On the algebraicity of the zero locus of an admissible normal function

Abstract

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic.In Part 2 of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the sl 2 -splitting of a real mixed Hodge structure. Corollary 1.3. If S is algebraic then the zero locus of an admissible normal function ν : S → J(H) is an algebraic subvariety of S.

Log Néron models over surfaces, II

“…Since (2) is the special case of (1), we prove only (1). We see that, for any element N of σ, there are non-negative rational numbers a j (1 ≤ j ≤ m) such that N = a j N j and that N (e) = a j h j .…”

“…Here, as an application of Corollary 1.14, we give an alternative proof of the following theorem, which is a special case of a theorem by P. Brosnan and G. Pearlstein [1] (independently proved by C. Schnell [13] and by K. Kato, C. Nakayama, and S. Usui [6]). [6] corresponding to this step: In the second last paragraph in Introduction of [6], "k = 0, 1" should be "k = 0, w, where w is the weight of H in 0.2".…”

“…(1) Since N 1 is injective, there is a positive ε 0 ≤ 1 such that for any positive rational ε ≤ ε 0 , the operator (1 − ε)N 1 + εN 2 is also injective. Hence, by replacing N 2 by (1 − ε 0 )N 1 + ε 0 N 2 , and X by H 1 + H 1+ε 0 , we may assume that (1 − ε)N 1 + εN 2 is injective for any 0 ≤ ε ≤ 1.…”

“…As an application, we give an alternative, simple proof of the case of the surface base of the theorem by P. Brosnan and G. Pearlstein [1] on the algebraicity of zero loci of admissible normal functions (independently proved by C. Schnell [13] and by K. Kato, C. Nakayama, and S. Usui [6]). …”

Book Review: Mumford–Tate groups and domains: their geometry and arithmetic

“…One of the main applications of the compactificationsJ →S to date has been to show that the zero locus Z(ν) of ν is an algebraic subvariety of S [8,39,63]. The vanishing of ν can also be described in terms of having an extra Hodge tensor, and the analogous question about the field of definition of Z(ν) is closely connected to some of the conjectural properties of the Bloch-Beilinson filtration.…”