Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture

TL;DR

This paper proves Hu and Yau's conjecture that any bimeromorphic map of hyperkahler manifolds can be decomposed into Mukai's elementary transformations in arbitrary dimensions, extending known 4-dimensional results.

Contribution

It establishes the decomposition of bimeromorphic maps into Mukai's elementary transformations for compact hyperkahler manifolds of maximal holonomy.

Findings

Decomposition of bimeromorphic maps into wall-crossing flops.

Extension of 4D results to arbitrary dimensions.

Validation of Hu and Yau's conjecture.

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.