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Chow group of 1-cycles of the moduli of parabolic bundles over a curve. (English) Zbl 1475.14007

Given a nonsingular projective curve \(X\) of genus \(g \ge 3\) over \(\mathbb{C}\), let \(\mathcal{M}(r, \mathcal{L})\) denote the moduli space of stable bundles over \(X\) of rank \(r\) with fixed determinant \(\mathcal{L}\). Moreover let \(\mathcal{M}(r, \mathcal{L}, \alpha )\) denote the moduli space of parabolic bundles of full flags along the fixed parabolic points and generic weights \(\alpha\). The author proves the following result on the Chow group of 1-cycles with rational coefficient:
Theorem 1.1. (Theorem 3.12) For any two generic weights \(\alpha\) and \(\beta\), there exists a canonical isomorphism \[ \mathrm{CH}_1^{\mathbb{Q}} (\mathcal{M}_\alpha) \cong \mathrm{CH}_1^{\mathbb{Q}} (\mathcal{M}_\beta) \]
On the other hand for \(\mathcal{L} = \mathcal{O}_X(x)\), it has been shown in [I. Choe and J.-M. Hwang, Math. Z. 253, No. 2, 281–293 (2006; Zbl 1102.14002)] that \[ \mathrm{CH}_1^{\mathbb{Q}} ( \mathcal{M} (2, \mathcal{L}) \cong \mathrm{CH}_0^{\mathbb{Q}} (X). \tag{1} \] Combining with this result, the author shows:
Theorem 1.2 (Theorem 4.6) In case of rank 2 and determinant \(\mathcal{O}_X(x)\), for any generic weight \(\alpha\), we have \[ \mathrm{CH}_1^{\mathbb{Q}} (\mathcal{M}_\alpha) \cong \mathbb{Q}^n \oplus \mathrm{CH}_0^{\mathbb{Q}} (X), \] where \(n\) is the number of points of the parabolic data.
Main idea is to show that for a sufficiently small generic \(\alpha\), the space \(\mathcal{M}_\alpha\) has a structure of a \((\mathbb{P}^1)^n\)-bundle over \(\mathcal{M}(2, \mathcal{O}_X(x))\).
Remark: Recently the isomorphism (1) has been improved to an isomorphism with integer coefficient by D. Li, Y. Lin and X. Pan [D. Li et al., C. R., Math., Acad. Sci. Paris 357, No. 2, 209–211 (2019; Zbl 1408.14043)]. It seems that Theorem 1.2 above can also be improved in view of this result.

MSC:

14C15 (Equivariant) Chow groups and rings; motives
14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces
14H60 Vector bundles on curves and their moduli
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References:

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