify the filtration
F1pKdwith the second Fp
1Kd= F p
2Kd, which leads to the identification F p
1HdRd (K) =
F2pHd
dR(K). We say that the filtration (F p
1HdRd (K)) is a Hodge filtration of weight d
if
HdRd = M
p+q=d
F1pHdRd (K) ∩ F1qHd dR(K).
There exists another equivalent expression for Hodge filtration. The filtration (F1pHdRd (K)) is a Hodge filtration of weight d if and only if
HdRd = F1pHdRd (K) ⊕ F1qHdRd (K)
for every (p, q) with p + q = d + 1. Moreover, in this case there is a canonical isomorphism
for every (p, q) with p+q = d. For convenience, we use F1p, F1qto represent F1pHdRd (K), F1qHdRd (K), respectively, and set
Vp,q := F1pHdRd (K) ∩ F1qHd dR(K).
For more details, we refer the reader to [ASTT, § 1.5].
Theorem 2.9 ([DGMS75, ATo13, CS15, CHT15, ATa17, PSU20, Al19]). For any bounded double complex K = (K•,•, ∂1, ∂2) with a real structure, the following
char-acterizations are equivalent: for any p, q, k,
(1) K is a sum of double complexes of Types of single dots and squares; (2) im ∂1p−1,q∩ ker ∂2p,q = im ∂1∂2p−1,q−1, i.e., Bp,q= 0; (3) ker ∂1∂2p,q = im ∂ p−1,q 1 + ker ∂ p,q 2 , i.e., Cp,q= 0; (4) ker ∂1p,q∩ im ∂2p,q−1= im ∂1∂2p−1,q−1, i.e., Dp,q= 0; (5) ker ∂1∂2p,q = ker ∂ p,q 1 + im ∂ p,q−1 2 , i.e., Ep,q = 0;
(6) all Varouchas’ cohomology groups in (2.1) vanish, i.e., Ap,q= Bp,q = Cp,q=
Dp,q= Ep,q= Fp,q= 0; (7) hkBC + hkA= 2bk;
(8) hkBC = hkA; (9) hkBC = bk;
(10) hkA= bk;
(11) The Fr¨olicher spectral sequence defined by the filtration F1• :=Lp≥•Kp,q
de-generates at E1-page and the induced filtration on Hk gives a Hodge structure
of weight k, i.e., Hk=L
p+q=kVp,q for every k ≥ 0;
Here all the dimensions of the various cohomology groups are assumed to be finite. Proof. We first prove the equivalence of (2) and (1), while (1) ⇒ (2) is easy. Assume that (2) holds. Since any zigzags of Types
2.1, 2.2, 3.1, 4.1, 4.2, 5.1, 5.2
have a vector space Kp,qwith bp,q> 0, these zigzags can not exist. Moreover, that the
double complex has a real structure means that the zigzag of Type 3.2 in the complex decomposition also vanishes since if S•,• is an indecomposable complex of Type 3.2, then σS•,• is an indecomposable complex of Type 2.2. The proof for equivalence of (3), (4), (5), (7) to (1) is similar.
Then let us prove the equivalence of (8) and (1) while (1) ⇒ (8) is obvious. Now we assume that (8) holds. If there is a zigzag with length more than 2, the zigzag double complex is nonzero only in degrees d and d + 1. Since hdBC < hdA and hd+1BC > hd+1A always hold, there must exist a zigzag with degrees d − 1 and d to ensure the equality hdBC = hdA. By the same token, the zigzags with degrees d − 2 and d − 1 also exist to ensure the validity of equality hd−1BC = hd−1A . We can derive the contradiction because the double complex is bounded. The proof for the equivalence of (9),(10) to (1) is similar.
Finally, we consider the equivalence between (11) and (1). In fact, the even zigzags are the obstacle for the degeneration of Fr¨olicher spectral sequence at E1-page. In
[St18b, Corollary 7], J. Stelzig proved that the Fr¨olicher spectral sequences degenerate at Er-page if and only if, only shapes of even zigzags of lengths strictly less than 2r
have nonzero multiplicity. The odd zigzags with lengths ≥ 3 invalidate the existence of Hodge structure of weight k on Hk hold for every k ≥ 0, cf. also [St18b, Corollary
It is easy to see from the proof of Theorem 2.9 that Theorem 2.9.(1) plays an essential role and thus we give:
Definition 2.10. We say that a general bounded double complex is a ∂1∂2-complex
as it is isomorphic to a sum of double complexes of Types of single dots and squares; in particular, if the Dolbeault complex on a complex manifold satisfies this, it will be called a ∂∂-complex and the complex manifold is called a ∂∂-manifold.
Remark 2.11. Besides the equivalent characterization for the ∂1∂2-complex in [DGMS75],
D. Angella–A. Tomassini [ATo13] prove a Fr¨olicher-type inequality in Corollary 2.12.(2) between Bott–Chern, Aeppli and de Rham cohomologies, and show that the equality (7) holds if and only if the complex manifold X satisfies the ∂∂-lemma. In [ATa17], D. Angella–N. Tardini get the similar equality (8) between the Bott–Chern and Aeppli cohomologies. In [CHT15], T-W. Chen–C-I. Ho–J-H. Teh define Aeppli and Bott– Chern cohomologies for bi-generalized complex manifolds and show that the validity of ∂1∂2-lemma is equivalent to the same dimension of several cohomology groups. In
[CS15], K. Chan–Y-H. Suen prove another Fr¨olicher-type inequality for a compact generalized complex manifold and show that the equality holds if and only if the manifold satisfies the generalized ∂1∂2-lemma. For the generalization to the higher
pages of the Fr¨olicher spectral sequence, one may refer [PSU20].
