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Hypercohomologies of Truncated Twisted Holomorphic De Rham Complexes 3

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Hypercohomologies of Truncated Twisted Holomorphic De Rham Complexes 3 HYPERCOHOMOLOGIES OF TRUNCATED TWISTED HOLOMORPHIC DE RHAM COMPLEXES LINGXU MENG Abstract. We investigate the hypercohomologies of truncated twisted holomorphic de Rham complexes on (not necessarily compact) complex manifolds. In particular, we general- ize Leray-Hirsch, K¨unneth and Poincar´e-Serre duality theorems on them. At last, a blowup formula is given, which affirmatively answers a question posed by Chen, Y. and Yang, S. in [3]. 1. Introduction All complex manifolds mentioned in this paper are connected and not necessarily compact unless otherwise specified. Let X be a complex manifold. The de Rham theorem says that the cohomology Hk(X, C) with complex coefficients can be computed by the hypercohomology Hk • • (X, ΩX ) of the holomorphic de Rham complex ΩX . Moreover, if X is a compact K¨ahler F pHk X, C Hr,k−r X, C manifold, the Hodge filtration dR( )= ∂¯ ( ) on the de Rham cohomology rL≥p k C Hk ≥p HdR(X, ) is just the hypercohomology (X, ΩX ) of the truncated holomorphic de Rham ≥p complex ΩX , which plays a significant role in the Hodge theory. The Deligne cohomology is an important object in the research of algebraic cycles, since it connects with the intermediate Jacobian and the integeral Hodge class group. It has a strong relation with the singular ∗ Z H∗ ≤p−1 cohomology H (X, ) and the hypercohomology (X, ΩX ) of the truncated holomorphic ≤p−1 de Rham complex ΩX via the long exact sequence ··· / k−1 Z / Hk−1 ≤p−1 / k Z / k Z / Hk ≤p−1 / ··· H (X, ) (X, ΩX ) HD (X, (p)) H (X, ) (X, ΩX ) . So it seems natural to investigate the hypercohomology of truncated holomorphic de Rham complexes. H∗ [s,t] We consider the more general cases - the hypercohomology (X, ΩX (L)) of truncated twisted holomorphic de Rham complexes Ω[s,t](L) (see Section 3.2 or [3, Section 4.2] for arXiv:1909.00368v2 [math.AG] 22 Mar 2020 X the definition) and obtain the generalizations of several classical theorems for the de Rham cohomology with values in a local system as follows. Theorem 1.1 (Leray-Hirsch theorem). Let π : E → X be a holomorphic fiber bundle with C compact fibers over a complex manifold X and L a local system of X -modules of finite rank on X. Assume that there exist d-closed forms t1, . ., tr of pure degrees on E such that the restrictions of their Dolbeault classes t , . ., t to E is a basis of H•,• E [ 1]∂¯ [ r]∂¯ x ∂¯ ( x)= 2010 Mathematics Subject Classification. Primary 32C35; Secondary 14F43. Key words and phrases. hypercohomology; truncated twisted holomorphic de Rham complex; Leray-Hirsch theorem; K¨unneth theorem; Poincar´e-Serre duality theorem; blowup formula. 1 2 LINGXU MENG Hp,q E for every x X. Then there exists an isomorphism ∂¯ ( x) ∈ p,qL≥0 r − − Hk−ui−vi [s ui,t ui] Hk [s,t] −1 (X, ΩX (L))→ ˜ (E, ΩE (π L)) Mi=1 for any k, s, t, where (ui, vi) is the degree of ti for 1 ≤ i ≤ r. Theorem 1.2 (K¨unneth theorem). Let X, Y be complex manifolds and L, H local systems C C of X -, Y -modules of finite ranks on X, Y respectively. If X or Y is compact, then there is an isomorphism Ha [u,v] Hb [w,w] ∼ Hc [s,t] ⊠ (X, ΩX (L)) ⊗C (Y, ΩY (H)) = (X × Y, ΩX×Y (L H)) aM+b=c u+w=s v+w=t for any c, s, t. Theorem 1.3 (Poincar´e-Serre duality theorem). Let X be an n-dimensional compact complex C manifold and L a local system of X -modules of finite rank on X. Then there exists an isomorphism Hk [s,t] ∼ H2n−k [n−t,n−s] ∨ ∗ (X, ΩX (L)) = ( (X, ΩX (L ))) for any k, s, t, where ∗ denotes the dual of a C-vector space. Recently, the blowup formulas for the de Rham cohomology with values in a local system were studied with different approaches [3, 5, 6, 8, 16, 17]. More generally, Chen, Y. and Yang, S. posed a question on the existence of a blowup formula for the hypercohomology of a truncated twisted holomorphic de Rham complex ([3, Question 10]). Now, we prove that it truly exists. Theorem 1.4. Let π : X → X