Complex Monge-Ampère equation for positive $(p,p)$-forms on compact Kähler manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909909782626304 |
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| author | George, Mathew |
| author_facet | George, Mathew |
| contents | A complex Monge-Ampère equation for differential $(p,p)$-forms is introduced on compact Kähler manifolds. For any $1 \leq p < n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for $p=n-1$, this gives the Monge-Ampère equation for $(n-1)$ plurisubharmonic functions studied by Tosatti-Weinkove. For other $p$ values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to $(p,p)$-forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_06497 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex Monge-Ampère equation for positive $(p,p)$-forms on compact Kähler manifolds George, Mathew Analysis of PDEs Algebraic Geometry Differential Geometry 35J15, 35J60, 58J05 A complex Monge-Ampère equation for differential $(p,p)$-forms is introduced on compact Kähler manifolds. For any $1 \leq p < n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for $p=n-1$, this gives the Monge-Ampère equation for $(n-1)$ plurisubharmonic functions studied by Tosatti-Weinkove. For other $p$ values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to $(p,p)$-forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown. |
| title | Complex Monge-Ampère equation for positive $(p,p)$-forms on compact Kähler manifolds |
| topic | Analysis of PDEs Algebraic Geometry Differential Geometry 35J15, 35J60, 58J05 |
| url | https://arxiv.org/abs/2411.06497 |