We now return to the general double complex. After decomposing the double complex, we can easily find quantitative and qualitative properties of cohomology groups or vector spaces. For instance, we can easily find the Fr¨olicher-type inequalities via the tables above, also see [ATa17, ATo13, CS15, CHT15].
Corollary 2.12. For any general double complex, its associated cohomology groups in (2.1) satisfy the (in)equalities: for any p, q, k,
(1) hp,qBC + hp,qA = hp,q∂ 2 + h p,q ∂1 + a p,q+ fp,q; (2) hkBC + hkA≥ 2bk; (3) bk ≤ Pp+q=kh p,q
∂2 , where the equality holds only when the indecomposable
double complexes of Type 2 do not exist in the complex decomposition, or equivalently, the Fr¨olicher spectral sequence (S1) degenerates at 1E1.
(4) bk ≤Pp+q=kh p,q
∂1, where the equality holds only when indecomposable double
complexes of Type 3 do not exist in the complex decomposition, or equivalently, the Fr¨olicher spectral sequence (S2) degenerates at2E1;
(5) ap,q ≤ bp,q, where the equality holds only when indecomposable double
com-plexes of Types 2, 5 do not exist in the complex decomposition;
(6) fp,q ≤ ep,q, where the equality holds only when indecomposable double
com-plexes of Types 2, 4 do not exist in the complex decomposition;
(7) ap,q ≤ dp,q, where the equality holds only when indecomposable double
com-plexes of Types 3, 5 do not exist in the complex decomposition;
(8) fp,q ≤ cp,q, where the equality holds only when indecomposable double
com-plexes of Types 3, 4 do not exist in the complex decomposition.
It seems that Corollaries 2.12.(5)–2.12.(8) are new. As in Theorem 2.9, when the Fr¨olicher-type inequality hkBC + hkA ≥ 2bk becomes an equality, the double complex turns out to be a ∂1∂2-complex. Besides the characterizations in Theorem 2.9, we
find more new characterizations of ∂1∂2-complex:
Theorem 2.13. For any general bounded double complex K = (K•,•, ∂1, ∂2), the
(1) K is a ∂1∂2-complex; (2) ap,q= bp,q, cp,q = fp,q; (3) ap,q= dp,q, ep,q = fp,q; (4) ap,q= bp,q, bk=Pp+q=kh∂p,q1, hp,q∂2 = hp,q∂1; (5) cp,q= fp,q, b k =Pp+q=kh p,q ∂2, h p,q ∂2 = h p,q ∂1 ; (6) ep,q= fp,q, bk=Pp+q=kh p,q ∂1 , h p,q ∂2 = h p,q ∂1; (7) ap,q= dp,q, bk= P p+q=kh p,q ∂2, h p,q ∂2 = h p,q ∂1; (8) cp,q= fp,q, bk =Pp+q=kh∂p,q2, hp,q∂2 = hp,q∂1 .
Moreover, the following properties are also equivalent to either of the above properties when the double complex K has a real structure: for any p, q,
(9) ap,q= bp,q, ep,q= fp,q; (10) ap,q= dp,q, cp,q= fp,q; (11) ap,q= bp,q, hp,q∂ 2 = h q,p ∂2; (12) cp,q= fp,q, hp,q∂ 2 = h q,p ∂2; (13) ep,q= fp,q, hp,q∂ 2 = h q,p ∂2; (14) ap,q= dp,q, hp,q∂ 2 = h q,p ∂2; (15) cp,q= fp,q, hp,q∂ 2 = h q,p ∂2.
Proof. It is a direct consequence of Corollary 2.12 and the explicit calculations of the cohomology dimensions in the tables for the odd zigzags.
Notice that the property hp,q∂
2 = h
p,q
∂1 can exclude the existence of Type 4 (resp.
Type 5) in the complex decomposition if we have excluded the existence of Types 2, 3 and Type 5 (resp. Type 4) since the ends of the zigzags in Type 4 (resp. Type 5) have the different Dolbeault and ∂1-cohomology groups, although of the possible
existence of Type 6 on the ends of the zigzags in Type 4 (resp. Type 5).
And similarly, when the double complex K has a real structure, the property hp,q∂
2 = h
q,p
∂2 can exclude the existence of Type 4 (resp. Type 5) in the complex
decomposition if we have excluded the existence of Types 2, 3 and Type 5 (resp. Type 4) since the ends of the zigzags in Type 4 (resp. Type 5) have the different Dolbeault cohomology groups, although of the possible existence of Type 6 on the
ends of the zigzags in Type 4 (resp. Type 5).
Remark 2.14. Angella–Suwa–Tardini–Tomassini also proved in [ASTT, § 1.5] that Theorem 2.9.(11) is equivalent to the following property: The Hodge symmetry and Hodge decomposition for each p, q, k,
(2.3) H∂p,q 2 ∼ = H∂q,p2 and Hk∼= M p+q=k H∂p,q 2 hold, where H∂q,p
2 denotes the vector space conjugate to H
q,p
∂2 , and additionally,
(H1) every class in H∂p,q2 has a d-closed representative α with ∂2α = 0 and ∂1α = 0,
i.e., dα = 0. Moreover, the assignment α 7→ ¯α induces the first isomorphism in (2.3), or equivalently, the assignment α 7→ α induces an isomorphism H∂p,q
2
∼ = H∂p,q
1 .
