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Indecomposable vector bundles and stable Higgs bundles over smooth projective curves. (English) Zbl 1342.14076

The principal object of this paper is to compute the Betti numbers of the moduli spaces of stable Higgs bundles of rank \(r\) and degree \(d\) over a compact Riemann surface \(X\). As in the work of T. Hausel (e. g., [Prog. Math. 235, 193–217 (2005; Zbl 1099.14026)]) and others, the method is to calculate the number of points of these moduli spaces defined over finite fields and use the Weil conjectures.
Let \(X\) be a smooth projective geometrically connected curve of genus \(g\) defined over a finite field \(\mathbb{F}_q\). Denote by \(\mathcal{A}_{r,d}(X)\) the number of isomorphism classes of geometrically indecomposable vector bundles of rank \(r\) and degree \(d\) on \(X\). Then (Theorem 1.1), there exists an explicit polynomial \(A_{g,r,d}\) in the Weil numbers \(\sigma_1,\cdots,\sigma_{2g}\) of \(X\) such that \(\mathcal{A}_{r,d}(X)=A_{g,r,d}(\sigma_1,\cdots,\sigma_{2g})\). (Here \(\sigma_1,\cdots,\sigma_{2g}\) are the eigenvalues of the Frobenius acting on \(H^1(\overline{X},\overline{\mathbb{Q}_l})\), where \(l\) is a prime not dividing \(q\) and \(\overline{X}=X\times_{\mathrm {Spec}(\mathbb{ F}_q)}\mathrm {Spec}(\overline{\mathbb{F}_q})\).) Moreover, when \(\mathrm{gcd}(r,d)=1\), let \(\mathrm {Higgs}^{\mathrm{st}}_{r,d}(X)\) denote the moduli space of stable Higgs bundles of rank \(r\) and degree \(d\) over \(X\). Then (Theorem 1.2), there exists an explicit constant \(C=C(r,d)\) such that, whenever \(\mathrm {char}(\mathbb{F}_q)>C\), we have \(\left|\mathrm {Higgs}^{\mathrm{st}}_{r,d}(X)(\mathbb{F}_q)\right|=q^{1+(g-1)r^2}\mathcal{A}_{r,d}(X)\). As a first corollary (Corollary 1.3), if \(\mathrm {char}(\mathbb{F}_q)>C\), the Poincaré polynomial \(\sum(-1)^n\dim H^n_c(\mathrm {Higgs}^{\mathrm{st}}_{r,d}(\overline{X}),\overline{\mathbb{Q}_l})t^n\) is equal to \(t^{2(1+(g-1)r^2)}A_{g,r,d}(t,\cdots,t)\). The same formula holds for the Poincaré polynomial of the moduli space of Higgs bundles on a compact Riemann surface \(X_{\mathbb{C}}\) of genus \(g\). In both cases, \(H^n_c\) denotes singular cohomology with compact supports. A further corollary (Corollary 1.4) gives a formula for the number of points of the nilpotent cone \(\Lambda^{\mathrm{st}}_{r,d}(\mathbb{F}_q)\) of the Hitchin map. Moreover the number of irreducible components of \(\Lambda^{\mathrm{st}}_{r,d}\) is equal to \(A_{g,r,d}(0)\) for \(X\) defined over \(\mathbb{F}_q\) and for \(X_{\mathbb{C}}\).
In addition to background material and the statements of the main theorems and their corollaries, Section 1 includes a precise expression for the polynomials \(A_{g,r,d}\) (Theorem 1.6), a conjecture (Conjecture 1.7) that \(A_{g,r,d}\) is independent of \(d\) and a generating series for \(A_{g,r,d}(0)\) (Corollary 1.9). Section 2 concerns stacks of pairs (a pair here is a coherent sheaf together with a filtration) and a truncation of the Harder-Narasimhan stratification, which is necessary in order to interpret \(\mathcal{A}_{r,d}(X)\) in terms of the volume of a certain stack. In section 3, a further stratification by Jordan type is defined, followed in Section 4 by a discussion of Hall algebras and the proof of Theorem 1.1. In Section 5, the volume of the stack of pairs is computed and Theorem 1.6 is proved. The proof of Theorem 1.2 is contained in Section 6. Section 7 contains a result analogous to Theorem 1.1 for bundles with parabolic structure, followed by some refinements and conjectures in Section 8. There are three appendices, on the volume of moduli stacks of torsion sheaves, the density of Weil numbers and the proof of Conjecture 1.7 when \(r\) is prime.

MSC:

14H60 Vector bundles on curves and their moduli
14G15 Finite ground fields in algebraic geometry

Citations:

Zbl 1099.14026
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References:

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