Kähler-Ricci shrinkers and Fano fibrations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912266306191360 |
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| author | Sun, Song Zhang, Junsheng |
| author_facet | Sun, Song Zhang, Junsheng |
| contents | In this paper, we build connections between Kähler-Ricci shrinkers, i.e., complete (possibly non-compact) shrinking gradient Kähler-Ricci solitons, and algebraic geometry. In particular, we (1). prove that a Kähler-Ricci shrinker is naturally a quasi-projective variety, using birational algebraic geometry; (2). formulate a conjecture relating the existence of Kähler-Ricci shrinkers and K-stability of polarized Fano fibrations, which unifies and extends the YTD type conjectures for Kähler-Einstein metrics, Ricci-flat Kähler cone metrics and compact Kähler-Ricci shrinkers; (3). formulate conjectures connecting tangent flows at singularities of Kähler-Ricci flows and algebraic geometry, via a 2-step degeneration for the weighted volume of a Fano fibration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09661 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kähler-Ricci shrinkers and Fano fibrations Sun, Song Zhang, Junsheng Differential Geometry Algebraic Geometry In this paper, we build connections between Kähler-Ricci shrinkers, i.e., complete (possibly non-compact) shrinking gradient Kähler-Ricci solitons, and algebraic geometry. In particular, we (1). prove that a Kähler-Ricci shrinker is naturally a quasi-projective variety, using birational algebraic geometry; (2). formulate a conjecture relating the existence of Kähler-Ricci shrinkers and K-stability of polarized Fano fibrations, which unifies and extends the YTD type conjectures for Kähler-Einstein metrics, Ricci-flat Kähler cone metrics and compact Kähler-Ricci shrinkers; (3). formulate conjectures connecting tangent flows at singularities of Kähler-Ricci flows and algebraic geometry, via a 2-step degeneration for the weighted volume of a Fano fibration. |
| title | Kähler-Ricci shrinkers and Fano fibrations |
| topic | Differential Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2410.09661 |