(H2) every class in HdRk has a representative β = Pp+q=kβp,q, where βp,q ∈ Kp,q
is d-closed. Moreover, the assignment β 7→ (βp,q)p+q=k induces the second
isomorphism in (2.3).
It seems that only additional (H1) is already equivalent to Theorem 2.13.(1), i.e., K
(2.2) and the condition for the representative in (H1) give rise to ep,q= fp,q and the
isomorphism in (H1) gives hp,q∂2 = hq,p∂2 or ap,q= bp,q.
Remark 2.15. Our argument for Theorem 2.13 can’t give rise to the equivalent property bk= X p+q=k hp,q∂ 1 = X p+q=k hp,q∂ 2 and h p,q ∂2 = h p,q ∂1
since we can’t exclude the simultaneous existence of Types 4 and 5, even when the double complex K has a real structure. Actually, there is an algebraic counter-example due to Angella–Tomassini at the end of [ATo13] and also a geometric one by M. Ceballos–A. Otal–L. Ugarte–R. Villacampa [COUV16, Proposition 4.3].
Other than the ∂1∂2-complex, the simplest double complex is the one that can be
decomposed into the direct sum of single dots and zigzags with lengths less than 2: Proposition 2.16. For any general bounded double complex K = (K•,•, ∂1, ∂2), the
following properties are equivalent:
(1) K is isomorphic to a sum of double complexes of the types: square and zigzags with lengths ≤ 2;
(2) the associated cohomologies to K satisfy hp,qBC+ hp,qA = hp,q∂
2 + h
p,q
∂1 for any p, q;
(3) the associated cohomologies to K satisfy Ap,q = 0 and Fp,q= 0 for any p, q. Proof. One can prove these equivalent relationships by checking all the
indecompos-able double complexes one by one.
A bounded double complex is called a weak ∂1∂2-complex if it is isomorphic to a
sum of double complexes of the following types: square and zigzags with lengths ≤ 2; in particular, if the Dolbeault complex on a complex manifold satisfies this, it will be called a weak ∂∂-complex.
In [PSU20, Theorem 2.15]v2, Popovici–Stelzig–Ugarte study the so-called
‘page-r-∂1∂2-property’ for r ≥ 0 and give the analogous characterization of Proposition
2.16.(2).
If the double complex has a real structure, then B = 0 means that the complex is a ∂1∂2-complex. However, in the case of double complex without a real structure,
they are not equivalent.
Lemma 2.17. For any bounded double complex K = (K•,•, ∂1, ∂2), the following
properties are equivalent: for all p, q,
(1) im ∂1p−1,q∩ ker ∂2p,q = im ∂1∂2p−1,q−1, i.e., Bp,q= 0;
(2) K is the direct sum of the types: 1, 3.2, 6.
Moreover, one also has the characterization of A = B, which plays very important roles in Section 4.
Lemma 2.18. For any bounded double complex K = (K•,•, ∂1, ∂2), the following
properties are equivalent: for all p, q,
(1) im ∂1p−1,q∩ ker ∂2p,q = im ∂1p−1,q∩ im ∂2p,q−1, i.e., Ap,q= Bp,q; (2) K is the direct sum of the types: 1, 3.1, 3.2, 4.1, 4.2, 6.
Proof. In the indecomposable double complexes of Types 2.1, 2.2, 5.1, 5.2, there exist p, q such that Ap,q = 0 but Bp,q 6= 0. Moreover, Ap,q ⊆ Bp,q. This completes the
As we have mentioned, in [St18b, Corollary 7], Stelzig proved that the Fr¨olicher spectral sequences degenerate at Er-page if and only if, only shapes of even zigzags of
lengths strictly less than 2r have nonzero multiplicity. Therefore we have the following corollary.
Corollary 2.19. For any bounded double complex, if for all p, q, Ap,q = Bp,q, then the Fr¨olicher spectral sequence (S1) degenerates at E1-page.
Proof. We can also prove this corollary by Lemma 2.18 and Corollary 2.12.(3). Remark 2.20. Corollary 2.19 is more or less equivalent to the idea behind the proof of [LRW19, Theorem 3.2], where they use a general description on the terms in the Fr¨olicher spectral sequence in [CFGU97, Theorems 1 and 3]. Take the Dolbeault complex on a complex manifold X for example. One can interpret
Erp,q ∼= Zrp,q/Bp,qr as follows: Z1p,q= {α ∈ Ap,qX | ∂α = 0}, B1p,q= {α ∈ Ap,qX | α = ∂β, β ∈ Ap,q−1X }. For r ≥ 2, (2.4) Z p,q
r = {αp,q∈ Ap,qX | ∂αp,q = 0, and there exist αp+i,q−i ∈ Ap+i,q−iX
such that ∂αp+i−1,q−i+1+ ∂αp+i,q−i = 0, 1 ≤ i ≤ r − 1},
Bp,qr = {∂βp−1,q+ ∂βp,q−1∈ Ap,qX | there exist βp−i,q+i−1∈ Ap−i,q+i−1X , 2 ≤ i ≤ r − 1,
such that ∂βp−i,q+i−1+ ∂βp−i+1,q+i−2= 0, ∂βp−r+1,q+r−2 = 0},
and the map dr : Erp,q−→ Erp+r,q−r+1 is given by
dr[αp,q] = [∂αp+r−1,q−r+1],
where [αp,q] ∈ Erp,q and αp+r−1,q−r+1 appears in (2.4). To get
di = 0, for any i ≥ 1,
we just need that for any α ∈ Ap,qX with ∂∂α = 0, there exists a solution x ∈ Ap+1,q−1X for ∂x = ∂α, which is equivalent to Ap,q= Bp,q, as in Example 2.3.
