Holomorphic Connection Research Articles

G-connections on principal bundles over complete G-varieties

Let X be a complete variety over an algebraically closed field k of characteristic zero, equipped with an action of an algebraic group G. Let H be a reductive group. We study the notion of G-connection on a principal H-bundle. We give necessary and sufficient criteria for the existence of G-connections extending the Atiyah-Weil type criterion for holomorphic connections obtained by Azad and Biswas. We also establish a relationship between the existence of G-connection and equivariant structure on a principal H-bundle, under the assumption that G is semisimple and simply connected. These results have been obtained by Biswas et al. when the underlying variety is smooth.

Holomorphic Connections and Problems of Lifts

Holomorphic Connections and Problems of Lifts

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface of genus g⩾2 with the explicit parametrization in terms of the Tyurin data. The ‘non-abelian’ theta divisor consists of bundles such that h1⩾1 . On the complement of this divisor we construct a non-abelian (i.e. matrix) Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which also depends holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the GLn character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h 1.

Principal bundles with holomorphic connections over a Kähler Calabi-Yau manifold

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact Kähler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and generalizes a result previously obtained in [6] for simply connected compact Kähler Calabi-Yau manifolds. We give some applications of it in the framework of Cartan geometries and foliated Cartan geometries on Kähler Calabi-Yau manifolds.

Vector bundles on real abelian varieties

This paper is about real holomorphic vector bundles on real abelian varieties. The main result of the paper gives several conditions that are necessary and sufficient for the existence of a holomorphic connection on a real holomorphic vector bundle over a real abelian variety. Also proved is an analogue, for real abelian varieties, of a result of Simpson, which gives a criterion for a holomorphic vector bundle to arise by successive extensions of stable vector bundles with vanishing Chern classes.

The Monodromy Map from Differential Systems to the Character Variety Is Generically Immersive

Let G be a connected reductive affine algebraic group defined over \mathbb C and \mathfrak g its Lie algebra. We consider all pairs of the form (Y, D) , where Y is a complex structure on a compact oriented C^\infty surface \Sigma , and D is a holomorphic connection on the trivial holomorphic principal G -bundle Y\times G on Y ; these are known as \mathfrak g -differential systems. We study the monodromy map from the space of \mathfrak g -differential systems to the character variety of G -representations of the fundamental group of \Sigma . If the complex dimension of G is at least three, and \operatorname{genus}(\Sigma) \geq 2 , we show that the monodromy map is an immersion at the generic point.

Counting holomorphic connections with a prescribed Ricci tensor

How many holomorphic connections are there with a prescribed Ricci tensor? How many torsion-free holomorphic connections are there with a prescribed Ricci tensor? These questions are answered by using the holomorphic version of the Cauchy–Kowalevski theorem.

Relative connections on principal bundles and relative equivariant structures

We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.

Parabolic connections and stack of roots

Given a scheme over a field endowed with a strict normal crossings divisor, we define strongly parabolic connections, consistently with the current terminology for Higgs bundles. When the weights are rational with prescribed denominators, we show that strongly parabolic connections correspond to holomorphic connections on the corresponding stack of roots. We use this correspondence to establish that a holomorphic connection on a stack of roots can be reconstructed from its direct image to the moduli space.

The Atiyah class of generalized holomorphic vector bundles

We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah class can be defined in three different ways: using Čech cohomology, using the first-jet short exact sequence, or adopting the Lie pair point of view.

On some natural torsors over moduli spaces of parabolic bundles

A moduli space N of stable parabolic vector bundles, of rank r and parabolic degree zero, on a n-pointed smooth complex projective curve has two naturally occurring holomorphic T⁎N–torsors over it. One of them is given by the moduli space of pairs of the form (E⁎,D), where E⁎∈N and D is a connection on E⁎. This T⁎N–torsor has a C∞ section that sends any E⁎∈N to the unique connection on it with unitary monodromy. The other T⁎N–torsor is given by the sheaf of holomorphic connections on a theta line bundle over N. This T⁎N–torsor also has a C∞ section given by the Hermitian structure on the theta bundle constructed by Quillen. We prove that these two T⁎N–torsors are isomorphic by a canonical holomorphic map. This holomorphic isomorphism interchanges the above two C∞ sections.

