Compact Riemann Surface Of Genus Research Articles

A remark on the moduli space of Lie algebroid λ-connections

Let X be a compact Riemann surface of genus g ≥ 3 . Let L = ( L , [ . , . ] , ♯ ) be a holomorphic Lie algebroid over X of rank one and degree ( L ) < 2 − 2 g . We consider the moduli space of holomorphic L λ -connections over X, where λ ∈ C . We compute the Picard group of the moduli space of L λ -connections by constructing an explicit smooth compactification of the moduli space of those L λ -connections whose underlying vector bundle is stable such that the complement is a smooth divisor. We also show that the automorphism group of the moduli space of L λ -connections fits into a short exact sequence that involves the automorphism group of the moduli space of stable vector bundle over X. For λ = 1, we get Lie algebroid de Rham moduli space of L -connections and we determine its Chow group. Communicated by Manuel Reyes

Integrable vortices on compact Riemann surfaces of genus one

The Jackiw-Pi equation, which is one of the integrable vortex equations, is studied on a torus, a compact Riemann surface of genus one. The solutions are given in terms of doubly periodic functions, i.e., the elliptic functions. We reconsider the Jackiw-Pi vortex on a torus and provide the analytical method for determining the vortex number with explicit examples.

Phase diagram and topological expansion in the complex quartic random matrix model

AbstractWe use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers of 4‐valent connected graphs with j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann surfaces of genus 0,1,2, and 3. Also, for a Riemann surface of an arbitrary genus g, we calculate the leading term in the asymptotics of as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David. These phase transitions are of the following four types: (a) one‐cut to two‐cut through the splitting of the cut at the origin, (b) two‐cut to three‐cut through the birth of a new cut at the origin, (c) one‐cut to three‐cut through the splitting of the cut at two symmetric points, and (d) one‐cut to three‐cut through the birth of two symmetric cuts.

On the structure of connected components of the moduli spaces of compact Riemann surfaces

Using the Broughton’s equisymmetric stratification of the moduli space of compact Riemann surfaces of genus g ge 2, Bartolini, Costa and Izquierdo have shown that its singular locus S_{g} is disconnected except for g = 3, 4, 7, 13, 17, 19, 59. One of a lot of problems motivated by this result concerns the study of its connected components, both from the quantitative and qualitative points of view. In 2012, Bartolini and Izquierdo have shown that all strata corresponding to the actions of Z_{2} and Z_{3} are contained in a single connected component C^{2,3}_g. The other components which were known to these authors were those composed by single strata and they asked whether there are components different from C^{2,3}_ g being the union at least two equisymmetric strata. The aim of this paper is to give an affirmative answer to this question, by showing that there are connected components composed of precisely two strata in S_{g} for g = rp(p -1)/2 where r ge 6 and p > 5r + 3 is a prime for which pr + 2 is also a prime.

Orbifolds, the Regular Solids, and Hurwitz’s 84(g – 1) Theorem

This paper presents an exposition of the analysis via orbifolds of the isometries with isolated fixed points of compact surfaces, giving a view of the classification of the five regular solids, following the treatment by William Thurston. Then it is shown that a closely related method can be used to recover the classical theorem of Hurwitz that the group of holomorphic automorphisms of a compact Riemann surface of genus g>1 has order at most 84(g-1).

Period matrices of some hyperelliptic Riemann surfaces

Let X be a compact Riemann surface of genus \(g\ge 2\) and \(\Omega (X)\) the space of holomorphic 1-forms on X. A period matrix \(\Pi \) is a \(g\times g\) matrix defined by the pairing of a symplectic basis of \(H_1(X, {\mathbb {Z}})\) and a basis of \(\Omega (X)\). The matrix \(\Pi \) is symmetric and its imaginary part \(\textrm{Im}(\Pi )\) is positive definite and depends only on symplectic bases of \(H_1(X, {\mathbb {Z}})\). Period matrices are important tools for studying complex structures of Riemann surfaces. However, only few formulas for period matrices are known for Riemann surfaces of genus \(g \ge 2\). For Riemann surfaces of lower genus, there are only six Riemann surfaces whose period matrices are described by some formula. In this paper, our targets are algebraic curves defined by \(w^2=z(z^2-1)(z^2-a_1^2)(z^2-a_2^2)\cdots (z^2-a_{g-1}^2)\) for any \(g\ge 2\) and \(a_1, a_2, \dots , a_{g-1}\in {\mathbb {R}}\) which satisfy \(1<a_1<a_2<\cdots <a_{g-1}\). We describe period matrices of these algebraic curves by collections of complex integrals. To do this, we construct the curves from Euclidean polygons. A symplectic basis of these curves are given from the polygons.

Addendum to “On the orders of largest groups of automorphisms of compact Riemann surfaces” [J. Pure Appl. Algebra 225 (12) (2021) 106758]

Let μ ( g ) be the largest possible order of a group of automorphisms of a compact Riemann surface of genus g ≥ 2 . In the earlier paper we have showed together with Czesław Bagiński that there exist a rational sequence ( r n ) convergent to 8 and infinite sets R n so that μ ( g ) = r n ( g − 1 ) for g ∈ R n . Moreover, assuming the Twin Prime conjecture we have showed the same for 12 i.e. there exist a rational sequence ( s n ) convergent to 12 and infinite sets S n so that μ ( g ) = s n ( g − 1 ) for g ∈ S n . Here we prove the second assertion using Chen theorem instead of relying on TPC.