3. Morphism and tensor product of double complexes
We introduce E0-, E1-isomorphism and tensor product of double complexes, and
also present a new double complex from deformation of complex structures as an example of E0-isomorphism. The (weak) E1-isomorphism plays an important role in
constructing new ∂∂-complexes in Section 4.
3.1. E0- and E1-isomorphisms of double complexes. By a morphism of double
complexes, we mean a K-linear morphism of the underlying vector spaces, compatible with the bigrade and two endomorphisms ∂1, ∂2. If this morphism is an isomorphism,
we call it an E0-isomorphism, or simply isomorphism, denoted by ∼=. A morphism of
bounded double complexes from K to L is called an E1-isomorphism, if it induces an
isomorphism at E1-pages, i.e., 1E1p,q = H p,q ∂2 (K) = H p,q ∂2 (L) and2E p,q 1 = H p,q ∂1 (K) = H p,q ∂1 (L).
Write K '1 L, if there exists an E1-isomorphism K → L. Then '1 is an equivalence
at Er-pages for 1 ≤ r ≤ ∞. Of course, it induces isomorphisms FpHk(K) → FpHk(L)
of Hodge filtrations of de Rham cohomologies for all k, p, since the Fr¨olicher spectral sequences of bounded double complexes are biregular.
By [St18b, Lemma 8, Proposition 11], Stelzig observes that two complexes are E1
-isomorphic if and only if all zigzags occur with the same multiplicities. According to more careful analysis of each indecomposable double complex and the calculation of various cohomologies in Types 1–6, we can see more about it.
Lemma 3.1 ([St18b, Proposition 11]). For any bounded double complexes K and L, the following properties are equivalent2:
(1) K '1 L;
(2) There exists a morphism from K to L that induces an isomorphism on both Bott–Chern and Aeppli cohomologies;
(3) multS(K) = multS(L) for all shapes S of zigzags in the complex
decomposi-tions. In other words, whenever one picks direct sum decompositions of K and L into squares and zigzags, the same zigzags exactly occur with the same multiplicities.
Proof. As explained by Stelzig in [St18b], the dimensions of Dolbeault (resp. H∂1)
cohomology count the zigzags starting or ending with the horizontal ∂1-arrows (resp.
vertical ∂2-arrows) in a certain bidegree. The shape of zigzags can be decided by its
starting and ending. Hence, the information of both H∂1 and H∂2, i.e., E1-pages,
can decide the multiplicity of all shapes of zigzags. The case of Bott–Chern and Aeppli cohomologies is similar, since the dimensions of Bott–Chern (resp. Aeppli) cohomology count corners, i.e., the zigzags meeting a certain bidegree with incoming (resp. outgoing) arrows. If f is the E1-isomorphism in (1), then this f induces the
isomorphism on both Bott–Chern and Aeppli cohomologies in (2), and the converse
is also true.
As the double complex has a real structure, the isomorphism of Dolbeault cohomol-ogy is equivalent to the E1-isomorphism. Hence in this case the Dolbeault cohomology
determines the other three kinds of cohomologies, i.e., de Rham, Bott–Chern, Aeppli, see [St18a, Lemma 2], [St18b, Lemmata 11, 12] and also [Ag13, AK17] for special cases.
A morphism of two bounded double complexes K, L over K is called a weak E1
-isomorphism, if it induces an isomorphism only at 1E1p,q and write
K '10 L,
if there exists a weak E1-isomorphism K → L.
Lemma 3.2. For any bounded double complexes K and L possibly without real struc-tures, if K '10 L, then multS(K) = multS(L) for all shapes S of zigzags of Types 2.1,
2.2, that is, whenever one picks direct sum decompositions of K and L into squares and zigzags, the above types of zigzags exactly occur with the same multiplicities.
The converse of Lemma 3.2 does not hold necessarily.
Remark 3.3. With the theory of double complex, we can simplify the proof of many problems related with the ∂∂-lemma and Fr¨olicher spectral sequence. Let us introduce an example to be dated back to [De68, § 5.2]. For a compact K¨ahler manifold X,
2The equivalence between (1) and (2) was pointed out by Stelzig in private communication and will
we have two double complexes with real structures. The first one is the Dolbeault complex
AX = (A•,•X , ∂, ∂)
of smooth complex forms on X as in Example 2.2. The second one is the double complex of harmonic forms with respect to the ∂-Laplacian ∂, denoted by
HX = (H•,•∂ (X), ∂, ∂).
It is a double subcomplex of AX. On X, the K¨ahler identity d= 2∂ = 2∂ holds.
In the double complex HX, both two operators ∂ and ∂ are zero morphisms because
of the K¨ahler identities. There is a double complex injection morphism fromHX to
AX. Obviously, these two double complexes AX and HX have the same Dolbeault
cohomology group. The double complex HX can be seen as the direct sum of dots,
i.e., Types 6, and Lemma 3.1 implies that the indecomposable double complexes in the decomposition of AX can only be Types 1 and 6 since the double complex AX
has the real structure. It follows that AX is a ∂∂-complex.
