Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2206.14778 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915149763313664 |
|---|---|
| author | Huang, Jesse Zhou, Peng |
| author_facet | Huang, Jesse Zhou, Peng |
| contents | Let $T=(\C^*)^k$ act on $V=\C^N$ faithfully and preserving the volume form, i.e. $(\C^*)^k \into \text{SL}(V)$. On the B-side, we have toric stacks $Z_W$ (see Eq. \ref{eq:ZW})labelled by walls $W$ in the GKZ fan, and $Z_{/F}$ labelled by faces of a polytope corresponding to minimal semi-orthogonal decomposition (SOD) components. The B-side multiplicity $n^B_{W,F}$, well-defined by a result of Kite-Segal \cite{kite-segal}, is the number of times $\Coh(Z_{/F})$ appears in a complete SOD of $\Coh(Z_W)$. On the A-side, we have the GKZ discriminant loci components $\nabla_F \In (\C^*)^k$, and its tropicalization $\nabla^{trop}_{F} \In \R^k$. The A-side multiplicity $n^A_{W, F}$ is defined as the multiplicity of the tropical complex $\nabla^{trop}_{F}$ on wall $W$. We prove that $n^A_{W,F} = n^B_{W,F}$, confirming a conjecture in Kite-Segal \cite{kite-segal} inspired by \cite{aspinwall2017mirror}. Our proof is based on the result of Horja-Katzarkov \cite{horja2022discriminants} and a lemma about B-side SOD multiplicity, which allows us to reduce to lower dimension just as in A-side \cite{GKZ-book}[Ch 11]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_14778 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | GKZ discriminant and Multiplicities Huang, Jesse Zhou, Peng Algebraic Geometry Mathematical Physics 14J33, 14F08 Let $T=(\C^*)^k$ act on $V=\C^N$ faithfully and preserving the volume form, i.e. $(\C^*)^k \into \text{SL}(V)$. On the B-side, we have toric stacks $Z_W$ (see Eq. \ref{eq:ZW})labelled by walls $W$ in the GKZ fan, and $Z_{/F}$ labelled by faces of a polytope corresponding to minimal semi-orthogonal decomposition (SOD) components. The B-side multiplicity $n^B_{W,F}$, well-defined by a result of Kite-Segal \cite{kite-segal}, is the number of times $\Coh(Z_{/F})$ appears in a complete SOD of $\Coh(Z_W)$. On the A-side, we have the GKZ discriminant loci components $\nabla_F \In (\C^*)^k$, and its tropicalization $\nabla^{trop}_{F} \In \R^k$. The A-side multiplicity $n^A_{W, F}$ is defined as the multiplicity of the tropical complex $\nabla^{trop}_{F}$ on wall $W$. We prove that $n^A_{W,F} = n^B_{W,F}$, confirming a conjecture in Kite-Segal \cite{kite-segal} inspired by \cite{aspinwall2017mirror}. Our proof is based on the result of Horja-Katzarkov \cite{horja2022discriminants} and a lemma about B-side SOD multiplicity, which allows us to reduce to lower dimension just as in A-side \cite{GKZ-book}[Ch 11]. |
| title | GKZ discriminant and Multiplicities |
| topic | Algebraic Geometry Mathematical Physics 14J33, 14F08 |
| url | https://arxiv.org/abs/2206.14778 |