Asian Journal of Mathematics

Volume 22 (2018)

Number 2

Special issue in honor of Ngaiming Mok (1 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

Isotrivial VMRT-structures of complete intersection type

Pages: 333 – 354

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n2.a10

Authors

Baohua Fu (Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)

Jun-Muk Hwang (Korea Institute for Advanced Study, Seoul, South Korea)

Abstract

The family of varieties of minimal rational tangents on a quasi-homogeneous projective manifold is isotrivial. Conversely, are projective manifolds with isotrivial varieties of minimal rational tangents quasi-homogenous? We will show that this is not true in general, even when the projective manifold has Picard number $1$. In fact, an isotrivial family of varieties of minimal rational tangents needs not be locally flat in differential geometric sense. This leads to the question for which projective variety $Z$, the $Z$-isotriviality of varieties of minimal rational tangents implies local flatness. Our main result verifies this for many cases of $Z$ among complete intersections.

Keywords

equivalence problem, minimal rational curve, complete intersection

2010 Mathematics Subject Classification

14M10, 53B99, 58A15

Received 25 July 2016

Accepted 27 June 2017

Published 15 June 2018