3.2. A new double complex from deformation of complex structures. Let X be a compact complex manifold and π :X → B a holomorphic family of complex manifolds with X0 = X over a connected manifold, which determines a holomorphic
family of integrable Beltrami differentials ϕ(t) ∈ A0,1(X, TX1,0) with the holomorphic tangent bundle TX1,0 of X as t is small. On each fiber Xt:= π−1(t) for t ∈ B, one has
the natural Dolbeault complex
AXt = (A
•,• Xt, ∂t, ∂t)
of (p, q)-forms on Xtwith a real structure. In [ZR15, RZ18], Q. Zhao–the first author
introduce a morphism
eιϕ(t)|ιϕ(t) : Ap,q
X0 → A
p,q Xt,
which is a linear isomorphism as t is arbitrarily small, and its inverse e−ιϕ(t)|−ιϕ(t) as
follows. Here ιϕ(t)denotes the contraction operation with ϕ(t), and similarly for ιϕ(t).
For σ ∈ Ap,qX 0, define eιϕ(t)|ιϕ(t) (σ) = σi1···ip¯j1···¯jq(z) eιϕ(t) dzi1 ∧ · · · ∧ zip ∧eιϕ(t) dzj1 ∧ · · · ∧ dzjq , where σ is locally written as
σ = σi1···ip¯j1···¯jq(z)dzi1 ∧ · · · ∧ dzip∧ dzj1 ∧ · · · ∧ dzjq
and the operators eιϕ(t), eιϕ(t) follow the convention:
e♠= ∞ X k=0 1 k!♠ k,
where ♠kdenotes k-time action of the operator ♠. Since the dimension of X is finite, the summation in the above formulation is always finite.
In general, eιϕ(t)|ιϕ(t) is not commutative with the differential operators ∂, ∂, ∂
t, ∂t,
and thus not a morphism from AX0 to AXt. Here ∂t and ∂t are the natural Cauchy–
Riemann operator and its conjugate on Xt, respectively. The extension morphism
eιϕ(t)|ιϕ(t)
Now we can construct a family of new double complexes with real structures A(t) = (A•,•X , ∂(t), ∂(t)), where ∂(t) = e−ιϕ(t)|−ιϕ(t) ◦ ∂ t◦ e ιϕ(t)|ιϕ(t), ∂(t) = e−ιϕ(t)|−ιϕ(t)◦ ∂ t◦ eιϕ(t) |ι ϕ(t).
By the construction, the morphism
eιϕ(t)|ιϕ(t) : A(t) → A
Xt
is an E0-isomorphism since in addition A •,• X and A
•,•
Xt are isomorphic as complex vector
spaces.
Corollary 3.4 ([Vo02, Proposition 9.20]). Assume that the Fr¨olicher spectral se-quence (S1) of X0 degenerates at E1-page. Then for t near 0, we have
hp,q
∂ (Xt) = h p,q ∂ (X0).
Moreover, the Fr¨olicher spectral sequence (S1) of Xt degenerates at E1-page.
Proof. For the reader’s convenience, we include a proof here. As shown in Cor 2.12.(3), a general compact complex manifold X admits
bk(X) ≤
X
p+q=k
hp,q
∂ (X)
and its equality only when the Fr¨olicher spectral sequence (S1) degenerates at E1 since
the degeneration of the Fr¨olicher spectral sequence of Xt at E1-page is equivalent
to the vanishing of the indecomposable double complexes with Types 2.1 and 2.2 after one decomposes (A•,•(X), ∂, ∂). For t near 0, Kodaira–Spencer’s upper semi-continuity shows hp,q(Xt) ≤ hp,q(X0). As bk(Xt) is independent of t, the inequalities
bk(Xt) ≤ X p+q=k hp,q(Xt) ≤ X p+q=k hp,q(X0) = bk(X0),
are actually equalities, which yields the desired results.
According to Corollary 3.4, we have
Corollary 3.5. If the Fr¨olicher spectral sequence (S1) : E1p,q= H
p,q
∂(0)(A(0)) ⇒ H p+q dR (A(0))
degenerates at E1, then H∂(t)p,q(A(t)) is independent of small t and in particular,
(S1) : E1p,q = H p,q ∂(t)(A(t)) ⇒ H p+q dR (A(t)) degenerates at E1, too.
Here appears a basic but possibly difficult question:
Question 3.6. Can we construct an explicit E1-isomorphism or even E0-isomorphism
between AX = (A•,•X , ∂, ∂) of X and A(t) = (A•,•X , ∂(t), ∂(t)) or between AX and the
Dolbeault complex AXt of Xt when the Dolbeault complex AX is a ∂∂-complex or the
Fr¨olicher spectral sequence (S1) of X degenerates at E1-page?
be nice ones to show the necessity of morphism between double complexes in the definitions of E1- and E0-isomorphisms.
Stelzig also suggests a variant of Question 3.6 whether there is a finite dimensional subcomplex A0X0, E1-isomorphic to AX0, such that the deformations A
0
Xt of this
subcomplex are E1-isomorphic to AXt. For example, this is conjectured to be true
for the left-invariant forms A0X
0 on nilmanifolds and known in many cases, cf. [CF01,
Theorem 1]. If we start with an arbitrary subcomplex A0X0 '1 AX0, what is the
relation between A0X
t and AXt?
3.3. Tensor product of double complexes. Following [Mn20a], we introduce the tensor product of double complexes and its weak ∂1∂2-property in this subsection.
For the simple complexes (K•, ∂K), (L•, ∂L) and an integer m, the double complex
(K•⊗KL•)m is defined as (K•⊗KL•)p,qm = Kp⊗KLq, ∂1p,q = ∂Kp ⊗ 1Lq, ∂p,q 2 = (−1) m+p1 Kp⊗ ∂q L
for any (p, q) ∈ Z2. Clearly, (K•⊗ KL
•)
m is equal to (K•⊗KL •)
0 for even m and to
(K•⊗KL•)1 for odd m. Let (K•,•, ∂1, ∂2) be a double complex. The complex sK•,•
associated to K•,• is defined as (sK•,•)k = M p+q=k Kp,q, dk= X p+q=k ∂1p,q+ X p+q=k ∂2p,q.
