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Positivity of Schur forms for Griffiths positive vector bundles of rank three over complex threefolds

Xueyuan Wan

Abstract

In this paper, we prove the positivity of the double mixed discriminant associated with a positive linear map between spaces of \(3\times 3\) complex matrices, thereby settling the three-dimensional case of Finski's open problem. As an application, we show that all Schur forms are weakly positive for Griffiths positive Hermitian holomorphic vector bundles of rank three over complex threefolds. This yields a complete affirmative answer, in the case where both the rank and the dimension are three, to the question posed by Griffiths in 1969.

Positivity of Schur forms for Griffiths positive vector bundles of rank three over complex threefolds

Abstract

In this paper, we prove the positivity of the double mixed discriminant associated with a positive linear map between spaces of complex matrices, thereby settling the three-dimensional case of Finski's open problem. As an application, we show that all Schur forms are weakly positive for Griffiths positive Hermitian holomorphic vector bundles of rank three over complex threefolds. This yields a complete affirmative answer, in the case where both the rank and the dimension are three, to the question posed by Griffiths in 1969.
Paper Structure (7 sections, 10 theorems, 106 equations)

This paper contains 7 sections, 10 theorems, 106 equations.

Table of Contents

  1. Positivity of differential forms and vector bundles
  2. Positivity of differential forms
  3. Griffiths positive vector bundles
  4. Positivity of Schur forms
  5. Finski's open problem
  6. The case of rank two
  7. The case of rank three

Key Result

Theorem 3

Let $B_{i\bar{j}}\in M_3(\mathbb C)$, $1\le i,j\le 3$, satisfy eqn1 and normalization. Then In particular, $\mathrm{D}_W\circ H^{\otimes \dim W}\circ \mathrm{D}_V^*\in \mathbb{R}$ is positive if $H:\mathrm{End}(V)\to \mathrm{End}(W)$ is a positive linear map and $\dim V=\dim W=3$. Moreover, all Schur forms are weakly positive for Griffiths positive Hermitian holomorphic vector bundles of r

Theorems & Definitions (23)

  • Theorem 3
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Theorem 2.3: Fin
  • Lemma 2.5
  • Lemma 2.6
  • ...and 13 more