Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Harder-Narasimhan slopes of direct images

Siarhei Finski

Abstract

For a polarized family of complex projective manifolds, we study the asymptotic distribution of Harder-Narasimhan slopes of direct image sheaves associated with high tensor powers of the polarization. We establish a theorem of Mehta-Ramanathan type, showing that this asymptotic distribution can be recovered from the analogous asymptotic distributions associated with base changes of the family over generic curves.

On Harder-Narasimhan slopes of direct images

Abstract

For a polarized family of complex projective manifolds, we study the asymptotic distribution of Harder-Narasimhan slopes of direct image sheaves associated with high tensor powers of the polarization. We establish a theorem of Mehta-Ramanathan type, showing that this asymptotic distribution can be recovered from the analogous asymptotic distributions associated with base changes of the family over generic curves.
Paper Structure (5 sections, 38 theorems, 49 equations)

This paper contains 5 sections, 38 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Submultiplicative nature of Harder-Narasimhan filtrations
  3. Relative slopes on degenerating families of varieties
  4. Generic base changes and asymptotic distribution of slopes
  5. Numerical expressions for asymptotic slopes

Key Result

Theorem 1.1

The sequence of measures $\eta_k^{HN}$ converges weakly, as $k \to \infty$, to a probability measure $\eta^{HN}$ on $\mathbb{R}$, which is absolutely continuous with respect to the Lebesgue measure, except perhaps for a point mass at $\mathop{\mathrm{ess\,sup}}\limits \eta^{HN}$. Also, the limits be

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Theorem 2.2: Chen ChenHNolyg, Boucksom-Chen BouckChen
  • Remark 2.3
  • Theorem 2.4: Boucksom-Chen BouckChen
  • Proposition 2.5
  • Lemma 2.6
  • ...and 63 more