Now for double complexes (K•,•, ∂1K, ∂2K) and (L•,•, ∂1L, ∂2L), set Ap,q;r,s = Kp,r⊗KLq,s, ∂1p,q;r,s = (∂1K)p,r⊗ 1Lq,s : Ap,q;r,s → Ap+1,q;r,s, ∂2p,q;r,s = (−1)p+r1Kp,r ⊗ (∂1L)q,s: Ap,q;r,s → Ap,q+1;r,s, ∂3p,q;r,s = (∂2K)p,r⊗ 1Lq,s : Ap,q;r,s → Ap,q;r+1,s, ∂4p,q;r,s = (−1)p+r1Kp,r ⊗ (∂2L) q,s : Ap,q;r,s → Ap,q;r,s+1, for any p, q, s, r.
Definition 3.7. With the above setting, one defines the tensor product ss(K•,•⊗K L•,•) of the double complexes (K•,•, ∂1K, ∂2K) and (L•,•, ∂L1, ∂2L) as
ss(K•,•⊗ L•,•)k,l= M p+q=k r+s=l Ap,q;r,s with Dk,l1 = X p+q=k r+s=l ∂1p,q;r,s+ X p+q=k r+s=l ∂p,q;r,s2 : ss(K•,•⊗KL•,•)k,l → ss(K•,•⊗KL•,•)k+1,l, D2k,l= X p+q=k r+s=l ∂3p,q;r,s+ X p+q=k r+s=l ∂4p,q;r,s: ss(K•,•⊗KL•,•)k,l→ ss(K•,•⊗KL•,•)k,l+1
for any k, l. It is easy to check that ss(K•,•⊗ L•,•) is a double complex.
In general, the tensor product of double complexes are complicated, and we give some simple examples here. Denote by
[], [•], [→], [↑],
Example 3.8. (i) For the two double complexes K : K1 K2, L : L1 L2, ∂K 1 ∂L1 one has K ⊗ L : K1⊗ L1 K1⊗ L2⊕ K2⊗ L1 K2⊗ L2. D1 D1 We denote it by [→] ⊗ [→] = [→→]. (ii) For the two double complexes
K : K1 K2, ∂K 1 L : L1 L2 L3 L4, ∂L 1 ∂L 2 ∂L 1 ∂L 2 one has K ⊗ L : K1⊗ L1 K1⊗ L2⊕ K2⊗ L1 K2⊗ L2 K1⊗ L3 K1⊗ L4⊕ K2⊗ L3 K2⊗ L4. D1 D1 D2 D1 D2 D1 D2
(iii) For the two double complexes
K : K1 K2 K3 K4, ∂K 1 ∂K 2 ∂K 1 ∂K 2 L : L1 L2 L3 L4, ∂L 1 ∂L 2 ∂L 1 ∂L 2 one has K ⊗ L : K1⊗ L1 K1⊗ L2⊕ K2⊗ L1 K2⊗ L2 K1⊗ L3⊕ K3⊗ L1 K1⊗ L4⊕ K2⊗ L3⊕ K3⊗ L2⊕ K4⊗ L1 K2⊗ L4⊕ K4⊗ L2 K1⊗ L3 K3⊗ L4⊕ K4⊗ L3 K4⊗ L4. D1 D1 D2 D1 D2 D1 D2 D2 D1 D2 D1 D2 We denote it by [] ⊗ [] = [].
Now, we present the relationships of cohomologies of tensors of double complexes with the cohomologies of original double complexes.
(3) For any k, l ∈ Z, M
p+q=k r+s=l
Hp(K•,r) ⊗KHq(L•,s) ∼= Hk(ss(K•,•⊗KL•,•)•,l).
Proposition 3.10. If both (K•,•, ∂1K, ∂2K) and (L•,•, ∂1L, ∂2L) are ∂1∂2-complexes, then
so is ss(K•,•⊗ L•,•).
Proof. If both (K•,•, ∂K
1 , ∂2K) and (L•,•, ∂1L, ∂2L) are indecomposable complexes of
Type 6, i.e., the type of dot. It is easy to see that ss(K•,• ⊗ L•,•) is also Type 6.
If (K•,•, ∂K1 , ∂2K) is an indecomposable complex of Type 1, that is square. And (L•,•, ∂1L, ∂2L) is an indecomposable complex of Type 6, i.e., dot. Then it is obvious that ss(K•,•⊗ L•,•) is of the square type.
If both (K•,•, ∂1K, ∂2K) and (L•,•, ∂1L, ∂2L) are indecomposable complexes of Type 1, that is square. According to Lemma 3.9, all cohomologies of ss(K•,•⊗ L•,•) vanish. Then the direct sum decomposition of it are all squares.
In conclusion, the direct sum decomposition of ss(K•,• ⊗ L•,•) consists of only
squares and dots.
Proposition 3.10 implies that if X and Y are ∂∂-manifolds, then so is X × Y , which was given by Meng in [Mn19b, Theorem 1.5]. See a much related result on page-r-∂∂-manifold in [PSU20, Theorem 4.2.1]. Moreover, we get:
Proposition 3.11. If both (K•,•, ∂1K, ∂2K) and (L•,•, ∂1L, ∂2L) are weak ∂1∂2-complexes,
then so is ss(K•,•⊗ L•,•).
