Coleman-Oort's conjecture for degenerate irreducible curves
Let \(A_g\) be the moduli space of principally polarized abelian varieties, \(J_g\) the locus of Jacobians of smooth curves, and \(\bar J_g\) its closure. A locally symmetric subvariety (or special subvariety or Shimura subvariety) of \(A_g\) is a closed subvariety which is itself locally symmetric, and such that the natural inclusion is defined by an inclusion of algebraic groups. Frans Oort conjectured that such subvarieties should avoid \(J_g\); more precisely, that a positive dimensional locally symmetric subvariety contained in \(\bar J_g\) should not intersect \(J_g\) for \(g\) large. Combined with the André-Oort conjecture (proven assuming the GRH by [\textit{B. Klingler} and \textit{A. Yafaev}, ``The André-Oort conjecture'', Preprint, \url{arXiv:1209.0936}]) this immediately implies the Coleman conjecture, that there are only finitely many CM points in \(J_g\) for \(g\) large. It is known that \(g\) needs to be at least \(8\) (see e.g. [\textit{J. C. Rohde}, Cyclic coverings, Calabi-Yau manifolds and complex multiplication. Lecture Notes in Mathematics 1975. Berlin: Springer (2009; Zbl 1168.14001)]). The paper under review formulates and proves an analogue of Oort's conjecture inside the boundary of the second Voronoi compactification \(\bar{A}_g\) (Mumford and Namakawa proved that the Torelli map extends from the Deligne-Mumford compactification of \(M_g\) to \(\bar A_g\)). In this analogy, nonsingular curves are replaced by curves not of compact type with a single node, and locally symmetric subvarieties of \(A_g\) are replaced by their sets of boundary points in the compactification. Specifically, the author shows that if \(Y\) is a locally symmetric subvariety of \(A_g\) contained in \(\bar J_g\), and \(g \geq 4\), then its closure inside \(\bar A_g\) does not intersect the Torelli image of the locus of irreducible nodal curves. The author announces that this result will be used in further work to prove certain cases of Oort's conjecture using degeneration methods. The strategy is to prove that the boundary of a locally symmetric subvariety of \(A_g\) is always in a certain sense ``linear'' in any toroidal compactification, whereas the locus of Jacobians of stable curves in \(\bar A_g \setminus A_g\) is ``nonlinear''; for the precise meaning of this we must refer to the paper under review.
scientific article; zbMATH DE number 6108421
scientific article