2014/05/13 by Burgos, José, Ulf Kühn, Kühn, Ulf +3
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Algebraic Geometry and Number Theory #Plant-based Medicinal Research #Alkaloids: synthesis and pharmacology
paper · pdf · doi:10.48550/arxiv.1405.3075
A theorem by Mumford implies that every automorphic line bundle on a pure\nopen Shimura variety, equipped with an invariant smooth metric, can be uniquely\nextended as a line bundle on a toroidal compactification of the variety, in\nsuch a way that the metric acquires only logarithmic singularities. This result\nis the key of being able to compute arithmetic intersection numbers from these\nline bundles. Hence it is natural to ask whether Mumford's result remains valid\nfor line bundles on mixed Shimura varieties.\n In this paper we examine the simplest case, namely the sheaf of Jacobi forms\non the universal elliptic curve. We show that Mumford's result cannot be\nextended directly to this case and that a new interesting kind of singularities\nappears.\n By using the theory of b-divisors, we show that an analogue of Mumford's\nextension theorem can be obtained. We also show that this extension is\nmeaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type of\nformula.\n