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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.00321 |
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| _version_ | 1866909302197846016 |
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| author | Ballal, Aashirwad N. Pingali, Vamsi P. |
| author_facet | Ballal, Aashirwad N. Pingali, Vamsi P. |
| contents | We study MA-positivity, a notion of positivity relevant to a vector bundle version of the complex Monge--Ampère equation introduced in an earlier work, and show that for rank-two holomorphic bundles over complex surfaces, MA-semi-positive solutions of the vector bundle Monge--Ampère (vbMA) equation are also MA-positive. For vector bundles of rank-three and higher, over complex manifolds of dimension greater than one, we show that this positivity-preservation property need not hold for an algebraic solution of the vbMA equation treated as a purely algebraic equation at a given point. Finally, we set up a continuity path for certain classes of highly symmetric rank-two vector bundles over complex three-folds and prove a restricted version of positivity preservation which is nevertheless sufficient to prove openness along this continuity path. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positivity properties of the vector bundle Monge-Ampère equation Ballal, Aashirwad N. Pingali, Vamsi P. Differential Geometry We study MA-positivity, a notion of positivity relevant to a vector bundle version of the complex Monge--Ampère equation introduced in an earlier work, and show that for rank-two holomorphic bundles over complex surfaces, MA-semi-positive solutions of the vector bundle Monge--Ampère (vbMA) equation are also MA-positive. For vector bundles of rank-three and higher, over complex manifolds of dimension greater than one, we show that this positivity-preservation property need not hold for an algebraic solution of the vbMA equation treated as a purely algebraic equation at a given point. Finally, we set up a continuity path for certain classes of highly symmetric rank-two vector bundles over complex three-folds and prove a restricted version of positivity preservation which is nevertheless sufficient to prove openness along this continuity path. |
| title | Positivity properties of the vector bundle Monge-Ampère equation |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2409.00321 |