The Cohomology Ring of the Moduli Space of Stable Vector Bundles with Odd Determinant

The multiplicative structure of the cohomology ring of the moduli space of stable rank 2 bundles on a smooth projective curve is computed. INTRODUCTION Let C be a complete smooth curve of genus g ≥ 2 over the field of complex numbers C. Consider the moduli space N of stable rank 2 vector bundles on C with fixed determinant L of odd degree 2k− 1 . Then N is a smooth projective variety of dimension 3g − 3. On the direct product N × C one has a universal bundle V with Chern classes (cf. [1]) c1(V ) = pr∗ N (A1) + (2k − 1)pr∗ C(ω), c2(V ) = pr∗ N (A2) + 2g ∑ i=1 pr∗ N (Di) ∪ pr∗ C(βi) + k · pr∗ N (A1) ∪ pr∗ C(ω), where Aj ∈ H(N,C), Di ∈ H(N,C), ω ∈ H(C,C), and (βi) is a symplectic basis of H(C,C) (i.e., βi ∪ βg+s = δisω and βi ∪ βs = 0 for i, s = 1, . . . , s). One can show that Ai and Dj generate H∗(N,C) as a ring (cf. [1]). Hence to describe the cohomology ring it suffices to compute the intersection numbers ∫ N A1A m 2 ∏2g i=1D ri i where ri ∈ {0, 1} and all the degrees add up to 6g − 6 (the real dimension of N), and to determine relations between A1, A2 and Di. Let W = H(N,C), and let Λ∗(W ) be the exterior algebra over W . Note that the symplectic group Sp(2g,C) acts on Λ∗(W ) (a symplectic transformation of H(C,C) induces a transformation of W ). The intersection form on N is Spinvariant and, if we denote D = ∑g i=1Di∪Dg+i, the cohomology ring H∗(N,C) as a module over C[A1, A2, D] splits into direct sum of submodules generated by primitive components of the Lefschetz decomposition for Λ∗(W ). We will show that each of these submodules can be recovered from the cohomology of the moduli spaces corresponding to smaller genera. ∗The original version of this paper has appeared in Izv. Ross. Akad. Nauk Ser. Mat. 58, No 4. (1994) (English translation in Russian Acad. Sci. Izv. Math. Vol. 45 , No. 4 (1995)). Later it was revised by the author in an attempt to make it more readable.