On the modified $J$-equation
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866918356853981184 |
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| author | Takahashi, Ryosuke |
| author_facet | Takahashi, Ryosuke |
| contents | In this paper, we study the modified $J$-equation introduced by Li-Shi. We first show that, on compact Kähler manifolds, the solvability of the modified $J$-equation is equivalent to the coercivity of the modified $J$-functional. Motivated by this characterization, we formulate a Nakai-Moishezon type criterion for the existence of solutions to the modified $J$-equation on general compact Kähler manifolds. We then verify this conjectural criterion in the case of smooth projective toric varieties. This extends the work of Collins-Székelyhidi and provides further evidence for the expected algebro-geometric nature of the modified $J$-equation. As a potential application, we combine our results with Delcroix-Jubert. Assuming our conjectural Nakai-Moishezon type criterion holds in general, we obtain a numerical sufficient condition for the existence of extremal Kähler metrics on arbitrary compact Kähler manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_04953 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the modified $J$-equation Takahashi, Ryosuke Differential Geometry Primary 53C55, Secondary 35A01 In this paper, we study the modified $J$-equation introduced by Li-Shi. We first show that, on compact Kähler manifolds, the solvability of the modified $J$-equation is equivalent to the coercivity of the modified $J$-functional. Motivated by this characterization, we formulate a Nakai-Moishezon type criterion for the existence of solutions to the modified $J$-equation on general compact Kähler manifolds. We then verify this conjectural criterion in the case of smooth projective toric varieties. This extends the work of Collins-Székelyhidi and provides further evidence for the expected algebro-geometric nature of the modified $J$-equation. As a potential application, we combine our results with Delcroix-Jubert. Assuming our conjectural Nakai-Moishezon type criterion holds in general, we obtain a numerical sufficient condition for the existence of extremal Kähler metrics on arbitrary compact Kähler manifolds. |
| title | On the modified $J$-equation |
| topic | Differential Geometry Primary 53C55, Secondary 35A01 |
| url | https://arxiv.org/abs/2207.04953 |