Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908617237594112 |
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| author | Amerik, Ekaterina Soldatenkov, Andrey Verbitsky, Misha |
| author_facet | Amerik, Ekaterina Soldatenkov, Andrey Verbitsky, Misha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24454 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture Amerik, Ekaterina Soldatenkov, Andrey Verbitsky, Misha Algebraic Geometry 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. |
| title | Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.24454 |