New systems of log-canonical coordinates on $SL(2,\mathbb{C})$ character varieties of compact Riemann surfaces

Abstract

We construct new sets of log-canonical coordinates on the $SL(2,\mathbb{C})$ character variety of compact Riemann surfaces. These are labelled by families of $1\le m\le 3g-3$ non-intersecting simple loops on the Riemann surface and are obtained by combining complexified shear-type with length/twist-type coordinates. In the case $m=3g-3$ the loops define a trinion decomposition of the Riemann surface, and our coordinates are closely related to the (complexified) Fenchel-Nielsen ones.

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