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Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture

Ekaterina Amerik, Andrey Soldatenkov, Misha Verbitsky

TL;DR

The paper proves the Hu–Yau conjecture for compact hyperkähler manifolds of maximal holonomy by showing that any bimeromorphic map can be decomposed into a finite sequence of Hu–Yau transformations, i.e., Mukai elementary transformations away from codimension >2, via a wall-crossing analysis of MBM contractions. It develops a robust framework using MBM classes, Teichmüller theory, and the wall–chamber structure of the positive cone to decompose arbitrary maps into wall-crossing flops, and then reduces codimension-two cases to Mukai-type transforms using the symplectic rank. A key technical advancement is the precise control of MBM loci and their ranks under flops, enabling a step-by-step decomposition into Hu–Yau transformations and a proof of the main theorem. The work also provides a counterexample showing the limits of representing such compositions purely by Mukai flops on open subsets, highlighting the necessity of the Hu–Yau approach. Overall, the results yield a dimension-agnostic description of the birational geometry of hyperkähler manifolds, with potential implications for understanding their birational automorphism groups and moduli.

Abstract

Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They defined ``Mukai's elementary transformation'' as the blow-up of a subvariety ruled by complex projective spaces, composed with the contraction of the ruling. Hu and Yau conjectured that any bimeromorphic map of hyperkahler manifolds can be decomposed into a sequence of Mukai's elementary transformations, after possibly removing subvarieties of codimension greater than $2$. We prove this conjecture for compact hyperkahler manifolds of maximal holonomy by decomposing any bimeromorphic map into a composition of wall-crossing flops associated with MBM contractions.

Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture

TL;DR

The paper proves the Hu–Yau conjecture for compact hyperkähler manifolds of maximal holonomy by showing that any bimeromorphic map can be decomposed into a finite sequence of Hu–Yau transformations, i.e., Mukai elementary transformations away from codimension >2, via a wall-crossing analysis of MBM contractions. It develops a robust framework using MBM classes, Teichmüller theory, and the wall–chamber structure of the positive cone to decompose arbitrary maps into wall-crossing flops, and then reduces codimension-two cases to Mukai-type transforms using the symplectic rank. A key technical advancement is the precise control of MBM loci and their ranks under flops, enabling a step-by-step decomposition into Hu–Yau transformations and a proof of the main theorem. The work also provides a counterexample showing the limits of representing such compositions purely by Mukai flops on open subsets, highlighting the necessity of the Hu–Yau approach. Overall, the results yield a dimension-agnostic description of the birational geometry of hyperkähler manifolds, with potential implications for understanding their birational automorphism groups and moduli.

Abstract

Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They defined ``Mukai's elementary transformation'' as the blow-up of a subvariety ruled by complex projective spaces, composed with the contraction of the ruling. Hu and Yau conjectured that any bimeromorphic map of hyperkahler manifolds can be decomposed into a sequence of Mukai's elementary transformations, after possibly removing subvarieties of codimension greater than . We prove this conjecture for compact hyperkahler manifolds of maximal holonomy by decomposing any bimeromorphic map into a composition of wall-crossing flops associated with MBM contractions.

Paper Structure

This paper contains 17 sections, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Mukai flops
  4. Hu--Yau transformations and the Hu--Yau conjecture
  5. Hyperkähler manifolds and MBM classes
  6. Hyperkähler manifolds and their Teichmüller spaces
  7. MBM classes and the Kähler cone
  8. Wall-crossing flops
  9. The projective case
  10. The non-projective case
  11. Decomposition
  12. Wall-crossing flops in codimension two
  13. The symplectic rank and bimeromorphic maps
  14. Definition of the symplectic rank
  15. The symplectic rank of MBM loci and flops
  16. ...and 2 more sections

Figures (1)

  • Figure 1: The birational ample cone of $S^{[3]}$

Theorems & Definitions (19)

  • proof
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  • Definition 5.1
  • proof
  • ...and 9 more