Proof. Since K and L are weak ∂1∂2-complexes, their direct sum decomposition
con-sists of squares, dots, zigzags with length 2, which are denoted by [], [•], [→], [↑], respectively.
Case 1: It is easy to see that
[] ⊗ [↑] = [ ] and [] ⊗ [] = [],
as Example 3.8 shows. Again by Lemma 3.9, all cohomologies of the new double complex vanish and then the direct sum decompositions of it are all squares.
Case 2: One has [→] ⊗ [↑] = [].
Case 3: According to Example 3.8, we have [→] ⊗ [→] = [→→]. In this case, let
K : Kp,q ∂ K 1 −→ Kp+1,q and L : Lm,n ∂ L 1 −→ Lm+1,n. We assume that the dimension of all vectors is 1 and then have K ⊗ L:
Kp,q⊗ Lm,n D1
−→ Kp+1,q⊗ Lm,n⊕ Kp,q⊗ Lm+1,n D1
−→ Kp+1,q⊗ Lm+1,n. Since D1 ◦ D1 = 0 and the dimensional relationship, this sequence is D1-exact at
middle and HD•,• 1 = 0, H p+m,q+n D2 = 1, H p+m+1,q+n D2 = 2, H p+m+2,q+n D2 = 1.
Therefore, K ⊗ L is isomorphic to the direct sum of the following two zigzags with length 2:
K1 : Kp+m,q+n→ Kp+m+1,q+n, K2 : Kp+m+1,q+n→ Kp+m+2,q+n.
This is to say, K ⊗ L ∼= K1⊕ K2.
4. ∂∂-lemma, Fr¨olicher spectral sequence and L2 cohomology We will present several applications of the structure theory of double complexes to obtain the validity of the ∂∂-lemma on strongly pseudoconvex K¨ahler manifolds and the degeneration of the Fr¨olicher (or Hodge to de Rham) spectral sequence of logarithmic forms with a unitary local system at E1-page by solving a ∂-equation via
L2 cohomology on K¨ahler manifolds.
4.1. ∂∂-lemma on strongly pseudoconvex K¨ahler manifolds. In [KT99, KT00], H. Kazama–S. Takayama gave the counter-examples, toroidal groups of non-Hausdorff type and a simply connected 1-convex K¨ahler manifold, to the rather special Kodaira’s ∂∂-lemma on weakly pseudoconvex K¨ahler manifolds. Recall that a complex mani-fold X is called weakly pseudoconvex (or weakly 1-complete) if there exists a smooth plurisubharmonic exhaustive function on X. It is easy to see that a compact complex manifold, a strongly 1-convex manifold and every Stein manifold are weakly pseudo-convex. In this subsection, we will present the positive results on the validity of the ∂∂-lemma on strongly pseudoconvex K¨ahler manifolds.
Definition 4.1 ([Ko63, Definition 3.1]). A finite manifold is a pair {X, X0} where X0 is a C∞ manifold and X is an open submanifold of X0 such that:
(1) ¯X, the closure of X is compact.
(2) bX, the boundary of X, is a C∞ submanifold of X0.
(3) If P ∈ bX, there exists a coordinate neighborhood U of P with coordinates x1, · · · , xn such that xm(Q) < 0 if Q ∈ U ∩ X, and xm(Q) > 0 if Q ∈ U ∩ (X0− ¯X).
Definition 4.2 ([Ko63, Definition 4.1]). A finite complex manifold {X, X0} is called strongly pseudoconvex if for every holomorphic coordinate system with domain U , there exists a real C∞ function f on U such that:
(1) f (P ) < 0 if P ∈ X and f (P ) > 0 if P 6∈ ¯X. (2) (df )P 6= 0 if P ∈ bX. (3) If (a1, · · · , an) ∈ Cn, (a1, · · · , an) 6= 0, and ifP ∂f ∂zi(P )ai= 0 for some P ∈ bX, thenP ∂2f ∂zi∂ ¯zj(P )ai¯aj > 0.
Let X ⊂ X0 be a finite complex manifold of complex dimension n and Ap,qX the space of smooth (p, q)-forms on X. Here we denote by Lp,q(2)(X) as the completion of the norm-finite (p, q)-forms in Ap,qX under the canonical inner product. With these notations, one can state Kohn’s remarkable solution of ∂-Neumann problem:
Theorem 4.3 ([Ko63, Theorem 8.9]). If X ⊂ X0is a strongly pseudoconvex hermitian manifold and, if the hermitian metric on X0 is K¨ahler in a neighborhood of bX (by [Ko63, Proposition 4.4] such a metric always exists), then for 0 ≤ p, q ≤ n, there exists a compact operator
Here H(p,q) and ∂ ∗
are the harmonic projection and the adjoint operator of ∂, respec-tively.
Consider the complex vector spaces:
Hp,q ∂ (X) = {φ ∈ Ap,qX : ∂φ = 0} {∂Ap,q−1X } HLp,q(X) = {φ ∈ Dom(∂ p,q ) : ∂φ = 0} {∂Dom(∂p,q−1)} .
Then [Ko63, Theorem 10.7] gives
(4.1) Hp,q ∂ (X) = H p,q L (X) =H p,q ∂ (X), by Theorem 4.3, where Hp,q
∂ (X) is the harmonic space of (p, q)-forms on X with
respect to the ∂-Laplacian ∂ in the domain of ∂. Similarly, denote by H p,q
∂ (X) the
harmonic space with respect to the ∂-Laplacian ∂ in the domain of ∂. Notice that
although X is a K¨ahler manifold and K¨ahler identities d= 2∂ = 2∂ hold on the
space of smooth complex forms with compact support, the domains of ∂ and ∂
don’t necessarily coincide.
