Compact Riemann Surface Research Articles

On certain Tannakian categories of integrable connections over Kähler manifolds

AbstractGiven a compact Kähler manifold X, it is shown that pairs of the form $(E,\, D)$ , where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on E, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

Nilpotent Higgs Bundles and the Hodge Metric on the Calabi–Yau Moduli

Abstract We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi–Yau moduli.

Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions e λ e_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ → ∞ \lambda\to\infty if 0 < | ν | λ < 1 - δ 0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta for any fixed 0 < δ < 1 0<\delta<1 . Our result implies, for instance, that the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) .

On the orders of largest groups of automorphisms of compact Riemann surfaces

Let μ ( g ) be the largest possible order of a group of automorphisms of a compact Riemann surface of genus g ≥ 2 . The celebrated results of Accola-Maclachlan and Hurwitz-Macbeath, taken together say that μ ( g ) ranges between 8 ( g + 1 ) and 84 ( g − 1 ) and these two bounds are attained for infinitely many g . Here we prove that given a prime q > 167 congruent to −1 modulo 3, not congruent +1 modulo 5 and a prime p ≥ 84 q , μ ( g ) = ( 8 ( q + 4 ) / q ) ( g − 1 ) for g = q p m + 1 for infinitely many integers m . Furthermore we also prove that for any prime q for which p = q + 2 is a prime, μ ( g ) = ( 12 ( q + 2 ) / q ) ( g − 1 ) for g = q p m + 1 for infinitely many m . These mean that there exist an infinite rational sequence ( r n ) convergent to 8 and, assuming the truth of the Twin Prime Conjecture, an infinite rational sequence ( s n ) convergent to 12 together with infinite sets R n and S n of integers for which μ ( g ) = { r n ( g − 1 ) if g ∈ R n s n ( g − 1 ) if g ∈ S n and 8 , 12 are the unique numbers for which this can happen. A consequence of these results is the full understanding of the asymptotic's of μ . Namely if instead of μ = μ ( g ) we consider its normalization μ ˜ ( g ) = μ ( g ) / g then { 8 } ⊆ ( M d ) d ⊆ { 8 , 12 } , where M stands for the set of values of μ ˜ and the operator d for the set of accumulation points and, assuming the truth of the Twin Prime Conjecture, 12 ∈ ( M d ) d .

Smooth covers of moduli stacks of Riemann surfaces with symmetry

We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.

HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES

In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → P is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.

Torelli theorem for the moduli space of symplectic parabolic Higgs bundles

Let (X,D) and (X′,D′) be two compact Riemann surfaces of genus g≥4 with the set of marked points D⊂X and D′⊂X′. Fix a parabolic line bundle L with trivial parabolic structure. Let NSp(2m,α,L) and NSp′(2m,α,L) be the moduli spaces of stable symplectic parabolic Higgs bundles over X and X′ respectively, with rank 2m and fixed parabolic structure α, with the symplectic form taking values in L. We prove that if NSp(2m,α,L) is isomorphic to NSp′(2m,α,L), then there exist an isomorphism between X and X′ sending D to D′.

Distribution value of algebraic curves and the Gauss maps on algebraic minimal surfaces

In this paper, we first establish a degeneracy second main theorem for algebraic curves from compact complex Riemann surfaces into projective varieties intersecting hypersurfaces in subgeneral position. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in [Formula: see text] with finite total curvature in the case where the Gauss map may be algebraic degenerate. Our results generalize and improve the previous results in the field.

Complete Classification of Degree 7 for Genus 1

Assume that is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r-tuples ( of permutations in the symmetric group , in which and s generate a transitive subgroup G of This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.

A Hodge theoretic projective structure on compact Riemann surfaces

Given any compact Riemann surface C, there is a canonical meromorphic 2–form ηˆ on C×C, with pole of order two on the diagonal Δ⊂C×C, constructed in [4]. This meromorphic 2–form ηˆ produces a canonical projective structure on C. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface C. We prove that these two projective structures differ in general. This is done by comparing the (0,1)–component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. The (0,1)–component of the differential of the section corresponding to the projective structure given by the uniformization theorem was computed by Zograf and Takhtadzhyan in [16] as the Weil–Petersson Kähler form ωwp on the moduli space of curves. We prove that the (0,1)–component of the differential of the section of the moduli space of projective structures corresponding to ηˆ is the pullback of a nonzero constant scalar multiple of the Siegel form, on the moduli space of principally polarized abelian varieties, by the Torelli map.

