Global Igusa zeta function and $K$-equivalence
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909693667966976 |
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| author | Lee, Shuang-Yen |
| author_facet | Lee, Shuang-Yen |
| contents | Let $\mathfrak{X}$ and $\mathfrak{X}'$ be two smooth projective varieties over the ring of integers of a $p$-adic field $\textbf{K}$ with generic fibers being $X$ and $X'$. We introduce a (family of) good $s$-norms on the pluricanonical spaces of $X$ and $X'$, which are called global Igusa zeta functions in $s$, and show that if the $r$-canonical maps send $X$ and $X'$ birationally to their images respectively, then any isometry between $H^0(X, rK_X)$ and $H^0(X', rK_{X'})$ with respect to this $s$-norm (for some $s > 0$ and $s \neq 1/r$) induces $K$-equivalence on the $\textbf{K}$-points between $X$ and $X'$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_06767 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Global Igusa zeta function and $K$-equivalence Lee, Shuang-Yen Algebraic Geometry Let $\mathfrak{X}$ and $\mathfrak{X}'$ be two smooth projective varieties over the ring of integers of a $p$-adic field $\textbf{K}$ with generic fibers being $X$ and $X'$. We introduce a (family of) good $s$-norms on the pluricanonical spaces of $X$ and $X'$, which are called global Igusa zeta functions in $s$, and show that if the $r$-canonical maps send $X$ and $X'$ birationally to their images respectively, then any isometry between $H^0(X, rK_X)$ and $H^0(X', rK_{X'})$ with respect to this $s$-norm (for some $s > 0$ and $s \neq 1/r$) induces $K$-equivalence on the $\textbf{K}$-points between $X$ and $X'$. |
| title | Global Igusa zeta function and $K$-equivalence |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2210.06767 |