  We determine complex saddles of three-dimensional gravity with a positive
cosmological constant by applying the recently proposed holography. It is
sometimes useful to consider a complexified metric to study quantum gravity as
in the case of the no-boundary proposal by Hartle and Hawking. However, there
would be too many saddles for complexified gravity, and we should determine
which saddles to take. We describe the gravity theory by three-dimensional
SL$(2,\mathbb{C})$ Chern-Simons theory. At the leading order in the Newton
constant, its holographic dual is given by Liouville theory with a large
imaginary central charge. We examine geometry with a conical defect, called a
de Sitter black hole, from a Liouville two-point function. We also consider
geometry with two conical defects, whose saddles are determined by the
monodromy matrix of Liouville four-point function. Utilizing Chern-Simons
description, we extend the similar analysis to the case with higher-spin
gravity.
