On the $4$-dimensional minimal model program for Kähler varieties
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914745240518656 |
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| author | Das, Omprokash Hacon, Christopher Păun, Mihai |
| author_facet | Das, Omprokash Hacon, Christopher Păun, Mihai |
| contents | In this article we establish the following results: Let $(X, B)$ be a dlt pair, where $X$ is a $\mathbb Q$-factorial Kähler $4$-fold -- (i) if $X$ is compact and $K_X+B\sim_{\mathbb Q} D\geq 0$ for some effective $\mathbb Q$-divisor, then $(X, B)$ has a log minimal model, (ii) if $(X/T, B)$ is a semi-stable klt pair, $W\subset T$ a compact subset and $K_X+B$ is effective over $W$ (resp. not effective over $W$), then we can run a $(K_X+B)$-MMP over $T$ (in a neighborhood of $W$) which ends with a minimal model over $T$ (resp. a Mori fiber space over $T$). We also give a proof of the existence of flips for analytic varieties in all dimensions and the relative MMP for projective morphisms between analytic varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12205 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the $4$-dimensional minimal model program for Kähler varieties Das, Omprokash Hacon, Christopher Păun, Mihai Algebraic Geometry Complex Variables In this article we establish the following results: Let $(X, B)$ be a dlt pair, where $X$ is a $\mathbb Q$-factorial Kähler $4$-fold -- (i) if $X$ is compact and $K_X+B\sim_{\mathbb Q} D\geq 0$ for some effective $\mathbb Q$-divisor, then $(X, B)$ has a log minimal model, (ii) if $(X/T, B)$ is a semi-stable klt pair, $W\subset T$ a compact subset and $K_X+B$ is effective over $W$ (resp. not effective over $W$), then we can run a $(K_X+B)$-MMP over $T$ (in a neighborhood of $W$) which ends with a minimal model over $T$ (resp. a Mori fiber space over $T$). We also give a proof of the existence of flips for analytic varieties in all dimensions and the relative MMP for projective morphisms between analytic varieties. |
| title | On the $4$-dimensional minimal model program for Kähler varieties |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2205.12205 |