| 学 术 报 告 | |
| 报告题目 | 1. On the Dynamical Manin-Mumford problem for plane polynomial endomorphisms 2. Polynomial skew-products with small relative degree |
| 报告时间 | 2025/11/25&11/26 14:00:00 |
| 报告地点 | 闵行校区数学楼102报告厅 |
| 主讲人 | Matteo Ruggiero副教授（巴黎西岱大学） |
| 主持人 | 陈张弛 研究员 |
| 报告简介 | 1. The Dynamical Manin-Mumford problem is a dynamical question inspired by classical results from arithmetic geometry. Given an algebraic dynamical system (X,f), where X is a projective variety and f is a polarized endomorphism on X, we want to determine if a subvariety Y containing "unusually many" periodic points must be itself preperiodic. In a work in collaboration with Romain Dujardin and Charles Favre, we prove this property to hold when f is a regular endomorphism of P^2 coming from a polynomial endomorphism of C^2 of degree d>=2, under the additional condition that the action of f at the line at infinity doesn't have periodic superattracting points. The proof is an interesting blend of techniques coming from arithmetic geometry, holomorphic and non-archimedean dynamics. 2. Motivated by our recent works around the Dynamical Manin-Mumford problem for polynomial endomorphisms of C^2, we investigate the local dynamics of polynomial skew products of the form (z^d, w^c + zh(z,w)), under the condition 2 <= c < d (small relative degree). For these maps, the asymptotic contraction rate of any point p exists and is either c or d. We show that the locus W where the latter situation happens, analogous of the super-stable manifold in our setting, is the support of a pluripolar the Green current T introduced by Favre-Jonsson. Moreover, under a natural condition on the dynamics of the critical branches, we describe T as an average of currents of integration along a Cantor set of holomorphic curves. This structure can be elegantly interpreted through the induced dynamics on the Berkovich affine line over the field of Laurent series. This is a joint work with Romain Dujardin and Charles Favre. |
| 主讲人简介 | Professor Matteo Ruggiero obtained his PhD degree from Scuola Normale Superiore di Pisa, Italy. Then he was a post-doc of the FMJH at the Ecole Polytechnique, France. Now he is a Maître de conférences at Université Paris Cité, France. Professor Ruggiero made fundamental contributions to Local and semi-local dynamics of analytic maps, complex geometry and compactification of orbit spaces, birational aspects of singularities of analytic spaces and maps, links with toric, tropical and non-archimedean geometry, and arithmetic dynamics aspects of polynomial endomorphisms. |