Branched SL(r,C)-opers

We define the branched analog of SL(r,C)-opers and investigate their properties. For the usual SL(r,C)-opers, the underlying holomorphic vector bundle is independent of the opers. For the branched SL(r,C)-opers, the underlying holomorphic vector bundle depends on the oper. Given a branched SL(r,C)-oper, we associate to it another holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle does not depend on the branched oper. We characterize the branched SL(r,C)-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.

Deformation quantization of moduli spaces of Higgs bundles on a Riemann surface with translation structure

Let X be a compact connected Riemann surface of genus g > 0 equipped with a nonzero holomorphic 1-form. Let M denote the moduli space of semistable Higgs bundles on X of rank r and degree r(g-1)+1; it is a complex symplectic manifold. Using the translation structure on the open subset of X where the 1-form does not vanish, we construct a natural deformation quantization of a certain nonempty Zariski open subset of M.

Vector bundles and connections on Riemann surfaces with projective structure

Let ${mathcal B}_g(r)$ be the moduli space of triples of the form $(X,, K^{1/2}_X,, F)$, where $X$ is a compact connected Riemann surface of genus $g$, with $g, geq, 2$, $K^{1/2}_X$ is a theta characteristic on $X$, and $F$ is a stable vector bundle on $X$ of rank $r$ and degree zero. We construct a $T^*{mathcal B}_g(r)$--torsor ${mathcal H}_g(r)$ over ${mathcal B}_g(r)$. This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank $r$, on a fixed Riemann surface $Y$, given by the moduli space of holomorphic connections on the stable vector bundles of rank $r$ on $Y$, and on the other hand the torsor over the moduli space of Riemann surfaces given by the moduli space of Riemann surfaces with a projective structure. It is shown that ${mathcal H}_g(r)$ has a holomorphic symplectic structure compatible with the $T^*{mathcal B}_g(r)$--torsor structure. We also describe ${mathcal H}_g(r)$ in terms of the second order matrix valued differential operators. It is shown that ${mathcal H}_g(r)$ is identified with the $T^*{mathcal B}_g(r)$--torsor given by the sheaf of holomorphic connections on the theta line bundle over ${mathcal B}_g(r)$.

Quillen connection and the uniformization of Riemann surfaces

The Quillen connection on ${mathcal L} rightarrow {mathcal M}_g$, where ${mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${mathcal M}_g$, $g geq 5$, is uniquely determined by the condition that its curvature is the Weil--Petersson form on ${mathcal M}_g$. The bundle of holomorphic connections on ${mathcal L}$ has a unique holomorphic isomorphism with the bundle on ${mathcal M}_g$ given by the moduli stack of projective structures. This isomorphism takes the $C^infty$ section of the first bundle given by the Quillen connection on ${mathcal L}$ to the $C^infty$ section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.

Connections on trivial vector bundles over projective schemes

Over a smooth and proper complex scheme, the differential Galois group of an integrable connection may be obtained as the closure of the transcendental monodromy representation. In this paper, we employ a completely algebraic variation of this idea by restricting attention to connections on trivial vector bundles and replacing the fundamental group by a certain Lie algebra constructed from the regular forms. In more detail, we show that the differential Galois group is a certain ``closure of the aforementioned Lie algebra. This is then applied to construct connections on curves with prescribed differential Galois group.

On the moduli spaces of framed logarithmic connections on a Riemann surface

We describe some results on moduli space of logarithmic connections equipped with framings on a $n$-pointed compact Riemann surface.

Branched projective structures, branched SO(3,C)-opers and logarithmic connections on jet bundle

We study the branched holomorphic projective structures on a compact Riemann surface $X$ with a fixed branching divisor $S, =, sum_{i=1}^d x_i$, where $x_i ,in, X$ are distinct points. After defining branched ${rm SO}(3,{mathbb C})$--opers, we show that the branched holomorphic projective structures on $X$ are in a natural bijection with the branched ${rm SO}(3,{mathbb C})$--opers singular at $S$. It is deduced that the branched holomorphic projective structures on $X$ are also identified with a subset of the space of all logarithmic connections on $J^2((TX)otimes {mathcal O}_X(S))$ singular over $S$, satisfying certain natural geometric conditions.

Poisson structure on character varieties

We show that the character variety for a $n$-punctured oriented surface has a natural Poisson structure.

Moduli spaces of framed $G$--Higgs bundles and symplectic geometry

Let $X$ be a compact connected Riemann surface, $D, subset, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x, subsetneq, G_x$ a Zariski closed subgroup for every $x, in, D$. A framed principal $G$--bundle is a pair $(E_G,, phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $phi$ assigns to each $x, in, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,, phi)$ together with a section $theta, in, H^0(X,, text{ad}(E_G)otimes K_Xotimes{mathcal O}_X(D))$ such that $theta(x)$ is compatible with the framing $phi$ for every $x, in, D$. We construct a holomorphic symplectic structure on the moduli space $mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $mathcal{M}_{FH}(G)$ to the moduli space $mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,, theta)$ that forgets the framing, is Poisson. These results generalize cite{BLP} where $(G,, {H_x}_{xin D})$ is taken to be $(text{GL}(r,{mathbb C}),, {text{I}_{rtimes r}}_{xin D})$. We also investigate the Hitchin system for $mathcal{M}_{FH}(G)$ and its relationship with that for $mathcal{M}_{H}(G)$.

Hermitian symmetric space, flat bundle and holomorphicity criterion

Let $Kbackslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Gamma ,subset, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kahler manifold and $rho, :, pi_1(X, x_0),longrightarrow, Gamma$ a homomorphism such that the adjoint action of $rho(pi_1(X, x_0))$ on $text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $rho$ a harmonic map $X, longrightarrow, Kbackslash G/Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.