Title: Realizability of tropical canonical divisors,
Abstract: We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on
compactifications of strata of abelian differentials to give a comprehensive
solution to the realizability problem for effective tropical canonical divisors
in equicharacteristic zero. Given a pair $(\Gamma, D)$ consisting of a stable
tropical curve $\Gamma$ and a divisor $D$ in the canonical linear system on
$\Gamma$, we give a purely combinatorial condition to decide whether there is a
smooth curve $X$ over a non-Archimedean field whose stable reduction has
$\Gamma$ as its dual tropical curve together with a effective canonical divisor
$K_X$ that specializes to $D$. Along the way, we develop a moduli-theoretic
framework to understand Baker's specialization of divisors from algebraic to
tropical curves as a natural toroidal tropicalization map in the sense of
Abramovich-Caporaso-Payne.