Birational invariance of motivic zeta functions of $K$-trivial varieties, and obstructions to smooth fillings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929230257848320 |
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| author | Lunardon, Luigi Nicaise, Johannes |
| author_facet | Lunardon, Luigi Nicaise, Johannes |
| contents | The motivic zeta function of a smooth and proper $\mathbb{C}((t))$-variety $X$ with trivial canonical bundle is a rational function with coefficients in an appropriate Grothendieck ring of complex varieties, which measures how $X$ degenerates at $t=0$. In analogy with Igusa's monodromy conjecture for $p$-adic zeta functions of hypersurface singularities, we expect that the poles of the motivic zeta function of $X$ correspond to monodromy eigenvalues on the cohomology of $X$. In this paper, we prove that the motivic zeta function and the monodromy conjecture are preserved under birational equivalence, which extends the range of known cases. As a further application, we explain how the motivic zeta function acts as an obstruction to the existence of a smooth filling for $X$ at $t=0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_17772 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Birational invariance of motivic zeta functions of $K$-trivial varieties, and obstructions to smooth fillings Lunardon, Luigi Nicaise, Johannes Algebraic Geometry 14D06, 14E18, 14J32 The motivic zeta function of a smooth and proper $\mathbb{C}((t))$-variety $X$ with trivial canonical bundle is a rational function with coefficients in an appropriate Grothendieck ring of complex varieties, which measures how $X$ degenerates at $t=0$. In analogy with Igusa's monodromy conjecture for $p$-adic zeta functions of hypersurface singularities, we expect that the poles of the motivic zeta function of $X$ correspond to monodromy eigenvalues on the cohomology of $X$. In this paper, we prove that the motivic zeta function and the monodromy conjecture are preserved under birational equivalence, which extends the range of known cases. As a further application, we explain how the motivic zeta function acts as an obstruction to the existence of a smooth filling for $X$ at $t=0$. |
| title | Birational invariance of motivic zeta functions of $K$-trivial varieties, and obstructions to smooth fillings |
| topic | Algebraic Geometry 14D06, 14E18, 14J32 |
| url | https://arxiv.org/abs/2401.17772 |