Lagrangian classes in K-theory
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914412827246592 |
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| author | Choa, Dongwook Oh, Jeongseok |
| author_facet | Choa, Dongwook Oh, Jeongseok |
| contents | For a $(-1)$-shifted Lagrangian in a critical locus, we construct a homomorphism from the $K$-group of matrix factorisations of the critical locus to the $K$-group of the Lagrangian, partially answering the Joyce-Safronov conjecture. The key step is the construction of a specialisation functor for categories of matrix factorisations along the deformation to the normal cone.
Any $(-2)$-shifted symplectic space is a $(-1)$-shifted Lagrangian of a point, whose $K$-group is $\mathbb{Z}$. The image of $1\in \mathbb{Z}$ under the above homomorphism is the virtual structure sheaf.
We prove that two equivalent critical models of a given critical locus induce homomorphisms that commute via Knörrer periodicity. When a torus acts on the Lagrangian, we further prove a localisation formula, namely the commutativity of the homomorphisms associated with the Lagrangian and its fixed locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20660 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lagrangian classes in K-theory Choa, Dongwook Oh, Jeongseok Algebraic Geometry 14C17, 14F08, 14N35, 15A66 For a $(-1)$-shifted Lagrangian in a critical locus, we construct a homomorphism from the $K$-group of matrix factorisations of the critical locus to the $K$-group of the Lagrangian, partially answering the Joyce-Safronov conjecture. The key step is the construction of a specialisation functor for categories of matrix factorisations along the deformation to the normal cone. Any $(-2)$-shifted symplectic space is a $(-1)$-shifted Lagrangian of a point, whose $K$-group is $\mathbb{Z}$. The image of $1\in \mathbb{Z}$ under the above homomorphism is the virtual structure sheaf. We prove that two equivalent critical models of a given critical locus induce homomorphisms that commute via Knörrer periodicity. When a torus acts on the Lagrangian, we further prove a localisation formula, namely the commutativity of the homomorphisms associated with the Lagrangian and its fixed locus. |
| title | Lagrangian classes in K-theory |
| topic | Algebraic Geometry 14C17, 14F08, 14N35, 15A66 |
| url | https://arxiv.org/abs/2603.20660 |