The Hodge Chern character of holomorphic connections as a map of simplicial presheaves

We define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable non connection preserving isomorphisms of vector bundles with holomorphic connections an appropriate sequence of holomorphic forms. We apply this Chern character map to the Cech nerve of a good cover of a complex manifold and assemble the data by passing to the totalization to obtain a map of simplicial sets. In simplicial degree 0, this map gives a formula for the Chern character of a bundle in terms of the clutching functions. In simplicial degree 1, this map gives a formula for the Chern character of bundle maps. In each simplicial degree beyond 1, these invariants, defined in terms of the transition functions, govern the compatibilities between the invariants assigned in previous simplicial degrees. In addition to this, we also apply this Chern character to complex Lie groupoids to obtain invariants of bundles on them in terms of the simplicial data. For group actions, these invariants land in suitable complexes calculating various Hodge equivariant cohomologies. In contrast, the de Rham Chern character formula involves additional terms and will appear in a sequel paper. In a sense, these constructions build on a point of view of "characteristic classes in terms of transition functions" advocated by Raoul Bott, which has been addressed over the years in various forms and degrees, concerning the existence of formulae for the Hodge and de Rham characteristic classes of bundles solely in terms of their clutching functions.

A note on the moduli spaces of holomorphic and logarithmic connections over a compact Riemann surface

Let X be a compact Riemann surface of genus $$g \ge 3$$ . We consider the moduli space of holomorphic connections over X and the moduli space of logarithmic connections singular over a finite subset of X with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces and show that they are constant under certain conditions. We show the Torelli-type theorem for the moduli space of logarithmic connections. We also describe the rational connectedness of these moduli spaces.

The super Mumford form and Sato Grassmannian

We describe a supersymmetric generalization of the construction of Kontsevich and Arbarello, De Concini, Kac, and Procesi, which utilizes a relation between the moduli space of curves with the infinite-dimensional Sato Grassmannian. Our main result is the existence of a flat holomorphic connection on the line bundle λ3/2⊗λ1/2−5 on the moduli space of triples: a super Riemann surface, a Neveu-Schwarz puncture, and a formal coordinate system. We also prove a superconformal Noether normalization lemma for families of super Riemann surfaces.

Quillen connection and the uniformization of Riemann surfaces

The Quillen connection on \({\mathcal L}\, \longrightarrow \, {\mathcal M}_g\), where \({\mathcal L}^*\) is the Hodge line bundle over the moduli stack of smooth complex projective curves curves \({\mathcal M}_g\), \(g\, \ge \, 5\), 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.

Branched SL(r, ℂ)-Opers

Abstract Branched projective structures were introduced by Mandelbaum [22, 23], and opers were introduced by Beilinson and Drinfeld [2, 3]. We define the branched analog of ${\textrm SL}(r, {\mathbb C})$-opers and investigate their properties. For the usual ${\textrm SL}(r, {\mathbb C})$-opers, the underlying holomorphic vector bundle is actually determined uniquely up to tensoring with a holomorphic line bundle of order $r$. For the branched ${\textrm SL}(r, {\mathbb C})$-opers, the underlying holomorphic vector bundle depends more intricately on the oper. While the holomorphic connection for a branched ${\textrm SL}(r, {\mathbb C})$-oper is nonsingular, given a branched ${\textrm SL}(r, {\mathbb C})$-oper, we associate to it a certain holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle in question supporting a logarithmic connection does not depend on the branched oper. We characterize the branched ${\textrm SL}(r, {\mathbb C})$-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.

