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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).

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