## Kohn decomposition for forms on coverings of complex manifolds constrained along fibres

HTML articles powered by AMS MathViewer

- by A. Brudnyi and D. Kinzebulatov PDF
- Trans. Amer. Math. Soc.
**369**(2017), 167-186 Request permission

## Abstract:

The classical result of J.J. Kohn asserts that over a relatively compact subdomain $D$ with $C^\infty$ boundary of a Hermitian manifold whose Levi form has at least $n-q$ positive eigenvalues or at least $q+1$ negative eigenvalues at each boundary point, there are natural isomorphisms between the $(p,q)$ Dolbeault cohomology groups defined by means of $C^\infty$ up to the boundary differential forms on $D$ and the (finite-dimensional) spaces of harmonic $(p,q)$-forms on $D$ determined by the corresponding complex Laplace operator. In the present paper, using Kohn’s technique, we give a similar description of the $(p,q)$ Dolbeault cohomology groups of spaces of differential forms taking values in certain (possibly infinite-dimensional) holomorphic Banach vector bundles on $D$. We apply this result to compute the $(p,q)$ Dolbeault cohomology groups of some regular coverings of $D$ defined by means of $C^\infty$ forms constrained along fibres of the coverings.## References

- Alexander Brudnyi,
*Holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds*, Compos. Math.**142**(2006), no. 4, 1018–1038. MR**2249540**, DOI 10.1112/S0010437X06002156 - Alexander Brudnyi and Damir Kinzebulatov,
*Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds I*, Rev. Mat. Iberoam.**31**(2015), no. 3, 989–1032. MR**3420483**, DOI 10.4171/RMI/861 - Alexander Brudnyi and Damir Kinzebulatov,
*Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds. II*, Rev. Mat. Iberoam.**31**(2015), no. 4, 1167–1230. MR**3438387**, DOI 10.4171/RMI/866 - Lutz Bungart,
*On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers*, Topology**7**(1967), 55–68. MR**222338**, DOI 10.1016/0040-9383(86)90015-7 - Matjaž Erat,
*The cohomology of Banach space bundles over 1-convex manifolds is not always Hausdorff*, Math. Nachr.**248/249**(2003), 97–101. MR**1950717**, DOI 10.1002/mana.200310005 - G. B. Folland and J. J. Kohn,
*The Neumann problem for the Cauchy-Riemann complex*, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR**0461588** - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR**1288523**, DOI 10.1002/9781118032527 - M. R. Hestenes,
*Extension of the range of a differentiable function*, Duke Math. J.**8**(1941), 183–192. MR**3434** - Jaehong Kim,
*A splitting theorem for holomorphic Banach bundles*, Math. Z.**263**(2009), no. 1, 89–102. MR**2529489**, DOI 10.1007/s00209-008-0411-9 - J. J. Kohn,
*Harmonic integrals on strongly pseudo-convex manifolds. I*, Ann. of Math. (2)**78**(1963), 112–148. MR**153030**, DOI 10.2307/1970506 - J. J. Kohn,
*Harmonic integrals on strongly pseudo-convex manifolds. II*, Ann. of Math. (2)**79**(1964), 450–472. MR**208200**, DOI 10.2307/1970404 - J. J. Kohn and L. Nirenberg,
*Non-coercive boundary value problems*, Comm. Pure Appl. Math.**18**(1965), 443–492. MR**181815**, DOI 10.1002/cpa.3160180305 - B. S. Mitjagin,
*The homotopy structure of a linear group of a Banach space*, Uspehi Mat. Nauk**25**(1970), no. 5(155), 63–106 (Russian). MR**0341523** - B. V. Shabat,
*Introduction to complex analysis. Part II*, Translations of Mathematical Monographs, vol. 110, American Mathematical Society, Providence, RI, 1992. Functions of several variables; Translated from the third (1985) Russian edition by J. S. Joel. MR**1192135**, DOI 10.1090/mmono/110 - Walter Rudin,
*Principles of mathematical analysis*, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR**0385023** - M. G. Zaĭdenberg, S. G. Kreĭn, P. A. Kučment, and A. A. Pankov,
*Banach bundles and linear operators*, Uspehi Mat. Nauk**30**(1975), no. 5(185), 101–157 (Russian). MR**0415661**

## Additional Information

**A. Brudnyi**- Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada
- MR Author ID: 292684
- Email: abrudnyi@ucalgary.ca
**D. Kinzebulatov**- Affiliation: The Fields Institute, 222 College Street, Toronto, Ontario M5T 3J1, Canada
- Email: dkinzebu@fields.utoronto.ca
- Received by editor(s): March 5, 2014
- Received by editor(s) in revised form: November 4, 2014, and December 16, 2014
- Published electronically: February 12, 2016
- Additional Notes: The authors’ research was partially supported by NSERC
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 167-186 - MSC (2010): Primary 32A38, 32K99
- DOI: https://doi.org/10.1090/tran/6633
- MathSciNet review: 3557771