Holomorphic Morse inequalities. (English) Zbl 0588.32009

Algebraic and differential topology - global differential geometry, Occas. 90th Anniv. M. Morse’s Birth, Teubner-Texte Math. 70, 318-333 (1984).
[For the entire collection see Zbl 0568.00013.]
The author establishes a system of inequalities, analogous to the classical Morse inequalities, attached to a vector field V generating an isometry of the compact Kähler manifold M. A special case of Hopf’s theorem expressing the Euler characteristic of M in terms of the zeros of a vector field, \(\Sigma (-1)^ kM_ k = \Sigma (-1)^ kB_ k\) \((B_ k=k\)-th Betti number of M, \(M_ k=k\)-th Morse number of a Morse function on M), has a generalization for complex manifolds, the holomorphic Lefschetz formula or the Atiyah-Bott fixed point theorem. The holomorphic Morse inequalities imply the holomorphic Lefschetz formula just as the classical Morse inequalities imply the Hopf formula above for the Euler characteristic. Appropriate application of harmonic analysis on M provides the desired results. The question seems open whether holomorphic Morse inequalities exist in case V does not generate an isometry, and so far the non-Kählerian case has not been considered.
Reviewer: A.Aeppli


32Q99 Complex manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
32J99 Compact analytic spaces


Zbl 0568.00013