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NORMALISED TANGENT BUNDLE, VARIETIES WITH SMALL CODEGREE AND PSEUDOEFFECTIVE THRESHOLD

Part of: Projective and enumerative geometry Surfaces and higher-dimensional varieties Special varieties Analytic and descriptive geometry

Published online by Cambridge University Press:  15 July 2022

Baohua Fu  and
Jie Liu Open the ORCID record for Jie Liu [Opens in a new window]
Baohua Fu
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049 Beijing, China ([email protected])
Jie Liu*
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China
*
[email protected]
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Abstract

We propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalised tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Francesco Russo and Fyodor L. Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and, hence, the pseudoeffective cones of the projectivised tangent bundles of rational homogeneous spaces of Picard number $1$ are explicitly determined by studying the total dual variety of minimal rational tangents (VMRTs) and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number $1$.

Keywords

Fano varietiesrational homogeneous spacesdual varietyminimal rational curvesstratified Mukai flops

MSC classification

Primary: 14J45: Fano varieties
Secondary: 14M17: Homogeneous spaces and generalizations 14N05: Projective techniques 51N35: Questions of classical algebraic geometry
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 149 - 206
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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