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Two lemmas on double complexes and their applications to CR cohomology. (English) Zbl 1054.32023

Dragomir, Sorin (ed.), Selected topics in Cauchy-Riemann geometry. Rome: Aracne (ISBN 88-7999-409-3/hbk). Quad. Mat. 9, 125-137 (2003).
Let \((^A_*, d^A, d_A)\) be a double complex of Abelian groups. The authors establish certain algebraic homology results on the cohomology of the total complex \((\text{Tot}(A), \delta_A)\), where \(\text{Tot}^i(A)=\bigoplus_{r+s=i} A^r_s\). Precisely, let \(f: A^*_*\to B^*_*\) be a morphism of double complexes. Assume that \(A_j^*= 0\), \(B_j^*= 0\) for \(j< 0\) and \(j> k\), for some \(k\in\mathbb{Z}\), \(k\geq 0\). Let \(r\in\mathbb{Z}\) such that \(f_*: H^{r-h}(A^*_h)\to H^{r-h}(B^*_h)\) is onto and \(f_*: H^{r+1-h}(A^*_h)\to H^{r+1-h}(B^*_h)\) is one-one, for \(0\leq h\leq k\).
Then the authors show that 1) \(f_*: H^r(\text{Tot}(A),\delta_A)\to H^r(\text{Tot}(B),\delta_B)\) is onto and \(f_*: H^{r+1}(\text{Tot}(A),\delta_A)\to H^{r+1}(\text{Tot}(B),\delta_B)\) is one-one. Moreover, if the sequence \[ 0\to A^i_0@>d^A>> A^i_1 @>d^A>> A^i_2@>d^A>>\cdots @> d^A>> A^i_{k-1}@> d^A>> A^i_k \] is exact for any \(i\) then 2) \(H^r(\text{Tot}(A))\approx H^{r-k}(A^*_k/d^A(A^*_{k-1}),\widehat d_A)\), for any \(r\in\mathbb{Z}\), where \(\widehat dA\) is induced on quotients by \(d_A\).
The results obtained are then applied to the cohomology of the tangential Cauchy-Riemann complex of a CR manifold. The following is illustrative for the applications sought after by the authors. Let \(M\) be a \(q\)-pseudoconcave CR manifold of type \((n, k)\), generically embedded in an \((n + k)\)-dimensional complex manifold \(X\), such that the conormal bundle \(N^*_MX\) is trivial. Then the natural morphisms \[ H^j(M, \Omega^p_M)\to H^j([{\mathcal E}]^{p,*}(M), \overline\partial_M)\to H^j([{\mathcal D}']^{p,*}(M), \overline\partial_M) \] are onto for \(j= n- q+ 1\) and are isomorphisms for \(0\leq j< q\) and for \(j> n- q+ 1\). See also the papers by C. D. Hill and M. Nacinovich [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, No. 2, 315–339 (1995; Zbl 0848.32003)] and C. Laurent-Thiébaut and J. Leiterer [Math. Ann. 325, No. 1, 165–185 (2003; Zbl 1031.32012)].
For the entire collection see [Zbl 1015.00009].

MSC:

32V99 CR manifolds
55U15 Chain complexes in algebraic topology
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