Theta line bundles and the determinant of the Hodge bundle

Abstract

Abstract. We give an expression of the determinant of the push forward of a symmetric line bundle on a complex abelian fibration, in terms of the pull back of the relative dualizing sheaf via the zero section. Introduction Moreover, when L is totally symmetric (and therefore d is an even integer), we have Theorem B. Keeping the notation of Theorem A, assume in addition that L is a totally symmetric line bundle on X and thatThe theorems are proved by using a refinement of the theta transformation formula, see Propositions 2.1 and 2.2, in order to construct transition functions for det f * (L), see Lemma 3.1.In the last section, we apply Theorem B to the case of the universal Jacobian variety f g−1 : J g−1 −→ M g , where M g denotes the moduli space of smooth, irreducible curves of genus g ≥ 3, without automorphisms. This is an abelian torsor which parametrizes line bundles of degree g − 1 on the fibers of the universal curve ψ : C −→ M g . On J g−1 , there is a canonical theta divisor defined as the push forward of the universal symmetric product of degree g − 1, via the Abel-Jacobi

The transformation laws of algebraic theta functions

“…The question of finding the precise order of ∆ g,δ has been studied in [5], Thm. 5.1, [18], [8] and [10]. In particular, our approach to the question can be viewed as the algebraic version of [8], where the transformation laws of theta functions are employed to compute the order of ∆ g,δ .…”

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“…It is known from the transformation theory of theta-functions (see [10]) that this result is sharp for principal polarizations (d 1). In the case of analytic families of complex Abelian varieties A. Kouvidakis showed in [7] using theta functions that if the type of polarization is d 1 Y F F F Y d g with d 1 j F F F jd g then 4 Á DL 0 except when 3jd g and d gÀ1 T 0 mod3. In the latter case he proved that 12 Á DL 0.…”

“…Kouvidakis proved in [7] that for a totally symmetric line bundle L one always has 3 Á D H L 0 if g X 3 and the characteristic is zero (furthermore, D H L 0 if the 3-polarization type is not 1Y F F F Y 1Y 3 k , k b 0). It would be nice to extend this result to the case of positive characteristics.…”