Indecomposable vector bundles and stable Higgs bundles over smooth projective curves
type: Item
instance of: scholarly article
title: Indecomposable vector bundles and stable Higgs bundles over smooth projective curves
zbMATH Open document ID: 1342.14076
author: Olivier Schiffmann
published in: Annals of Mathematics. Second Series
publication date: 2016-02-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1406.3839
reviewed by: P. E. Newstead
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 6541587
zbMATH Keywords: vector bundles on curves
zbMATH Keywords: Higgs bundles
zbMATH Keywords: stacks
zbMATH Keywords: Betti numbers
MaRDI profile type: Publication
OpenAlex ID: W2267615388
arXiv ID: 1406.3839
cites work: A functorial construction of moduli of sheaves
cites work: Q3562557
cites work: On the Krull-Schmidt theorem with application to sheaves
cites work: On the motivic class of the stack of bundles
cites work: On the Hall algebra of an elliptic curve. I.
cites work: Stability conditions and Stokes factors
cites work: Absolutely indecomposable representations and Kac-Moody Lie algebras. (With an appendix by Hiraku Nakajima).
cites work: Q4073480
cites work: Q2945006
cites work: Counting local systems with principal unipotent local monodromy
cites work: The probability that a matrix be nilpotent
cites work: The \(y\)-genus of the moduli space of \(\mathrm{PGL}_n\)-Higgs bundles on a curve (for degree coprime to \(n\))
cites work: On the motives of moduli of chains and Higgs bundles
cites work: THE BETTI NUMBERS OF THE MODULI SPACE OF STABLE RANK 3 HIGGS BUNDLES ON A RIEMANN SURFACE
cites work: Hall algebras, hereditary algebras and quantum groups
cites work: Chevalley groups over function fields and automorphic forms
cites work: On the cohomology groups of moduli spaces of vector bundles on curves
cites work: Kac's conjecture from Nakajima quiver varieties.
cites work: Positivity for Kac polynomials and DT-invariants of quivers.
cites work: Mixed Hodge polynomials of character varieties. With an appendix by Nicholas M. Katz.
cites work: Cohomology of large semiprojective hyperkaehler varieties
cites work: Q3375888
cites work: Cohomology of the moduli stack of coherent sheaves on a curve
cites work: Q3919819
cites work: Stable bundles and integrable systems
cites work: The Self-Duality Equations on a Riemann Surface
cites work: Counting representations of quivers over finite fields
cites work: Q3579591
cites work: Infinite root systems, representations of graphs and invariant theory. II
cites work: Q3317214
cites work: Eisenstein series and quantum affine algebras
cites work: Q4226082
cites work: The Hall algebra of a curve
cites work: Instability in invariant theory
cites work: MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS
cites work: The fundamental lemma for unitary groups
cites work: Q3126815
cites work: Quivers, Perverse Sheaves, and Quantized Enveloping Algebras
cites work: Solutions of the Motivic ADHM Recursion Formula
cites work: Hitchin fibration and endoscopy
cites work: Noncommutative projective curves and quantum loop algebras
cites work: Q4981649
cites work: SPHERICAL HALL ALGEBRAS OF CURVES AND HARDER-NARASIMHAN STRATAS
cites work: Galois Cohomology
cites work: Q5589027
cites work: The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials
cites work: Hall algebras of curves, commuting varieties and Langlands duality
cites work: Cherednik algebras, \(W\)-algebras and the equivariant cohomology of the moduli space of instantons on \(\mathbb A^2\)
DOI: 10.4007/ANNALS.2016.183.1.6
Recommended article: Indecomposable Vector Bundles on the Projective Line
Recommended article: On indecomposable vector bundles
Recommended article: Q5490163
Recommended article: Decomposable subbundles of polystable vector bundles on projective curves
Recommended article: STABLE VECTOR BUNDLES ON PROJECTIVE SURFACES
Recommended article: Stable vector bundles on the families of curves
Recommended article: On semistable vector bundles over curves
Recommended article: Q5316171
Recommended article: Q4532366
Recommended article: Q3617867
review text: 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 \textit{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.NEWLINENEWLINELet \(X\) 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…