Line bundles on the moduli space of parabolic connections over a compact Riemann surface

Let X be a compact Riemann surface of genus g≥3 and S a finite subset of X. Let ξ be fixed a holomorphic line bundle over X of degree d. Let Mpc(r,d,α) (respectively, Mpc(r,α,ξ)) denote the moduli space of parabolic connections of rank r, degree d and full flag rational generic weight system α, (respectively, with the fixed determinant ξ) singular over the parabolic points S⊂X. Let Mpc′(r,d,α) (respectively, Mpc′(r,α,ξ)) be the Zariski dense open subset of Mpc(r,d,α) (respectively, Mpc(r,α,ξ)) parametrizing all parabolic connections such that the underlying parabolic bundle is stable. We show that there is a natural compactification of the moduli spaces Mpc′(r,d,α), and Mpc′(r,α,ξ) by smooth divisors. We describe the numerically effectiveness of these divisors at infinity. We determine the Picard group of the moduli spaces Mpc(r,d,α), and Mpc(r,α,ξ). Let C(L) denote the space of holomorphic connections on an ample line bundle L over the moduli space M(r,d,α) of parabolic bundles. We show that C(L) does not admit any non-constant algebraic function.

On the relative opers in dimension one

Abstract We investigate the relative opers over the complex analytic family of compact complex manifolds of relative dimension one. We introduce the notion of relative opers arising from the second fundamental form associated with a relative holomorphic connection. We also investigate the relative differential operators over the complex analytic family of compact complex manifolds whose symbol is the identity automorphism. We show that the set of equivalent relative opers arising from the second fundamental form is in bijective correspondence with the set of equivalent relative differential operators whose symbol is the identity automorphism.

On vector bundles over hyperkähler twistor spaces

We study the holomorphic vector bundles E over the twistor space \({{\,\mathrm{Tw}\,}}(M)\) of a compact simply connected hyperkähler manifold M. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection \(\pi \,:\, {{\,\mathrm{Tw}\,}}(M)\,\longrightarrow \, {\mathbb {CP}}^1\). It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on \({{\,\mathrm{Tw}\,}}(M)\) of prime rank, we prove that its restriction to the generic fibre of \(\pi \) is stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on \({{\,\mathrm{Tw}\,}}(M)\) whose restriction to every fibre of \(\pi \) is non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperkähler twistor spaces; this method is employed in constructing these examples.

Affine opers and conformal affine Toda

For $\mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $\text{Aut}\, \mathcal O$-equivariant identification between $\text{Fun}\,\text{Op}_{\mathfrak g}(D)$, the algebra of functions on the space of ${\mathfrak g}$-opers on the disc, and $W\subset \pi_0$, the intersection of kernels of screenings inside a vacuum Fock module $\pi_0$. This kernel $W$ is generated by two states: a conformal vector, and a state $\delta_{-1}\left|0\right>$. We show that the latter endows $\pi_0$ with a canonical notion of translation $T^{\text{(aff)}}$, and use it to define the densities in $\pi_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $\text{Aut}\,\mathcal O$-action defines a bundle $\Pi$ over $\mathbb P^1$ with fibre $\pi_0$. We show that the product bundles $\Pi \otimes \Omega^j$, where $\Omega^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $\nabla^{\text{(aff)}} - \alpha T^{\text{(aff)}}$, $\alpha\in \mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[\mathbf v_j dt^{j+1} ] \in H^1(\mathbb P^1, \Pi\otimes \Omega^j,\nabla^{\text{(aff)}})$ of the de Rham cohomology of $\nabla^{\mathrm{(aff)}}$. Any choice of ${\mathfrak g}$-Miura oper $\chi$ gives a connection $\nabla^{\mathrm{(aff)}}_\chi$ on $\Omega^j$. Using coinvariants, we define a map $\mathsf F_\chi$ from sections of $\Pi \otimes \Omega^j$ to sections of $\Omega^j$. We show that $\mathsf F_\chi \nabla^{\text{(aff)}} = \nabla^{\text{(aff)}}_\chi \mathsf F_\chi$, so that $\mathsf F_\chi$ descends to a well-defined map of cohomologies. Under this map, the classes $[\mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(\mathbb P^1, \Omega^j,\nabla^{\text{(aff)}}_\chi)$ defined by the ${\mathfrak g}$-oper underlying $\chi$.