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Geometry of the intersection ring and vanishing relations in the cohomology of the moduli space of parabolic bundles on a curve. (English) Zbl 1351.30032
The authors prove a vanishing result related to the subring of the cohomology ring of the moduli space of rank-\(N\) vector bundles on a compact Riemann surface \(\Sigma\) of genus \(g\) with parabolic structure at a given point \(P \in \Sigma\) that is generated by certain Chern classes. Let us introduce some notation to state the main result (Theorem 1.6). Let \(G=\mathrm{SU}(N)\), and let \(T\) be a maximal torus. Let \(t_1,\ldots,t_N \in (0,1)\) be distinct numbers such that \(t_1 + \cdots + t_N \in \mathbb{Z}\) but the sum of any proper subset of the \(t_j\) is not an integer. Denote \(t=\mathrm{Diag}(e^{2 \pi \sqrt{-1} t_1}, \ldots, e^{2 \pi \sqrt{-1} t_N}) \in T\). Then \(S_g(t)\) denotes the moduli space of rank-\(N\) vector bundles on \(\Sigma\) with parabolic structure at \(P\). Let \[ V_g(t) = \Big\{ (A_1, \ldots, A_g, B_1, \ldots, B_g) \in G^{2g} \;\Big|\; \prod_{i=1}^{g} [A_i,B_i]= t \Big\}. \] Then there is a \(T\)-bundle \(V_g(t) \to S_g(t)=V_g(t)/T\). Using the representation \(T \times \mathbb{C} \to \mathbb{C}\) given by \[ \Big(\mathrm{Diag}(e^{\sqrt{-1} \theta_1}, \ldots, e^{\sqrt{-1} \theta_N}), z\Big) \mapsto e^{\sqrt{-1} (\theta_i - \theta_j)} z \] we get line bundles \(L_{ij}\) on \(S_g(t)\). Theorem 1.6 states that for \(1 \leq i,j \leq N\) and \(i \neq j\), and nonnegative integers \(k_{ij}\), the cohomology class \[ \prod c_1(L_{ij})^{k_{ij}} \in H^* (S_g(t) , \mathbb{Q}) \] vanishes if \[ \sum k_{ij} \geq N(N-1)g-N+2. \] This limit is below the dimension of \(S_g(t)\) (Remark 1.7).
The idea (already used by J. Weitsman in [Topology 37, No. 1, 115–132 (1998; Zbl 0919.14018)]) is to exhibit sections of certain subsets of these line bundles without common vanishing points. This establishes a result similar to the Newstead-Ramanan conjecture on stable vector bundles of odd degree and rank 2 on a Riemann surface; and it continues a long line of similar results (a long list of references is provided in the introduction).

MSC:
30F99 Riemann surfaces
32G08 Deformations of fiber bundles
14H60 Vector bundles on curves and their moduli
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