Holomorphic One-forms Research Articles

Faber Series for $$L^2$$ Holomorphic One-Forms on Riemann Surfaces with Boundary

Faber Series for $$L^2$$ Holomorphic One-Forms on Riemann Surfaces with Boundary

Nowhere Vanishing Holomorphic One-Forms on Varieties of Kodaira Codimension One

Abstract Based on the celebrated result on zeros of holomorphic 1-forms on complex varieties of general type by Popa and Schnell, we study holomorphic 1-forms on $n$-dimensional varieties of Kodaira dimension $n-1$. We show that a complex minimal smooth projective variety $X$ of Kodaira dimension $\kappa (X)=\dim X-1$ admits a holomorphic 1-form without zero if and only if there is a smooth morphism from $X$ to an elliptic curve. Furthermore, for a general smooth projective variety (not necessarily minimal) $X$ of Kodaira codimension one, we give a structure theorem for $X$ given that $X$ admits a holomorphic 1-form without zero.

Constraints in the BV formalism: Six-dimensional supersymmetry and its twists

We formulate the abelian six-dimensional N=(2,0) theory perturbatively, in a generalization of the Batalin–Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimal twists at the perturbative level. This calculation hinges on the existence of an L∞ action of the supersymmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appears naturally in the pure spinor superfield formalism, but understanding it requires developing a presymplectic generalization of the BV formalism, inspired by Dirac's theory of constraints. The holomorphic twist consists of symplectic-valued holomorphic bosons from the N=(1,0) hypermultiplet, together with a degenerate holomorphic theory of holomorphic one-forms from the N=(1,0) tensor multiplet, which can be seen to describe the infinitesimal intermediate Jacobian variety. We check that our formulation and our results match with known ones under various dimensional reductions, as well as comparing the holomorphic twist to Kodaira–Spencer theory. Matching our formalism to five-dimensional Yang–Mills theory after reduction leads to some issues related to electric–magnetic duality; we offer some speculation on a nonperturbative resolution.

Positive line modules over the irreducible quantum flag manifolds

Noncommutative K\"ahler structures were recently introduced by the second author as a framework for studying noncommutative K\"ahler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant K\"ahler structures of irreducible type (those having an irreducible space of holomorphic one-forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds $\mathcal{O}_q(G/L_S)$. Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over $\mathcal{O}_q(G/L_S)$ can be identified as positive, negative, or flat, and hence we can conclude that each K\"ahler structure is of Fano type. Moreover, it proves possible to extend the Borel-Weil theorem for $\mathcal{O}_q(G/L_S)$ to a direct noncommutative generalisation of the classical Bott-Borel-Weil theorem for positive line bundles.

Kodaira dimension and zeros of holomorphic one-forms, revisited

Kodaira dimension and zeros of holomorphic one-forms, revisited

Holomorphic one-forms without zeros on threefolds

We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real 6-manifold fibres smoothly over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three.

Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds

AbstractWe prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

Lefschetz theorem for holomorphic one-forms on weakly 1-complete manifolds

For a holomorphic one-form \({\xi }\) on a weakly 1-complete manifold X with certain properties, we will discuss the connectivity of the pair \((\hat{X},F^{-1}(z))\), where \(\pi :\hat{X} \rightarrow X\) is a covering map and F is a holomorphic function on \(\hat{X}\) such that \(dF=\pi ^*{\xi }\). We will also discuss the criteria about when such a manifold X admits a proper holomorphic mapping onto a Riemann surface.

Global Conformal Parameterization via an Implementation of Holomorphic Quadratic Differentials.

We propose an algorithm to compute global conformal parameterizations of high-genus meshes, which is based on an implementation of holomorphic quadratic differentials. First, we design a novel diffusion method which is capable of computing a pole-free discrete harmonic measured foliation. Second, we propose a definition for discrete holomorphic quadratic differential which consists of a horizontal and a vertical harmonic measured foliation. Third, we present a practical algorithm to approximate the discrete natural coordinates for a holomorphic quadratic differential, which represents a flat metric with cones conformal to the original metric, i.e., a parameterization. Finally, we apply the discrete natural coordinates for parameterization of high genus meshes. Our parameterization method is global conformal and simple to implement. The advantage of our method over the approach based on holomorphic differential one-forms is that ours has a larger space of parameterizations. We demonstrate our approach with hundreds of configurations on dozens of meshes to show its robustness on conformal parameterization.

Zeros of Holomorphic One-Forms and Topology of Kähler Manifolds

Abstract A conjecture of Kotschick predicts that a compact Kähler manifold X fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao [10], we use our approach to prove Kotschick’s conjecture for smooth projective three-folds.

Logarithmic Comparison with Smooth Boundary Divisor in Mixed Hodge Modules

We use filtered log-$\mathscr{D}$-modules to represent the (dual) localization of Saito's Mixed Hodge Modules along a smooth hypersurface, and show that they also behave well under the direct image functor and the dual functor in the derived category of filtered log-$\mathscr{D}$-modules. The results of this paper can be used to generalize the result of M. Popa and C. Schnell about Kodaira dimension and zeros of holomorphic one-forms into the log setting.