Picard group of moduli of parabolic Higgs bundles

Let X be a compact Riemann surface of genus and let be the moduli space of semistable parabolic bundles over X. Let denote the moduli space of semistable parabolic Higgs bundles over X. In this article, we study the Picard group of and .

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.

Chen–Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surface

Let \((X,\,D)\) be an m-pointed compact Riemann surface of genus at least 2. For each \(x \,\in \, D\), fix full flag and concentrated weight system \(\alpha \). Let \(P \mathcal {M}_{\xi }\) denote the moduli space of semi-stable parabolic vector bundles of rank r and determinant \(\xi \) over X with weight system \(\alpha \), where r is a prime number and \(\xi \) is a holomorphic line bundle over X of degree d which is not a multiple of r. We compute the Chen–Ruan cohomology of the orbifold for the action on \(P \mathcal {M}_{\xi }\) of the group of r-torsion points in \(\mathrm{Pic}^0(X)\).

On large prime actions on Riemann surfaces

Abstract In this article, we study compact Riemann surfaces of genus 𝑔 with an automorphism of prime order g + 1 g+1 . The main result provides a classification of such surfaces. In addition, we give a description of them as algebraic curves, determine and realise their full automorphism groups and compute their fields of moduli. We also study some aspects of their Jacobian varieties such as isogeny decompositions and complex multiplication. Finally, we determine the period matrix of the Accola–Maclachlan curve of genus four.

Shatz and Bialynicki-Birula stratifications of the moduli space of Higgs bundles

Let $X$ be a compact Riemann surface of genus $g\geq 2$. In this paper we study how two memorable stratifications of the moduli space of Higgs bundles of rank $4$ over $X$, Shatz stratification and Bialynicki-Birula stratification, relate to each other and provide dimensions for the Harder-Narasimhan type of a semistable rank 4 Higgs bundle over $X$. In particular, we prove that the two stratifications do not coincide. Thus, we extend to rank 4 the work of Hausel [7], who proved that both stratifications coincide in rank 2, and of Gothen and Zúñiga-Rojas [6], who proved that such a thing no longer occurred in rank 3.

Flat conical Laplacian in the square of the canonical bundle and its regularized determinants

Let $X$ be a compact Riemann surface of genus $g\geq 2$ equipped with flat conical metric $|\Omega|$, where $\Omega$ be a holomorphic quadratic differential on $X$ with $4g-4$ simple zeroes. Let $K$ be the canonical line bundle on $X$. Introduce the Cauchy-Riemann operators $\bar \partial$ and $\partial$ acting on sections of holomorphic line bundles over $X$ ($K^2$ in the definition of $\Delta^{(2)}_{|\Omega|}$ below) and, respectively, anti-holomorphic line bundles ($\bar { K}^{-1}$ below). Consider the Laplace operator $\Delta^{(2)}_{|\Omega|}:=|\Omega| \partial |\Omega|^{-2}\bar\partial$ acting in the Hilbert space of square integrable sections of the bundle $K^2$ equipped with inner product $ _{K^2}=\int_X\frac {Q_1\bar Q_2}{|\Omega|}$. We discuss two natural definitions of the determinant of the operator $\Delta^{(2)}_{|\Omega|}$. The first one uses the zeta-function of some special self-adjoint extension of the operator (initially defined on smooth sections of $K^2$ vanishing near the zeroes of $\Omega$), the second one is an analog of Eskin-Kontsevich-Zorich (EKZ) regularization of the determinant of the conical Laplacian acting in the trivial bundle. In contrast to the situation of operators acting in the trivial bundle, for operators acting in $K^2$ these two regularizations turn out to be essentially different. Considering the regularized determinant of $\Delta^{(2)}_{|\Omega|}$ as a functional on the moduli space $Q_g(1, \dots, 1)$ of quadratic differentials with simple zeroes on compact Riemann surfaces of genus $g$, we derive explicit expressions for this functional for the both regularizations. The expression for the EKZ regularization is closely related to the well-known explicit expressions for the Mumford measure on the moduli space of compact Riemann surfaces of genus $g$.

One-relator Sasakian groups

We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus $g \gt 0$ with at most one orbifold poi

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 .

Finite simple groups acting with fixity 3 and their occurrence as groups of automorphisms of Riemann surfaces

The results in this article are based on the classification of finite simple groups that act with fixity 3. Motivated by the theory of Riemann surfaces, we investigate which ones of these groups act faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed points on each non-regular orbit and at most four fixed points in total. In each case, we give detailed information about the possible branching data of the surface.

Riemann surfaces of genus 1 + q2 with 3q2 automorphisms

In this article we classify compact Riemann surfaces of genus 1+q2 with a group of automorphisms of order 3q2, where q is a prime number. We also study decompositions of the corresponding Jacobian varieties.

Weierstrass points on modular curves X0(N) fixed by the Atkin–Lehner involutions

PurposeThe authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions WQ on X0(N) for N ≤ 50 are Weierstrass points or not.Design/methodology/approachThe design is by using Lawittes's and Schoeneberg's theorems.FindingsFinding all Weierstrass points on X0(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.Originality/valueThe Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X0(N).

Non-flat elliptic four-folds, three-form cohomology and strongly coupled theories in four dimensions

In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus g in the base. These non-flat fibers contribute Kähler moduli to the four-fold but also add to the three-form cohomology for g > 0. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional mathcal{N} = (1, 0) superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively the non-flat fibers can be avoided by performing birational base changes analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant.

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′.