Algebra of affine differential invariants and the moduli of affinely homogeneous surfaces
Dr. Zhangchi Chen
2022-03-01 9:30-11:30
MCM410
Speaker: Dr. Zhangchi Chen (MCM)
Time: 9:30-11:30 Mar. 1, 2022 (Tuesday)
Place: MCM410
Title: Algebra of affine differential invariants and the moduli of affinely homogeneous surfaces
Abstract: Consider a smooth graphed holomorphic surface {u = F(x, y)} in C3 x,y,u under the action of the affine transformation group A3(C). An affine differential invariant of order n is an analytic combination of differentials among Fxx, Fxy, Fyy,... that is invariant under the lifted action of A3(C) on the jet spaces Jn 2,1.
In the first part of the talk, I will introduce the algebraic structure of all affine differential invariants. All such invariants forms a finitely generated algebra. Fels-Olver's recurrence formula gives a finite generating set and the generating relations between differentials of lower order invariants and higher order ones.
A smooth surface in C3 is affinely homogeneous if there is a subgroup of A3(C) acting transversely on it. In 1999, Eastwood and Ezhov classifed all affinely homogeneous surfaces into a list by determining possible tangential vector fields. In 2020, with Merker J., we organise all homogeneous models in inequivalent branches. And we express the moduli of each branch as an algebraic variety.
In the second part of the talk, I will introduce how the recurrence formulas apply to describe the moduli of homogeneous surfaces. We obtain necessary conditions for homogeneity of algebraic nature by the principle that all differential invariants of a homogeneous surfaces are constant, hence the differentials of all invariants vanish.
This work has been submitted to Differential Geometry and its Applications.