Plemelj\u2013Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators

Let R be a compact Riemann surface and Gamma be a Jordan curve separating R into connected components Sigma _1 and Sigma _2. We consider Calderón–Zygmund type operators T(Sigma _1,Sigma _k) taking the space of L^2 anti-holomorphic one-forms on Sigma _1 to the space of L^2 holomorphic one-forms on Sigma _k for k=1,2, which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves Gamma , to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to L^2 anti-holomorphic one-forms on R with respect to the inner product on Sigma _1. We show that the restriction of the Schiffer operator T(Sigma _1,Sigma _2) to V is an isomorphism onto the set of exact holomorphic one-forms on Sigma _2. Using the relation between this Schiffer operator and a Cauchy-type integral involving Green’s function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

Vector flows and the analytic moduli of singular plane branches

We provide a geometric elementary proof of the fact that an analytic plane branch is analytically equivalent to one whose terms corresponding to contacts with holomorphic one-forms—except for Zariski’s $$\lambda $$-invariant— are zero (so called “short parametrizations”). This is the main step missed by Zariski in his attempt to solve the moduli problem.

Calibrated entanglement entropy

The Ryu-Takayanagi prescription reduces the problem of calculating entanglement entropy in CFTs to the determination of minimal surfaces in a dual anti-de Sitter geometry. For 3D gravity theories and BTZ black holes, we identify the minimal surfaces as special Lagrangian cycles calibrated by the real part of the holomorphic one-form of a spacelike hypersurface. We show that (generalised) calibrations provide a unified way to determine holographic entanglement entropy from minimal surfaces, which is applicable to warped AdS3 geometries. We briefly discuss generalisations to higher dimensions.

Erratum to Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces (J. reine angew. Math. 697 (2014), 1–14)

Erratum to Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces (J. reine angew. Math. 697 (2014), 1–14)

Teichmüller curves in genus three and just likely intersections in G m n × G a n $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmuller curves. For the stratum $\Omega\mathcal{M}_{3}(4)$ , consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmuller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmuller curves in terms of cross-ratios of six points on $\mathbf{P}^{1}$ . These constraints are akin to those that appear in Zilber and Pink’s conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$ (rather than the more standard $\mathbf{G}_{m}^{n}$ ). The ambient algebraic group lies outside the scope of Zilber’s Conjecture but we are nonetheless able to prove a sufficiently strong height bound. For the generic stratum $\Omega\mathcal{M}_{3}(1,1,1,1)$ , we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of $\mathbf{G}_{m}^{9}$ . These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.

Kodaira dimension and zeros of holomorphic one-forms

We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge modules on abelian varieties.

Applications of the deformation formula of holomorphic one-forms

This paper studies some geometric aspects of moduli of curves Mg, using as a tool the deformation formula of holomorphic one-forms. Quasi-isometry guarantees the L 2 convergence of deformation of holomorphic one-forms, which is a kind of global result. After giving the period map a full expansion, we can also write out the Siegel metric, curvature and second fundamental form of a nonhyperelliptic locus of Mg in a quite detailed manner, while gaining some understanding of a totally geodesic manifold in a nonhyperelliptic locus. This paper is a complement to our joint paper [Liu et al. 2012b] with Kefeng Liu, and explores more applications of the deformation formula of holomorphic oneforms to some problems related to moduli spaces of Riemann surfaces, including the full expansion of the period map, the Siegel metric and its curvature formulae, the second fundamental form of Torelli space’s nonhyperelliptic locus, and also a global result about the deformation of holomorphic one-forms. We start with the Kuranishi coordinate of the Teichmuller space Tg of Riemann surfaces of genus g and the deformation formula of holomorphic one-forms. t/, whose construction is contained in Section 2. The key points of the deformation formula lie in Theorem 2.1. To be more precise, on the Kuranishi family$VX!1 with a Riemann surface $ 1 .0/D X0 as its central fiber and a global holomorphic one-form of the central fiber 2 H 0 .X0; 1 X0 /, the deformation formula of holomorphic one-forms emerges as

Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces

Abstract The first goal of this paper is to study the question of finiteness of integral points on a cofinite non-compact complex two-dimensional ball quotient defined over a number field. Along the process we will also consider some negatively curved compact surfaces. Using some fundamental results of Faltings, the question is to reduce to a conjecture of Borel about existence of virtual holomorphic one-forms on cofinite non-cocompact complex ball quotients. The second goal of this paper is to study the conjecture on such non-compact surfaces.

Gunning–Narasimhan’s Theorem with a Growth Condition

Given a compact Riemann surface X and a point x_0 in X, we construct a holomorphic function without critical points on the punctured Riemann surface R = X - x_0 which is of finite order at the point x_0. This complements the result of Gunning and Narasimhan from 1967 who constructed a noncritical holomorphic function on every open Riemann surface, but without imposing any growth condition. On the other hand, if the genus of X is at least one, then we show that every algebraic function on R admits a critical point. Our proof also shows that every cohomology class in H^1(X;C) is represented as a de Rham class by a nowhere vanishing holomorphic one-form of finite order on the punctured surface X-x_0.