The zero locus of an admissible normal function. (English) Zbl 1184.32004

One of the deepest results in support of the Hodge conjecture is the algebraicity of the Hodge loci, proved in a fundamental paper of E. Cattani, P. Deligne and A. Kaplan [J. Am. Math. Soc. 8, No. 2, 483–506 (1995; Zbl 0851.14004)]. M. Green and P. Griffiths conjectured a similar behaviour for the zero locus of an admissible normal function (a definition of admissible normal function can be found in: [M. Saito, J. Algebr. Geom. 5, No. 2, 235–276 (1996; Zbl 0918.14018)]).
In the paper under review, it is proved that the zero locus of an admissible normal function over an algebraic curve is algebraic. The central idea is that a normal function vanishes at a point if and only if the canonical real grading of the corresponding mixed Hodge structure at the same point is integral. The main ingredients of the proof are Deligne’s results and Pearlstein’s \(\text{SL}_2\)-orbit theorem [G. Pearlstein, J. Differ. Geom. 74, No. 1, 1–67 (2006; Zbl 1107.14010); see also K. Kato, C. Nakayama and S. Usui; J. Algebr. Geom. 17, No. 3, 401–479 (2008; Zbl 1144.14005) for a several variable \(\text{SL}_2\)-orbit theorem with arbitrary weight filtration)].


32G20 Period matrices, variation of Hodge structure; degenerations
14D07 Variation of Hodge structures (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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