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Birational transformations on irreducible compact Hermitian symmetric spaces

Cong Ding

Abstract

We construct a sequence of explicit blow-ups and blow-downs on irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties.

Birational transformations on irreducible compact Hermitian symmetric spaces

Abstract

We construct a sequence of explicit blow-ups and blow-downs on irreducible compact Hermitian symmetric spaces which transforms it into a projective space of the same dimension. Moreover this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of and its higher secant varieties.
Paper Structure (14 sections, 54 equations, 1 figure, 3 tables)

This paper contains 14 sections, 54 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries in Hermitian symmetric spaces
  3. Hermitian symmetric spaces of tube type
  4. Balanced subspaces
  5. Hermitian characteristic symmetric subspaces
  6. The Białynicki-Birula decomposition and constructions of the centers
  7. The loci of infinity points of degree $k$ rational curves
  8. Stratifications of the centers
  9. Singularities and smoothing the stratification
  10. The Białynicki-Birula decomposition of irreducible compact Hermtian symmetric spaces
  11. Proof of the Main theorem
  12. Some lemmas
  13. Stratifications as linear sections of $X$
  14. Proof of the Main Theorem

Figures (1)

  • Figure 1: Marked Dyndin diagram of $E_6$ and the subdiagram associated to $Q^8$

Theorems & Definitions (26)

  • proof
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  • ...and 16 more