Examples of real stable bundles on K3 surfaces
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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_ | 1866917946277756928 |
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| author | Festi, Dino Platt, Daniel Singhal, Ragini Tanaka, Yuuji |
| author_facet | Festi, Dino Platt, Daniel Singhal, Ragini Tanaka, Yuuji |
| contents | Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalised Hoppe Criterion of Jardim-Menet-Prata-Sá Earp, which requires verifying an arithmetic condition for elements in the Picard group of the surfaces. We establish this by using computer aid in two critical steps: first, we construct K3 surfaces with small Picard group-one branched double cover of $\mathbb{P}^1 \times \mathbb{P}^1$ with Picard rank $2$ using a new method which may be of independent interest; and second, we verify the arithmetic condition for carefully chosen elements of the Picard group, which provides a systematic approach for constructing further examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Examples of real stable bundles on K3 surfaces Festi, Dino Platt, Daniel Singhal, Ragini Tanaka, Yuuji Algebraic Geometry Differential Geometry 14J28 Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalised Hoppe Criterion of Jardim-Menet-Prata-Sá Earp, which requires verifying an arithmetic condition for elements in the Picard group of the surfaces. We establish this by using computer aid in two critical steps: first, we construct K3 surfaces with small Picard group-one branched double cover of $\mathbb{P}^1 \times \mathbb{P}^1$ with Picard rank $2$ using a new method which may be of independent interest; and second, we verify the arithmetic condition for carefully chosen elements of the Picard group, which provides a systematic approach for constructing further examples. |
| title | Examples of real stable bundles on K3 surfaces |
| topic | Algebraic Geometry Differential Geometry 14J28 |
| url | https://arxiv.org/abs/2503.02937 |