# Special Kähler structures, cubic differentials and hyperbolic metrics

@article{Haydys2018SpecialKS, title={Special K{\"a}hler structures, cubic differentials and hyperbolic metrics}, author={Andriy Haydys and Bin Xu}, journal={Selecta Mathematica}, year={2018}, volume={26}, pages={1-21} }

We obtain necessary conditions for the existence of special Kähler structures with isolated singularities on compact Riemann surfaces. We prove that these conditions are also sufficient in the case of the Riemann sphere and, moreover, we determine the whole moduli space of special Kähler structures with fixed singularities. The tool we develop for this aim is a correspondence between special Kähler structures and pairs consisting of a cubic differential and a hyperbolic metric.

#### One Citation

Twisted and Singular Gravitating Vortices

- Mathematics, Physics
- The Journal of Geometric Analysis
- 2021

We introduce the notion of twisted gravitating vortex on a compact Riemann surface. If the genus of the Riemann surface is greater than 1 and the twisting forms have suitable signs, we prove an… Expand

#### References

SHOWING 1-10 OF 27 REFERENCES

Realisation of special Kahler manifolds as parabolic spheres

- Mathematics
- 1999

We prove that any simply connected special Kaihler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any… Expand

Local models of isolated singularities for affine special Kähler structures in dimension two

- Mathematics, Physics
- 2017

We construct local models of isolated singularities for special K\"ahler structures in real dimension two assuming that the associated holomorphic cubic form does not have essential singularities. As… Expand

Affine special Kaehler structures in real dimension two

- Mathematics, Physics
- 2017

We review properties of affine special Kaehler structures focusing on singularities of such structures in the simplest case of real dimension two. We describe all possible isolated singularities and… Expand

Isolated Singularities of Affine Special Kähler Metrics in Two Dimensions

- Mathematics, Physics
- 2015

We prove that there are just two types of isolated singularities of special Kähler metrics in real dimension two, provided the associated holomorphic cubic form does not have essential singularities.… Expand

Teichmüller Theory for Conic Surfaces

- Mathematics
- 2017

This paper develops the local deformation theory, and some aspects of the global Teichmuller theory, of constant curvature metrics on a surface Σ with a finite number of conic singularities, with all… Expand

Special Kähler Manifolds

- 1999

We give an intrinsic definition of the special geometry which arises in global supersymmetry. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any… Expand

A note on special Kähler manifolds

- Mathematics, Physics
- 1999

We proved a conjecture of D. Freed that there are no non-trivial complete special Kaehler manifolds.

Collapsing K3 Surfaces, Tropical Geometry and Moduli Compactifications of Satake, Morgan-Shalen Type

- Mathematics
- 2018

We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study… Expand

Large Complex Structure Limits of K3 Surfaces

- Mathematics
- 2000

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahler… Expand

THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE

- Mathematics
- 1987

In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean… Expand