be the blowup of a complex manifold X along a complex C submanifold Y and L a locale system of X -modules of finite rank on X. Assume that iY : Y → X is the inclusion and r = codimCY ≥ 2. Then there exists an isomorphism r−1 Hk [s,t] Hk−2i [s−i,t−i] −1 ∼ Hk [s,t] −1 (X, ΩX (L)) ⊕ (Y, ΩY (iY L)) = (X, Ω (π L)), (1.1) M Xe i=1 e for any k, s, t. In Section 2, we recall some basic notions on complexes and give some properties of them, •,• which may be well known for experts. In Section 3, the double complexes SX (L,s,t) and •,• TX (L,s,t) are defined and studied, which play important roles in the study of the hyperco- Hk [s,t] homology (X, ΩX (L)). In Section 4, Theorems 1.1-1.4 are proved. Acknowledgements. The author would like to thank Drs. Chen, Youming and Yang, Song for many useful discussions. The author sincerely thanks the referee for many valuable suggestions and careful reading of the manuscript. The author is supported by the Natural Science Foundation of Shanxi Province of China (Grant No. 201901D111141). 2. Complexes Let C be an Abelian category. All complexes and morphisms in this section are in C. HYPERCOHOMOLOGIES OF TRUNCATED TWISTED HOLOMORPHIC DE RHAM COMPLEXES 3 2.1. Complexes. A complex (K•, d) consists of objects Kk and morphisms dk : Kk → Kk+1 for all k ∈ Z satisfying that dk+1 ◦ dk = 0. Sometimes, we briefly denote (K•, d) by K• and denote its k-th cohomology by Hk(K•). A morphism f : K• → L• of complexes means a k k k Z k+1 k k k family of morphisms f : K → L for all k ∈ satisfying that f ◦ dK = dL ◦ f for all k ∈ Z. Moreover, if there exists a morphism g : L• → K• satisfying that f ◦ g = id and g ◦ f = id, then f is said to be an isomorphism. •,• p,q p,q p,q A double complex (K , d1, d2) consists of objects K and morphisms d1 : K → p+1,q p,q p,q p,q+1 Z2 p+1,q p,q p,q+1 p,q K , d2 : K → K for all (p,q) ∈ satisfying that d1 ◦d1 = 0, d2 ◦d2 = 0 p,q+1 p,q p+1,q p,q •,• Z and d1 ◦d2 +d2 ◦d1 = 0. Sometimes, it will be shortly written as K . Fixed p ∈ , p,• p,• •,• •,• (K , d2 ) is a complex. A morphism f : K → L of double complexes means a family p,q p,q p,q Z p+1,q p,q p,q p,q of morphisms f : K → L for all p, q ∈ satisfying that f ◦ dK1 = dL1 ◦ f and p,q+1 p,q p,q p,q Z •,• •,• f ◦ dK2 = dL2 ◦ f for all p, q ∈ . If there exists a morphism g : L → K satisfying that f ◦ g = id and g ◦ f = id, then f is said to be an isomorphism. Let K•,• be a double complex. The complex sK•,• associated to K•,• is defined as •,• k p,q k p,q p,q •,• • • (sK ) = K and d = d1 + d2 . Let (KEr , F H ) be the spec- p+Lq=k p+Pq=k p+Pq=k •,• p,q q p,• tral sequence associated to a double complex K . Then the first page K E1 = H (K ) and Hk = Hk(sK•,•). A morphism f : K•,• → L•,• of double complexes naturally induces a morphism sf : sK•,• → sL•,• of complexes, where (sf)k = f p,q for any k ∈ Z. p+Pq=k 2.2. Shifted complexes. For an integer m, denote by (K•[m], d[m]) the m-shifted com- plex of (K•, d), namely, K•[m]k = Kk+m and d[m]k = dk+m for all k ∈ Z. For a pair •,• of integers (m,n), denote by (K [m,n], d1[m,n], d2[m,n]) the (m,n)-shifted double com- •,• •,• p,q p+m,q+n p,q p+m,q+n plex of (K , d1, d2), namely, K [m,n] = K and di[m,n] = di for any (p,q) ∈ Z2 and i = 1, 2. Clearly, s(K•,•[m,n]) = (sK•,•)[m + n]. 2.3. Dual complexes. In this subsection, assume that C is the category of C-vector spaces. ∗ For a C-vector space V , let V = HomC(V, C) be its dual space. The dual complex ((K•)∗, d∗) of (K•, d) is defined as (K•)∗k = (K−k)∗ and d∗kf k = (−1)k+1f k ◦ d−k−1 for any k ∈ Z. Let f k ∈ (K•)∗k satisfy d∗kf k = 0, i.e., f k ◦ d−k−1 = 0. Then f k naturally induces a linear map f˜k : H−k(K•) → C. The morphism [f k] 7→ f˜k gives the following isomorphism, where [f k] denotes the class in Hk((K•)∗). Lemma 2.1 ([4, IV, (16.5)]). Hk((K•)∗) =∼ (H−k(K•))∗. •,• ∗ ∗ ∗ •,• •,• ∗p,q −p,−q ∗ The dual double complex ((K ) , d1, d2) of (K , d1, d2) is defined as (K ) = (K ) ∗p,q p,q p+q+1 p,q −p−1,−q ∗p,q p,q p+q+1 p,q −p,−q−1 and d1 f = (−1) f ◦ d1 , d2 f = (−1) f ◦ d2 for any (p,q) ∈ Z2 and f p,q ∈ (K−p,−q)∗.
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