Rank two quadratic pairs and surface group representations

Let $X$ be a compact Riemann surface. A quadratic pair on $X$ consists of a holomorphic vector bundle with a quadratic form which takes values in fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the $\mathrm{SO}_0(2,3)$-character variety of $X$.Comment: 37 pages, 1 figur

Moduli Spaces for Principal Bundles in Arbitrary Characteristic

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.Comment: V4: Final version, to appear in Advances in Mathematics, 69p

The cohomology rings of moduli stacks of principal bundles over curves

We prove that the cohomology of the moduli stack of G-bundles on a smooth projective curve is freely generated by the Atiyah--Bott classes in arbitrary characteristic. The main technical tool needed is the construction of coarse moduli spaces for bundles with parabolic structure in arbitrary characteristic. Using these spaces we show that the cohomology of the moduli stack is pure and satisfies base-change for curves defined over a discrete valuation ring. Thereby we get an algebraic proof of the theorem of Atiyah and Bott and conversely this can be used to give a geometric proof of the fact that the Tamagawa number of a Chevalley group is the number of connected components of the moduli stack of principal bundles