Mabuchi rays, test configurations and quantization for toric manifolds

Abstract

We consider Mabuchi rays of toric K\"ahler structures on symplectic toric manifolds which are associated to toric test configurations and that are generated by convex functions on themoment polytope, $P$, whose second derivative has support given by a compact subset $K<P$. Associated to the test configuration there is a polyhedral decomposition of $P$ whose components are approximated by the components of $P \setminus K$. Along such Mabuchi rays, the toric complex structure remains unchanged on the inverse image under the moment map of $(P \setminus \check {K})$, where $\check {K}$ denotes the interior of $K$. At infinite geodesic time, the K\"ahler polarizations along the ray converge to interesting new toric mixed polarizations. The quantization in these limit polarizations is given by restrictions of the monomial holomorphic sections of the K\"ahler quantization, for monomials corresponding to integral points in $P \setminus \check {K}$, and by sections on the fibers of the moment map over the integral points contained in $\check {K}$, which, along the directions parallel to $K$ are holomorphic and which along the directions transverse to $K$ are distributional. These quantizations correspond to quantizations of the central fiber of the test family, in the symplectic picture. We present the case of $S2$ in detail and then generalize to higher dimensional symplectic toric manifolds. Metrically, at infinite Mabuchi geodesic time, the sphere decomposes into two discs and a collection of cylinders, separated by infinitely long lines. Correspondingly, the quantization in the limit polarization decomposes into a direct sum of the contributions from the quantizations of each of these components.