Holomorphic Differentials Research Articles

The canonical representation of the Drinfeld curve

AbstractIf is a smooth projective curve over an algebraically closed field and is a group of automorphisms of , the canonical representation of is given by the induced ‐linear action of on the vector space of holomorphic differentials on . Computing it is still an open problem in general when the cover is wildly ramified. In this paper, we fix a prime power , we consider the Drinfeld curve, that is, the curve given by the equation over together with its standard action by , and decompose as a direct sum of indecomposable representations of , thus solving the aforementioned problem in this case.

Planar minimal surfaces with polynomial growth in the Sp(4,ℝ)-symmetric space

abstract: We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\Sp(4,\R)$-symmetric space. We describe a homeomorphism between the "Hitchin component" of wild $\Sp(4,\R)$-Higgs bundles over $\CP^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\h^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\R^4$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\h^{2,2}$ associated to $\Sp(4,\R)$-Hitchin representations along rays of holomorphic quartic differentials.

The Grunsky norm of univalent functions and abelian holomorphic differentials

The Grunsky norm of univalent functions and abelian holomorphic differentials

Bounded differentials on the unit disk and the associated geometry

For a harmonic diffeomorphism between the Poincaré disks, Wan [J. Differential Geom. 35 (1992), pp. 643–657] showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to r r -differentials. We study the relationship between bounded holomorphic r r -differentials/ ( r − 1 ) (r-1) -differential and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space S L ( r , R ) / S O ( r ) SL(r,\mathbb R)/SO(r) arising from cyclic/subcyclic Higgs bundles. Also, we show the equivalence between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in R 3 \mathbb {R}^3 , maximal surfaces in H 2 , n \mathbb {H}^{2,n} and J J -holomorphic curves in H 4 , 2 \mathbb {H}^{4,2} . Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods.

The Liouville current of holomorphic quadratic differential metrics

In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in Bankovic and Leininger (Trans Am Math Soc 370:1867–1884, 2018) that for a fixed closed surface, there is an injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain a new proof of a classical result that almost all simple geodesics of a quadratic differential metric will be dense in the surface. Furthermore, for other flat cone metrics, there is no simple dense geodesic.

The rank of the Cartier operator on Picard curves

For an algebraic curve [Formula: see text] defined over an algebraically closed field of characteristic [Formula: see text], [Formula: see text]-number [Formula: see text] is the dimension of the space of exact holomorphic differentials on [Formula: see text]. We computed the [Formula: see text]-number for a family of certain Picard curves using the action of the Cartier operator on [Formula: see text].

Algebraic de Rham theorem and Baker-Akhiezer function

For the case of algebraic curves (compact Riemann surfaces), it is shown that de Rham cohomology group $H^1_{\mathrm{dR}}(X,\mathbb{C})$ of a genus $g$ of the Riemann surface $X$ has a natural structure of a symplectic vector space. Every choice of a non-special effective divisor $D$ of degree $g$ on $X$ defines a symplectic basis of $H^1_{\mathrm{dR}}(X,\mathbb{C})$ consisting of holomorphic differentials and differentials of the second kind with poles on $D$. This result, which is the algebraic de Rham theorem, is used to describe the tangent space to Picard and Jacobian varieties of $X$ in terms of differentials of the second kind, and to define a natural vector fields on the Jacobian of the curve $X$ that move points of the divisor $D$. In terms of the Lax formalism on algebraic curves, these vector fields correspond to the Dubrovin equations in the theory of integrable systems, and the Baker-Akhierzer function is naturally obtained by the integration along the integral curves.

The Harmonic Map Compactification of Teichmüller Spaces for Punctured Riemann Surfaces

In the paper [The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479], Wolf provided a global coordinate system of the Teichmüller space of a closed oriented surface S S with the vector space of holomorphic quadratic differentials on a Riemann surface X X homeomorphic to S S . This coordinate system is via harmonic maps from the Riemann surface X X to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmüller space by adding a point at infinity to each endpoint of harmonic map rays starting from X X in the space. Wolf also showed this compactification coincides with the Thurston compactification. In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification.

A rigidity result for holomorphic quadratic differentials of finite norm in the unit disk

AbstractLet be a holomorphic quadratic differential of finite norm in the unit disk . Let be equipped with the metric . In this paper, we prove that a geodesic‐preserving map of the unit disk into itself is affine. A map is called geodesic‐preserving if the image of each geodesic is a geodesic. We make no assumptions on the continuity of such a map.

𝑝-group Galois covers of curves in characteristic 𝑝

We study cohomologies of a curve with an action of a finite p p -group over a field of characteristic p p . Assuming the existence of a certain “magical element” in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic p p -group cover has a “magical element”. As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order.

Length spectrum compactification of the SO0(2,3)-Hitchin component

We find a compactification of the SO0(2,3)-Hitchin component by studying the degeneration of the induced metric on the unique equivariant maximal surface in the 4-dimensional pseudo-hyperbolic space H2,2. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic quartic differentials on a Riemann surface. As an application, we describe the behavior of the entropy of the induced metric along rays of quartic differentials.

Holomorphic differentials of Klein four covers

Let k be an algebraically closed field of characteristic two, and let G be isomorphic to Z/2×Z/2. Suppose X is a smooth projective irreducible curve over k with a faithful G-action, and assume that the cover X→X/G is totally ramified, in the sense that it is ramified and every branch point is totally ramified. We study to what extent the lower ramification groups of the closed points of X determine the isomorphism types of the indecomposable kG-modules and the multiplicities with which they occur as direct summands of the space H0(X,ΩX/k) of holomorphic differentials of X over k. In the case when X/G=Pk1, we completely determine the decomposition of H0(X,ΩX/k) into a direct sum of indecomposable kG-modules. Moreover, we show that the isomorphism classes of indecomposable kG-modules that actually occur as direct summands belong to an infinite list of non-isomorphic indecomposable kG-modules that contain modules of arbitrarily large k-dimension. In particular, our results show that [14, Theorem 6.4] is incorrect.

