Moduli spaces of stable pairs and non-abelian zeta functions of curves via wall-crossing

Mozgovoy, Sergey; Reineke, Markus

Journal de l’École polytechnique — Mathématiques, Tome 1 (2014), pp. 117-146.

Dans cet article nous étudions et mettons en relation les fonctions zêta non abéliennes introduites par Weng et les invariants des espaces de modules de paires stables de rang arbitraire sur les courbes. Nous prouvons une formule « wall-crossing » pour ces invariants et obtenons une formule explicite pour ceux-ci en terme du motif de la courbe. Auparavant, des formules pour ces invariants n’étaient connues qu’en rang $2$ par Thaddeus et en rang $3$ par Muñoz. En utilisant ces résultats nous obtenons une formule explicite pour les fonctions zêta non abéliennes, nous vérifions la conjecture d’uniformité de Weng pour les rangs $2$ et $3$, et nous montrons sa conjecture de dénombrement miracle.

In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Muñoz. Using these results we obtain an explicit formula for the non-abelian zeta functions, we check the uniformity conjecture of Weng for the ranks 2 and 3, and we prove the counting miracle conjecture.

Classification : 14H60, 14D20
Keywords: Stable pairs, vector bundles, wall-crossing formulas, higher zeta functions
Mot clés : Paires stables, fibrés vectoriels, formules « wall-crossing », fonctions zêta supérieures

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