Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups
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arXiv
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2025
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| author | Delloque, Rémi Napame, Achim Scarpa, Carlo Tipler, Carl |
| author_facet | Delloque, Rémi Napame, Achim Scarpa, Carlo Tipler, Carl |
| contents | We introduce the notion of P-critical connections for hermitian holomorphic vector bundles over compact balanced manifolds: integrable hermitian connections whose curvature solves a polynomial equation. Such connections include HYM and dHYM connections, as well as solutions to higher rank Monge-Ampère or J-equations, and are a slight generalisation of Dervan-McCarthy-Sektnan's Z-critical connections motivated by Bayer's polynomial Bridgeland stability conditions. The associated equations come with a moment map interpretation, and we provide numerical conditions that are expected to characterise existence of solutions in suitable cases: P-positivity and P-stability. We then provide some devices to check those numerical conditions in practice. First, we observe that P-positivity is equivalent to its equivariant version over T-varieties. In the toric case, we thus obtain an explicit finite set of subvarieties to test P-positivity on, independently on the choice of the polynomial equation. We also introduce equivariant P-stability and discuss its relation to P-stability. Secondly, we show that a uniform version of P-positivity is preserved by pulling back along a blow-up of points. We apply those results to some examples, such as blow-ups of Hirzebruch surfaces, or a Fano 3-fold. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23842 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups Delloque, Rémi Napame, Achim Scarpa, Carlo Tipler, Carl Algebraic Geometry 53C07 (primary), 14D20 14J60 (secondary) We introduce the notion of P-critical connections for hermitian holomorphic vector bundles over compact balanced manifolds: integrable hermitian connections whose curvature solves a polynomial equation. Such connections include HYM and dHYM connections, as well as solutions to higher rank Monge-Ampère or J-equations, and are a slight generalisation of Dervan-McCarthy-Sektnan's Z-critical connections motivated by Bayer's polynomial Bridgeland stability conditions. The associated equations come with a moment map interpretation, and we provide numerical conditions that are expected to characterise existence of solutions in suitable cases: P-positivity and P-stability. We then provide some devices to check those numerical conditions in practice. First, we observe that P-positivity is equivalent to its equivariant version over T-varieties. In the toric case, we thus obtain an explicit finite set of subvarieties to test P-positivity on, independently on the choice of the polynomial equation. We also introduce equivariant P-stability and discuss its relation to P-stability. Secondly, we show that a uniform version of P-positivity is preserved by pulling back along a blow-up of points. We apply those results to some examples, such as blow-ups of Hirzebruch surfaces, or a Fano 3-fold. |
| title | Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups |
| topic | Algebraic Geometry 53C07 (primary), 14D20 14J60 (secondary) |
| url | https://arxiv.org/abs/2506.23842 |