Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Hitchin morphism for K-trivial varieties

Aryaman Patel, Dario Weissmann

Abstract

We study the Hitchin morphism for higher dimensional varieties and show that, for a certain class of varieties which we call r-small, the set-theoretic image of the Hitchin morphism from the Dolbeault moduli space coincides with the spectral base. In other words, a stronger version of the conjecture of Chen and Ngô holds for this class of varieties, which includes K-trivial varieties. As part of the proof, we slightly modify the construction of spectral covers to obtain normal spectral covers.

The Hitchin morphism for K-trivial varieties

Abstract

We study the Hitchin morphism for higher dimensional varieties and show that, for a certain class of varieties which we call r-small, the set-theoretic image of the Hitchin morphism from the Dolbeault moduli space coincides with the spectral base. In other words, a stronger version of the conjecture of Chen and Ngô holds for this class of varieties, which includes K-trivial varieties. As part of the proof, we slightly modify the construction of spectral covers to obtain normal spectral covers.

Paper Structure

This paper contains 7 sections, 19 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Spectral covers
  3. The spectral base
  4. Irreducible spectral data
  5. Normalized spectral covers
  6. Birational morphisms
  7. A stronger version of the conjecture

Key Result

Theorem 1.2

Let $X$ be a smooth projective variety whose canonical divisor $K_X$ is numerically trivial. Then for all $r\in\mathbb{N}$ the restricted Hitchin morphism $h^r_{X,\mathrm{Dol}}:M^r_{X,\mathrm{Dol}}\to\mathscr{B}^r_X$ is surjective. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (45)

  • Conjecture 1.1: CN20
  • Theorem 1.2: \ref{['ktrivial']}
  • Theorem 1.3: \ref{['generalprop']}
  • Definition 1.4
  • Remark 1.5
  • Conjecture 1.6: equivalent to Conjecture \ref{['conjecture']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 35 more