Complete curves in the strata of differentials

Gendron proved that the strata of holomorphic differentials with prescribed orders of zeros do not contain complete algebraic curves by applying the maximum modulus principle to saddle connections. Here we provide an alternative proof for this result by using positivity of divisor classes on moduli spaces of curves.

Poles of cubic differentials and ends of convex $\mathbb{RP}^2$-surfaces

On any oriented surface, the affine sphere construction gives a one-to-one correspondence between convex $\mathbb{RP}^2$-structures and holomorphic cubic differentials. Generalizing results of Benoist–Hulin, Loftin and Dumas–Wolf, we show that poles of order less than $3$ of cubic differentials correspond to finite-volume ends of convex $\mathbb{RP}^2$-surfaces, whereas poles of order $3$ and greater than $3$ correspond to geodesic and piecewise geodesic ends, respectively. Moreover, at each pole of order at least $3$, we construct a natural bordification of the surface such that the $\mathbb{RP}^2$-structure extends to the boundary circle in a metric preserving way.

Towards logarithmic GLSM : the r–spin case

In this article, we establish foundations for a logarithmic compactification of general GLSM moduli spaces via the theory of stable log maps. We then illustrate our method via the key example of Witten's $r$-spin class. In the subsequent articles, we will push the technique to the general situation. One novelty of our theory is that such a compactification admits two virtual cycles, a usual virtual cycle and a "reduced virtual cycle". A key result of this article is that the reduced virtual cycle in the $r$-spin case equals to the r-spin virtual cycle as defined using cosection localization by Chang--Li--Li. The reduced virtual cycle has the advantage of being $\mathbb{C}^*$-equivariant for a non-trivial $\mathbb{C}^*$-action. The localization formula has a variety of applications such as computing higher genus Gromov--Witten invariants of quintic threefolds and the class of the locus of holomorphic differentials.

The a-Number of Certain Hyperelliptic Curves

The a-number of an algebraic curve $${\mathcal {X}}$$ is the dimension of the space of exact holomorphic differentials on $${\mathcal {X}}$$ . In this paper, we give a formula for the a-number of family of hyperelliptic curves defined by the equations $$y^2=x^{m}+1$$ and $$y^2=x^{m}+x$$ for infinitely many values of m.

Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves

We study the combinatorial geometry of a random closed multicurve on a surface of large genus g and of a random square-tiled surface of large genus g. We prove that primitive components $$\gamma _1, \dots ,\gamma _k$$ of a random multicurve $$m_1\gamma _1+\dots +m_k\gamma _k$$ represent linearly independent homology cycles with asymptotic probability 1 and that all its weights $$m_i$$ are equal to 1 with asymptotic probability $$\sqrt{2}/2$$ . We prove analogous properties for random square-tiled surfaces. In particular, we show that all conical singularities of a random square-tiled surface belong to the same leaf of the horizontal foliation and to the same leaf of the vertical foliation with asymptotic probability 1. We show that the number of components of a random multicurve and the number of maximal horizontal cylinders of a random square-tiled surface of genus g are both very well approximated by the number of cycles of a random permutation for an explicit non-uniform measure on the symmetric group of $$3g-3$$ elements. In particular, we prove that the expected value of these quantities has asymptotics $$(\log (6g-6) + \gamma )/2 + \log 2$$ as $$g\rightarrow \infty $$ , where $$\gamma $$ is the Euler–Mascheroni constant. These results are based on our formula for the Masur–Veech volume $${\text {Vol}}{\mathcal {Q}}_g$$ of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A. Aggarwal and with the uniform asymptotic formula for intersection numbers of $$\psi $$ -classes on $${\overline{{\mathcal {M}}}}_{g,n}$$ for large g proved by A. Aggarwal in 2020.

The Heights Theorem for infinite Riemann surfaces

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the horizontal map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary Riemann surface with a conformal hyperbolic metric. This was established by Strebel in the case of the unit disk. When a hyperbolic surface has a bounded geodesic pants decomposition, the horizontal map assigns a bounded measured lamination to each integrable holomorphic quadratic differential. When surface has a sequence of closed geodesics whose lengths go to zero, then there exists an integrable holomorphic quadratic differential whose horizontal measured lamination is not bounded.

Tau Function and Moduli of Meromorphic Quadratic Differential

The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most n simple poles on genus g complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.

Towards a classification of connected components of the strata of $k$-differentials

A k -differential on a Riemann surface is a section of the k -th power of the canonical bundle. Loci of k -differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of k -differentials. The classification of connected components of the strata of k -differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich-Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the strata of k -differentials for general k . As an application, we give a complete classification of connected components of the strata of quadratic differentials with arbitrary poles. Moreover, we distinguish certain components of the strata of k -differentials by generalizing the hyperelliptic structure and spin parity for higher k . We also describe an approach to determine explicitly parities of k -differentials in genus zero and one, which inspires an amusing conjecture in number theory. A key viewpoint we use is the notion of multi-scale k -differentials introduced by Bainbridge-Chen-Gendron-Grushevsky-Möller for k = 1 and extended by Costantini-Möller-Zachhuber for all k .