Canonical generators of the cohomology of moduli of parabolic bundles on curves

Abstract

Let X be a compact Riemann surface, fix integers n and d with n positive, and let ∆ be a parabolic datum of rank n on X (see Section 3 below). Denote by U X (n, d, ∆) the moduli space of parabolic vector bundles of rank n and degree d, which are parabolic semistable with respect to ∆. Fix a holomorphic line bundle L of degree d on X, and let SU X (n, L, ∆) be the subvariety of U X (n, d, ∆) consisting of vector bundles with determinant isomorphic to L. We make the following two hypotheses on the parameters n, d and ∆.Assumption 1.1 Every parabolic vector bundle of rank n and degree d on X which is parabolic semistable with respect to the parabolic datum ∆ is in fact parabolic stable.Assumption 1.2 There exists a universal parabolic bundle (or briefly, a universal bundle) on U X (n, d, ∆) × X.Recall that a universal bundle on U X (n, d, ∆) × X is a vector bundle U on U X (n, d, ∆) × X together with a flag of subbundles j *x U for each x ∈ J (J being the set of parabolic points), where j x : U X (n, d, ∆)→U X (n, d, ∆)×X is the map E → (E, x), such that for each E ∈ U X (n, d, ∆), the restriction of U to {E}×X is parabolically isomorphic to E. We use the same symbols to denote the restrictions of U and U x,i etc. to SU X (n, L, ∆). Notation 1.3 • If S and T are topological spaces, and if α ∈ H * (S × T, Q), then σ(α) : H * (T, Q) → H * (S, Q) is the map σ(α)z = α/z, where / denotes the slant product.

Kirwan map and moduli space of flat connections

“…It is well-defined and so are the generators constructed from it. This was proved by Biswas and Raghavendra in [3] a j , b j,i , c j , a j , b K j,i , c j be equivariant classes defined as above. The following relations hold c j = κ(c j ) b j,i = κ(b K j,i ).…”

“…In Example 3.1, these classes are the known Atiyah-Bott -Biswas-Raghavendra generators of H * K (Y β ) (see [1] and [3]). A natural question is: what is the relation between these classes and the ones of Corollary 2.7?…”

Chen–Ruan cohomology of some moduli spaces of parabolic vector bundles

“…for j = 0, 1, 2. Then by [BR,Theorem 1.5], the cohomology algebra H * (P M ξ , Q) is generated by the Chern classes c j (Hom(U l…”

Restriction map in a regular reduction of $\mathbf{SU}(n)^{2g}$

“…In particular, Narasimhan and Seshadri's theorem (see Theorems 2.9 and 2.13) is used throughout this article to identify m β with µ −1 (β)/SU(n) and H * (m β ) with H * SU(n) (µ −1 (β)). In Section 3, we give a construction of a universal bundle on m β × X, we then recall how Biswas and Raghavendra [4] use this bundle to define a set {a k , b k,j , d k , 2 ≤ k ≤ n, 1 ≤ j ≤ 2g} of canonical multiplicative generators of the cohomology of m β (Theorem 3.4). In the next section we define a bundle on SU(n) 2g × X − {point} and use it to get a set {c k , σ k,j , 2 ≤ k ≤ n, 1 ≤ j ≤ g} of multiplicative generators for the equivariant cohomology of SU(n) 2g (Theorems 4.4 and 4.6).…”

“…Following Biswas and Raghavendra [4], we define in this section some characteristic classes of a projective bundle. We will see that when the projective bundle comes from a vector bundle of degree 0, these characteristic classes are the same as the Chern classes of the vector bundle.…”