Details
| Original language | English |
|---|---|
| Pages (from-to) | 161-202 |
| Number of pages | 42 |
| Journal | Inventiones Mathematicae |
| Volume | 222 |
| Issue number | 1 |
| Publication status | Published - 2020 |
| Externally published | Yes |
Abstract
Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often ξ takes a torsion value (for instance, Manin’s theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when ξ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira–Spencer map associated to (A, ξ) (assuming A without fixed part, and Zξ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension ≤ 3 , and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if A→ S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space A g has dimension at least g, then the Betti map of any non-torsion section ξ is generically a submersion, so that ξ-1Ators is dense in S(C).
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Inventiones Mathematicae, Vol. 222, No. 1, 2020, p. 161-202.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Appendix to "The Betti map associated to a section of an abelian scheme"
AU - Gao, Ziyang
AU - André, Yves
AU - Corvaja, Pietro
AU - Zannier, Umberto
N1 - Publisher Copyright: © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020
Y1 - 2020
N2 - Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often ξ takes a torsion value (for instance, Manin’s theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when ξ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira–Spencer map associated to (A, ξ) (assuming A without fixed part, and Zξ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension ≤ 3 , and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if A→ S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space A g has dimension at least g, then the Betti map of any non-torsion section ξ is generically a submersion, so that ξ-1Ators is dense in S(C).
AB - Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often ξ takes a torsion value (for instance, Manin’s theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when ξ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira–Spencer map associated to (A, ξ) (assuming A without fixed part, and Zξ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension ≤ 3 , and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if A→ S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space A g has dimension at least g, then the Betti map of any non-torsion section ξ is generically a submersion, so that ξ-1Ators is dense in S(C).
UR - http://www.scopus.com/inward/record.url?scp=85082687218&partnerID=8YFLogxK
U2 - 10.1007/s00222-020-00963-w
DO - 10.1007/s00222-020-00963-w
M3 - Article
VL - 222
SP - 161
EP - 202
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
SN - 0020-9910
IS - 1
ER -