Proposition 4.4. If X ⊂ X0 is a strongly pseudoconvex K¨ahler manifold and more-over the harmonic spaces Hp,q
∂ =H p,q
∂ for each p, q in the respective domains of the
Laplacians of ∂ and ∂, then the ∂∂-lemma holds on X.
Proof. Here one can use the idea in Remark 3.3. We have two double complexes with real structures and the first one is the Dolbeault complex
AX = (A•,•X , ∂, ∂), .. . ... · · · Ap,qX Ap+1,qX · · · · · · Ap,q+1X Ap+1,q+1X · · · .. . ... . ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Let HX = (H•,•∂ (X), ∂, ∂)
.. . ... · · · Hp,q ∂ H p+1,q ∂ · · · · · · Hp,q+1 ∂ H p+1,q+1 ∂ · · · .. . ... . ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
In the double complex HX, the operators ∂ and ∂ are zero morphisms because of
the K¨ahler identities. There is a double complex injection morphism from HX to
AX. Obviously, these two double complexes AX and HX have the same Dolbeault
cohomology groups by (4.1). Notice that the indecomposable double complexes in the decomposition of HX can only be Type 6. As these two double complexes AX
and HX are E1-isomorphic, we deduce that the indecomposable double complexes in
the decomposition of AX can only be Types 1, 6. This completes the proof of this
proposition.
4.2. ∂-equation for logarithmic forms with values in local system. First we introduce some basic facts and notations on the sheaf of logarithmic forms. For more details, one may refer to [EV92, GH78, No95].
We introduce some notations to be used in the whole section. Let (X, ω) be a compact K¨ahler manifold of dimension n, and D a simple normal crossing divisor on it, i.e., D = Pr
i=1Di, where the Di are distinct smooth hypersurfaces intersecting
transversely in X. Denote by j : U = X −D → X the natural holomorphic embedding and
ΩpX(∗D) = lim→
ν
ΩpX(ν · D) = j∗ΩpY.
Then (Ω•X(∗D), d) is a complex. The sheaf of logarithmic forms ΩpX(log D)
(introduced by Deligne in [De68]) is defined as the subsheaf of ΩpX(∗D) with logarith-mic poles along D, i.e., for any open subset V ⊂ X,
Γ(V, ΩpX(log D)) = {α ∈ Γ(V, ΩpX(∗D)) : α and dα have simple poles along D}. From [De70, II, 3.1-3.7] or [EV92, Properties 2.2], the log complex (Ω•X(log D), d) is a subcomplex of (Ω•X(∗D), d) and ΩpX(log D) is locally free,
ΩpX(log D) = ∧pΩ1X(log D).
For any z ∈ X, which k of these Di pass, we may choose local holomorphic
coor-dinates {z1, · · · , zn} in a small neighborhood V of z = (0, · · · , 0) such that D ∩ V = {z1· · · zk= 0}
forms and logarithmic differentials dzi/zi (i = 1, · · · , k), i.e., ΩpX(log D) = ΩpX dz 1 z1 , · · · , dzk zk . Denote by A0,q(Ωp X(log D))
the germ sheaf of smooth (0, q)-forms on X with values in ΩpX(log D). Set A0,q(X, ΩpX(log D))
as the space of smooth (0, q)-forms on X with values in ΩpX(log D), and call an element of A0,q(X, ΩpX(log D)) a logarithmic (p, q)-form. We call
(4.2) ADX := (A0,•(X, Ω•X(log D)), ∂, ∂) the logarithmic Dolbeault complex of (X, D).
We will consider the complement U = X − D of a simple normal crossing divisor D in a compact K¨ahler manifold X. It is well-known that we can choose a local coordinate chart (W ; z1, · · · , zn) of X such that the locus of D is given by z1· · · zk= 0
and
U ∩ W = W∗ = (∆∗)k× (∆)n−k,
where ∆ (resp. ∆∗) is the (resp. punctured) open disk of radius in the complex
plane and ∈ (0,12]. Instead of focusing on the compact complex manifold X, we shall give a K¨ahler metric ωU only on the open manifold U , which enjoys some special
asymptotic behaviors along D.
Definition 4.5. We say that the metric ωU on U is of Poincar´e type along D, if for
each local coordinate chart (W ; z1, · · · , zn) along D the restriction ωU|W∗ 1 2
is equivalent to the usual Poincar´e type metric ωP defined by
ωP = √ −1 k X j=1 dzj∧ dzj |zj|2· log2|zj|2 +√−1 n X j=k+1 dzj∧ dzj.
As a fundamental result along this line, it is well-known that there always exists a K¨ahler metric ωU on U = X − D which is of Poincar´e type along D. Furthermore,
this metric is complete and of finite volume. See [Zu79, § 3] for more details.
Let V be a local system on U . By definition, a local system over a topological space B is a sheaf of abelian groups locally isomorphic to a constant sheaf of stalk G, where G is a fixed abelian group. There is a bijective correspondence between isomorphism classes of smooth (or holomorphic in the case where B is a complex manifold) vector bundles equipped with an integrable connection and isomorphism classes of local systems of vector spaces. Assume that V is unitary, i.e., V has a positive definite hermitian form. In [De70], Deligne constructed the canonical extension (V, ∇) of V . Here V is a holomorphic vector bundle on X with an integrable connection ∇ with logarithmic poles along D such that ker ∇|U = V. The hypercohomology of the
logarithmic de Rham complex
DRV:= (Ω•X(log D) ⊗V, ∇) is the cohomology of V over U .
