Branched affine and projective structures on compact Riemann surfaces

A Hodge theoretic projective structure on compact Riemann surfaces

We study the set \mathcal{P}_S consisting of all branched holomorphic projective structures on a compact Riemann surface X of genus g \geq 1 and with a fixed branching divisor S := \sum_{i=1}^d n_i\cdot x_i , where x_i \in X . Under the hypothesis that n_i,=1 , for all i , with d a positive even integer such that d \neq 2g-2 , we show that \mathcal{P}_S coincides with a subset of the set of all logarithmic connections with singular locus S , satisfying certain geometric conditions, on the rank two holomorphic jet bundle J^1(Q) , where Q is a fixed holomorphic line bundle on X such that Q^{\otimes 2}= TX \otimes \mathcal{O}_X(S) . The space of all logarithmic connections of the above type is an affine space over the vector space H^0(X, K^{\otimes 2}_X \otimes\mathcal{O}_X(S)) of dimension 3g-3+d . We conclude that \mathcal{P}_S is a subset of this affine space that has codimenison d at a generic point.

We show that every extended Schottky group G which uniformizes a compact Riemann surface S is a geometric limit of Schottky groups G" which also uniformize S .That is, every element g EG is the limit of elements gn G" .Let T be a Fuchsian group covering a compact Riemann surface S, and let (f, x) be a projective structure on S. Suppose that / is a covering map and that x(f) is an extended Schottky group.(Definitions will be given in the text.For the moment, we note that our definition of extended Schottky group does not include Schottky groups.)In this paper, we will show that there exists a sequence (/" , %") of projective structures on S with the following properties: (i) Xn(T) is a Schottky group, and (ii) (fn,Xn) converges to (f, x) The precise statement, Theorem 1, and proof of this result are given in 3.In 4 we prove an analogue of Theorem 1 for compact surfaces of genus 1.We note that the homomorphisms Xn converge algebraically to /.In our proofs, we exhibit cyclic subgroups of Xn(T) which have a rank 2 parabolic subgroup of x(T) as a geometric limit (Lemma 1 and Theorem 2).This reproduces, within the context of algebraic convergence of projective structures on a fixed Riemann surface, the basic example of a geometric limit given by Jorgensen and Thurston (see [9] for details). PreliminariesThroughout, T will denote a torsion-free Fuchsian group acting on the upper half plane U and covering a compact surface U/T.We denote the space of holomorphic quadratic differentials (defined on U) for T by 772(r) and equip 772(r) with the norm \\cp\\ = supzfE(/{|p(z)|(2Imz)2}.For every cp B2(T) there exists a meromorphic local homeomorphism f? = /, where /: U -> C and S(f) = cp ; here S(f) is the Schwarzian derivative of / given by s(f) = (f"//')'-W'lf')2-

Let X be a connected Riemann surface equipped with a projective structure \(\mathfrak{p}\). Let E be a holomorphic symplectic vector bundle over X equipped with a flat connection. There is a holomorphic symplectic structure on the total space of the pullback of E to the space of all nonzero holomorphic cotangent vectors on X. Using \(\mathfrak{p}\), this symplectic form is quantized. A moduli space of Higgs bundles on a compact Riemann surface has a natural holomorphic symplectic structure. Using \(\mathfrak{p}\), a quantization of this symplectic form over a Zariski open subset of the moduli space of Higgs bundles is constructed.

Every compact Riemann surface X admits a natural projective structure pu as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on X, namely the Hodge projective structure ph, related to the second fundamental form of the period map. We then describe how projective structures correspond to (1, 1)-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that pu and ph are not the same structure.

Grafting is a surgery on Riemann surfaces introduced by Thurston which connects hyperbolic geometry and the theory of projective structures on surfaces. We will discuss the space of projective structures in terms of the Thurston's geometric parametrization given by grafting. From this approach we will prove that on any compact Riemann surface with genus greater than $1$ there exist infinitely many projective structures with Fuchsian holonomy representations. In course of the proof it will turn out that grafting is closely related to harmonic maps between surfaces.