Shared values of meromorphic functions on a compact Riemann surface

We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions.

On families of Riemann surfaces with automorphisms

In this article we determine the maximal possible order of the automorphism group of the form $ag + b$, where $a$ and $b$ are integers, of a complex three and four-dimensional family of compact Riemann surfaces of genus $g$, appearing for all genus. In addition, we construct and describe explicit complex three and four-dimensional families possessing these maximal number of automorphisms.

The volume of the quiver vortex moduli space

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vacuum expectation value of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey–Kirwan residue formula) leads to the Bradlow bounds ( upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss the properties of the moduli space volume in these theories. Our formulae are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with $\mathbb{C} P^N$ target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

4d F(4) gauged supergravity and black holes of class \u2131

We perform a consistent reduction of 6d matter-coupled F(4) supergravity on a compact Riemann surface {Sigma}_{mathfrak{g}} of genus mathfrak{g} , at the level of the bosonic action. The result is an mathcal{N} = 2 gauged supergravity coupled to two vector multiplets and a single hypermultiplet. The four-dimensional model is holographically dual to the 3d superconformal field theories of class ℱ, describing different brane systems in massive type IIA and IIB wrapped on {Sigma}_{mathfrak{g}} . The naive reduction leads to a non-standard 4d mixed duality frame with both electric and magnetic gauge fields, as well as a massive tensor, that can be only described in the embedding tensor formalism. Upon a chain of electromagnetic dualities, we are able to determine the scalar manifolds and electric gaugings that uniquely specify the model in the standard supergravity frame. We then use the result to construct the first examples of static dyonic black holes in AdS6 and perform a microscopic counting of their entropy via the 5d topologically twisted index. Finally, we show the existence of further subtruncations to the massless sector of the 4d theory, such as the Fayet-Iliopoulos gauged T3 model and minimal gauged supergravity. We are in turn able to find new asymptotically AdS4 solutions, providing predictions for the squashed S3 partition functions and the superconformal and refined twisted indices of class ℱ theories.

From the Hitchin section to opers through nonabelian Hodge

For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point $\mathbf{u}$ of the base of Hitchin’s integrable system for $(G,C)$. One family $\nabla_{\hbar ,\mathbf{u}}$ consists of $G$-opers, and depends on $\hbar \in \mathbb{C}^\times$. The other family $\nabla_{R, \zeta,\mathbf{u}}$ is built from solutions of Hitchin’s equations, and depends on $\zeta \in \mathbb{C}^\times , R \in \mathbb{R}^+$. We show that in the scaling limit $R \to 0, \zeta = \hbar R$, we have $\nabla_{R,\zeta,\mathbf{u}} \to \nabla_{\hbar,\mathbf{u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.

A Trudinger-Moser inequality with mean value zero on a compact Riemann surface with boundary

In this paper, on a compact Riemann surface $(\Sigma, g)$ with smooth boundary $\partial\Sigma$, we concern a Trudinger-Moser inequality with mean value zero. To be exact, let $\lambda_1(\Sigma)$ denotes the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition and $\mathcal{ S }= \left\{ u \in W^{1,2} (\Sigma, g) : \|\nabla_g u\|_2^2 \leq 1\right.$ and $\left.\int_\Sigma u \,dv_g = 0 \right \},$ where $W^{1,2}(\Sigma, g)$ is the usual Sobolev space, $\|\cdot\|_2$ denotes the standard $L^2$-norm and $\nabla_{g}$ represent the gradient. By the method of blow-up analysis, we obtain \begin{eqnarray*} \sup_{u \in \mathcal{S}} \int_{\Sigma} e^{ 2\pi u^{2} \left(1+\alpha\|u\|_2^{2}\right) }d v_{g} 0$. Based on the similar work in the Euclidean space, which was accomplished by Lu-Yang \cite{Lu-Yang}, we strengthen the result of Yang \cite{Yang2006IJM}.

Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary

<p style='text-indent:20px;'>In this paper, we establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals on a compact Riemann surface <inline-formula><tex-math id="M1">\begin{document}$ ( \Sigma,g) $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Sigma $\end{document}</tex-math></inline-formula>. To be exact, let <inline-formula><tex-math id="M3">\begin{document}$ \lambda_1(\partial\Sigma) $\end{document}</tex-math></inline-formula> denotes the first eigenvalue of the Laplace-Beltrami operator <inline-formula><tex-math id="M4">\begin{document}$ \Delta _ { g} $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M5">\begin{document}$ \partial \Sigma $\end{document}</tex-math></inline-formula>. Moreover, for any <inline-formula><tex-math id="M6">\begin{document}$ 0\leq\alpha&lt;\lambda_1(\partial\Sigma) $\end{document}</tex-math></inline-formula>, we set <inline-formula><tex-math id="M7">\begin{document}$ \mathcal { H } = \{ u \in W^{1,2} ( \Sigma, g) : \left(\int _{\Sigma} |\nabla_g u|^2 dv_g -\alpha \int _{\partial\Sigma} {u^2}ds_g \right)^{1/2}\leq 1 \ \, \mathrm{and}\, \int _{\partial\Sigma} {u}\,ds_g = 0 \} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M8">\begin{document}$ W^{1,2}(\Sigma, g) $\end{document}</tex-math></inline-formula> is the usual Sobolev space. By the method of blow-up analysis, we first prove the supremum <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \sup\limits_{ u \in \mathcal { H } }\int _ { \partial\Sigma } e ^ {\pi u^ 2} ds_g \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>is attained by some function <inline-formula><tex-math id="M9">\begin{document}$ u_\alpha \in \mathcal{H}\cap C^{\infty} \left(\overline{ \Sigma}\right) $\end{document}</tex-math></inline-formula>. Further, we extend the result to the case of higher order eigenvalues. The results generalize those of Li-Liu [<xref ref-type="bibr" rid="b9">9</xref>] and Yang [<xref ref-type="bibr" rid="b19">19</xref>, <xref ref-type="bibr" rid="b20">20</xref>].

Boundary and lens rigidity for non-convex manifolds

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the X-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations, allowing a non-empty trapped set.

Universal commensurability augmented Teichmüller space and moduli space

It is known that every unbranched finite covering \(\alpha\colon\widetilde{S}_{g(\alpha)}\rightarrow S\) of a compact Riemann surface \(S\) with genus \(g\geq 2\) induces an isometric embedding \(\Gamma_{\alpha}\) from the Teichmuller space \(T(S)\) to the Teichmuller space \(T(\widetilde{S}_{g(\alpha)})\). Actually, it has been showed that the isometric embedding \(\Gamma_{\alpha}\) can be extended isometrically to the augmented Teichmuller space \(\widehat{T}(S)\) of \(T(S)\). Using this result, we construct a direct limit \(\widehat{T}_{\infty}(S)\) of augmented Teichmuller spaces, where the index runs over all unbranched finite coverings of \(S\). Then, we show that the action of the universal commensurability modular group \(\operatorname{Mod}_{\infty}(S)\) can extend isometrically on \(\widehat{T}_{\infty}(S)\). Furthermore, for any \(X_{\infty}\in T_{\infty}(S)\), its orbit of the action of the universal commensurability modular group \(\operatorname{Mod}_{\infty}(S)\) on \(\widehat{T}_{\infty}(S)\) is dense. Finally, we also construct a direct limit \(\widehat{M}_{\infty}(S)\) of augmented moduli spaces by characteristic towers and show that the subgroup \(\operatorname{Caut}(\pi_{1}(S))\) of \(\operatorname{Mod}_{\infty}(S)\) acts on \(\widehat{T}_{\infty}(S)\) to produce \(\widehat{M}_{\infty}(S)\) as the quotient.

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $\mathfrak{gl}(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\cat{M}'_{lc}(n,d)$ and $\cat{M}_{lc}(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^nE, \tilde{D}) \cong (L,D_L)$ respectively. Let $\cat{M}'_{lc}(n,L) \subset \cat{M}_{lc}(n,L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{lc}(n,d)$ and $\cat{M}'_{lc}(n,L)$ and compute their Picard groups. We also show that $\cat{M}'_{lc}(n,L)$ and hence $\cat{M}_{lc}(n,L)$ do not have any non-constant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over $S$, with arbitrary residues.