A hyperbolic metric and stability conditions on $\mathrm{K}3$ surfaces with $\rho = 1$

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective $\mathrm{K}3$ surfaces $X$ with Picard rank $\rho (X) = 1$ and give some applications. 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 some topics. As an application of the metric, we give an explicit example of stable complexes in large volume limits.

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The author was partially supported by Grant-in-Aid for Scientific Research (S), No 22224001.

Received 27 September 2013

Accepted 6 September 2016

Published 12 December 2019