Metrics with minimal singularities and the Abundance conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929400646205440 |
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| author | Lazić, Vladimir |
| author_facet | Lazić, Vladimir |
| contents | The Abundance conjecture predicts that on a minimal projective klt pair $(X,Δ)$, the adjoint divisor $K_X+Δ$ is semiample. When $χ(X,\mathcal O_X)\neq0$, we give a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of $K_X+Δ$. Furthermore, we prove fundamental structural properties as well as regularity and weak convergence behaviour of an important class of currents with minimal singularities: the supercanonical currents. The results of the paper indicate strongly that supercanonical currents are central to the completion of the proof of the Abundance conjecture for minimal klt pairs $(X,Δ)$ with $χ(X,\mathcal O_X)\neq0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_18233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metrics with minimal singularities and the Abundance conjecture Lazić, Vladimir Algebraic Geometry Complex Variables Differential Geometry 14E30, 32U40, 32J25 The Abundance conjecture predicts that on a minimal projective klt pair $(X,Δ)$, the adjoint divisor $K_X+Δ$ is semiample. When $χ(X,\mathcal O_X)\neq0$, we give a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of $K_X+Δ$. Furthermore, we prove fundamental structural properties as well as regularity and weak convergence behaviour of an important class of currents with minimal singularities: the supercanonical currents. The results of the paper indicate strongly that supercanonical currents are central to the completion of the proof of the Abundance conjecture for minimal klt pairs $(X,Δ)$ with $χ(X,\mathcal O_X)\neq0$. |
| title | Metrics with minimal singularities and the Abundance conjecture |
| topic | Algebraic Geometry Complex Variables Differential Geometry 14E30, 32U40, 32J25 |
| url | https://arxiv.org/abs/2406.18233 |