Convergence of Narasimhan-Simha measures on degenerating families of Riemann surfaces

Abstract

Given a compact Riemann surface Y and a positive integer m, Narasimhan and Simha defined a measure on Y associated to the m-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by Narasimhan-Simha measure. For m = 1, both these measures coincide with the Bergman measure on Y . We also extend the definition of the Narasimhan-Simha measure to the singular curves on the boundary of Mg in such a way that these measures form a continuous family of measures on the universal curve over Mg.

Global pluripotential theory on hybrid spaces

“…Assume for instance that (µ t ) t∈D * is a continuous family of probability measures on X, such that µ t is supported on X t for each t ∈ D * . Since the hybrid space provides a canonical compactification of X over the puncture, it is a natural question to ask whether or not the family of measures converges on X hyb , at least in a weak sensemore concrete examples of such situations will be given in Sections 4.3 and 5.1; see also [Shi20a], [Shi20b].…”

Submultiplicative norms and filtrations on section rings

“…In family setting, the study of positivity of related (pseudo)norms is linked to Iitaka conjecture and invariance of plurigenera problem, see Kawamata [16], Berndtsson-Pȃun [9], Pȃun-Takayama [21]. See also Amini-Nicolussi [1], [2] and Shivaprasad [25], [24] for other related result in singular family setting.…”