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Algebraic and combinatoric aspects of the Harder-Narasimhan recursion and its inversion. (Algebraische und kombinatorische Aspekte der Harder-Narasimhan-Rekursion und ihrer Umkehrung.) (German) Zbl 0958.14020

Bonner Mathematische Schriften. 315. Bonn: Univ. Bonn, Mathematisch-Naturwissenschaftliche Fakultät, 271 S. (1999).
Let \({\mathcal M}(n,d)\) denote the moduli space of stable holomorphic vector bundles of coprime rank \(n\) and degree \(d\) over a fixed Riemann surface of genus \(g \geq 2\). The space \({\mathcal M}(n,d)\) is a non-singular complex projective variety whose geometry has been much studied. In particular, G. Harder and M. S. Narasimhan [Math. Ann. 212, 215-248 (1975; Zbl 0324.14006)] and U. V. Desale and S. Ramanan [Math. Ann. 216, 233-244 (1975; Zbl 0317.14005)] first described in 1975 an inductive method to determine the Betti numbers of \({\mathcal M}(n,d)\) using number theoretic methods and the Weil conjectures. Subsequently in 1982, M. F. Atiyah and R. Bott [Philos. Trans. R. Soc. Lond., Ser. A 308, 523-615 (1982; Zbl 0509.14014)] obtained the same recursion using gauge theory.
New developments in this direction can be found in the recent preprint by R. Earl and F. Kirwan [“The Hodge numbers of the moduli spaces of vector bundles over a Riemann surface”, math.AG/0012260; http://front.math.ucdavis.edu], where it is given a similar inductive method for determining the Hodge-Poincaré polynomial of \({\mathcal M}(n,d)\), which is the polynomial in two variables whose coefficients are the Hodge numbers of the moduli space.
The thesis under review studies the Harder-Narasimhan recursion with methods from combinatorics. A key role is played by the inversion formula for the Harder-Narasimhan recursion found by D. Zagier [in: Proc. Hirzebruch 65 Conf. Algebraic Geometry, Ramat Gan 1993, Isr. Math. Conf. Proc. 9, 445-462 (1996; Zbl 0854.14020)]. In particular, a proof with purely combinatorial methods is given of the fact that the inversion formula really gives a polynomial and not only a rational function for the Poincaré polynomial. At the end of this rather technical paper a nicely simple (non recursive) formula for the rank of the total cohomology of \({\mathcal M}(n,1)\) is presented.

MSC:

14H60 Vector bundles on curves and their moduli
05A15 Exact enumeration problems, generating functions
14H10 Families, moduli of curves (algebraic)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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