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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.15065 |
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| _version_ | 1866912673322500096 |
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| author | Shein, Vladimir |
| author_facet | Shein, Vladimir |
| contents | Let $K_0(\mathcal{V}_{K})$ be the Grothendieck ring of varieties over a field $K$ of characteristic zero, and let $\mathbb{L} = [\mathbb{A}^1_{K}]$ denote the Lefschetz class. We prove that if a $K$-variety has $\mathbb{L}$-rational singularities, then all its symmetric powers also have $\mathbb{L}$-rational singularities. We then use this result to show that, for a smooth complex projective variety $X$ of dimension greater than one, the rationality of its Kapranov motivic zeta function $Z(X, t)$ (viewed as a formal power series over $K_0(\mathcal{V}_{\mathbb{C}})$) implies that the Kodaira dimension of $X$ is negative and that $X$ does not admit global nonzero differential forms of even degree. This extends the irrationality part of the Larsen-Lunts rationality criterion from the surface case to arbitrary dimension. We also discuss some applications of these results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singularities of symmetric powers and irrationality of motivic zeta functions Shein, Vladimir Algebraic Geometry 14G10 (Primary) 14B05, 14E05 (Secondary) Let $K_0(\mathcal{V}_{K})$ be the Grothendieck ring of varieties over a field $K$ of characteristic zero, and let $\mathbb{L} = [\mathbb{A}^1_{K}]$ denote the Lefschetz class. We prove that if a $K$-variety has $\mathbb{L}$-rational singularities, then all its symmetric powers also have $\mathbb{L}$-rational singularities. We then use this result to show that, for a smooth complex projective variety $X$ of dimension greater than one, the rationality of its Kapranov motivic zeta function $Z(X, t)$ (viewed as a formal power series over $K_0(\mathcal{V}_{\mathbb{C}})$) implies that the Kodaira dimension of $X$ is negative and that $X$ does not admit global nonzero differential forms of even degree. This extends the irrationality part of the Larsen-Lunts rationality criterion from the surface case to arbitrary dimension. We also discuss some applications of these results. |
| title | Singularities of symmetric powers and irrationality of motivic zeta functions |
| topic | Algebraic Geometry 14G10 (Primary) 14B05, 14E05 (Secondary) |
| url | https://arxiv.org/abs/2508.15065 |