Projective structures on a Riemann surface, III

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

This big chapter is devoted to the theory of compact Riemann surfaces. Tori are examples of compact Riemann surfaces. This means that we generalize the theory of elliptic functions here. A compact Riemann surface can be associated with any algebraic function, and in this way we obtain all compact Riemann surfaces. The compact Riemann surfaces achieve the same result for the integration of algebraic functions as does the theory of elliptic functions for the elliptic integrals. The triumph of the theory of Riemann surfaces was that it made the “integrals of the first kind” understandable and solved the so-called Jacobi inversion problem. We have to go a long way to achieve this aim. At the end, we shall arrive at the best-known theorems of the theory of Riemann surfaces, such as the the Riemann–Roch theorem, Abel’s theorem, and the Jacobi inversion theorem. On the way, we must also understand the topology of compact Riemann surfaces. We shall treat the topological classification completely here.

Following results of Gunning on geometric realizations of projective structures on Riemann surfaces, we investigate more fully certain generalizations of such structures. We define the notion of a branched analytic cover on a Riemann surface $M$ (of genus $g$) and specialize this to the case of branched projective and affine structures. Establishing a correspondence between branched projective and affine structures on $M$ and the classical projective and affine connections on $M$ we show that if a certain linear homogeneous differential equation involving the connection has only meromorphic solutions on $M$ then the connection corresponds to a branched structure on $M$. Utilizing this fact we then determine classes of positive divisors on $M$ such that for each divisor $\mathfrak {D}$ in the appropriate class the branched structures having $\mathfrak {D}$ as their branch locus divisor form a nonempty affine variety. Finally we apply some of these results to study the structures on a fixed Riemann surface of genus 2.

Following results of Gunning on geometric realizations of projective structures on Riemann surfaces, we investigate more fully certain generalizations of such structures. We define the notion of a branched analytic cover on a Riemann surface $M$ (of genus $g$) and specialize this to the case of branched projective and affine structures. Establishing a correspondence between branched projective and affine structures on $M$ and the classical projective and affine connections on $M$ we show that if a certain linear homogeneous differential equation involving the connection has only meromorphic solutions on $M$ then the connection corresponds to a branched structure on $M$. Utilizing this fact we then determine classes of positive divisors on $M$ such that for each divisor $\mathfrak {D}$ in the appropriate class the branched structures having $\mathfrak {D}$ as their branch locus divisor form a nonempty affine variety. Finally we apply some of these results to study the structures on a fixed Riemann surface of genus 2.

Let (M,g) be a compact smooth Riemannian surface with boundary. In this paper, we use blowing-up analysis to prove that some Moser?Trudinger trace inequalities hold on certain function spaces, and that the extremal functions exist in those function spaces without any additional hypothesis on (M,g).

We give a general definition of the exponents of a meromorphic connection ∇ on a holomorphic vector bundle E of rank n over a compact Riemann surface X. We prove that they can be computed as invariants of a vector bundle E L canonically attached to E, which we construct and call the Levelt bundle of E, and whose degree (equal to the sum of the exponents) we estimate by upper and lower bounds (Fuchs' relations). We use this definition to construct, for every linear differential equation on a compact Riemann surface (with regular or irregular singularities), the companion bundle of the equation, a vector bundle endowed with a meromorphic connection that is equivalent to the given equation and has precisely the same singularities and the same set of exponents.

As we know from the Riemann-Roch theorem the topological classification of complex vector bundles E → M over compact Riemann surface is ‘very simple’: they are classified by one integer c 1(E), the first Chern class of E → M. The situation is quite different in the case of classification of holomorphic category. This could have been expected from our experience with the Teichmuller theory, the main chapter of the Riemann moduli problem.

This paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact Riemann surface having a prescribed Weil divisor (i.e., having prescribed zeros and poles with directional as well as multiplicity information included in the vector case). This new formalism unifies the earlier approach of Ball-Clancey (in the setting of trivial bundles over an abstract Riemann surface) with an earlier approach of the authors (where the Riemann surface was assumed to be the normalizing Riemann surface for an algebraic curve embedded in C 2 with determinantal representation, and the vector bundles were assumed to be presented as the kernels of linear matrix pencils). The main tool is a version of the Cauchy kernel appropriate for flat vector bundles over the Riemann surface. Our formula for the interpolating bundle map (in the special case of a single zero and a single pole) can be viewed as a generalization of the Fay trisecant identity from the usual line bundle case to the vector bundle case in terms of Cauchy kernels. In particular we obtain a new proof of the Fay trisecant identity.