A hyperbolic metric and stability conditions on K3 surfaces with ρ=1

Abstract

In this article we introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank $\rho (X)=1$. And we show that all walls are geodesic in the normalized space with respect to the hyperbolic metric. Furthermore we demonstrate how the hyperbolic metric is helpful for us by discussing mainly three topics. We first make a study of so called Bridgeland's conjecture. In the second topic we prove a famous Orlov's theorem without the global Torelli theorem. In the third topic we give an explicit example of stable complexes in large volume limits by using the hyperbolic metric. Though Bridgeland's conjecture may be well-known for algebraic geometers, we would like to start from the review of it.

arXiv comment27 pages, 4 figures. In version 2, the title was slightly changed. We added one reference article and change the proof of the first main theorem Theorem 1.2. In version 3, Appendix was removed and Section 7 was added

References in corpus (4)

• Markov convexity and local rigidity of distorted metrics
• Computing the Walls Associated to Bridgeland Stability Conditions on Projective Surfaces
• Cusps of the Kähler moduli space and stability conditions on K3 surfaces
• Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications