An injectivity theorem on snc compact Kähler spaces: an application of the theory of harmonic integrals on log-canonical centers via adjoint ideal sheaves
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917780650983424 |
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| author | Chan, Tsz On Mario Choi, Young-Jun Matsumura, Shin-ichi |
| author_facet | Chan, Tsz On Mario Choi, Young-Jun Matsumura, Shin-ichi |
| contents | Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact Kähler manifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic line bundle on $X$ equipped with a smooth metric $h_F = e^{-φ_F}$. Via the use of the adjoint ideal sheaves (constructed from $φ_F$ and $D$) and the associated residue morphisms, sections of $K_D \otimes \left. F\right|_D$ on $D$ (as well as those of $K_X \otimes D \otimes F$ on $X$) can be related to the $F$-valued holomorphic top-forms on each lc center of $(X,D)$ by an inductive use of a certain residue exact sequence derived from the adjoint ideal sheaves. The theory of harmonic integrals is valid on each lc center (which is compact Kähler), so this provides a pathway to apply the techniques in harmonic theory to the possibly singular Kähler space $D$. To illustrate the use of such apparatus in problems concerning lc pairs, we prove a Kollár-type injectivity theorem for the cohomology on $D$ when $F$ is semi-positive. This in turn also solves the conjecture by Fujino on the injectivity theorem for the compact Kähler lc pair $(X,D)$, providing an alternative proof of a recent result by Cao and Păun. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12025 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An injectivity theorem on snc compact Kähler spaces: an application of the theory of harmonic integrals on log-canonical centers via adjoint ideal sheaves Chan, Tsz On Mario Choi, Young-Jun Matsumura, Shin-ichi Complex Variables Algebraic Geometry 32J25 (primary), 32Q15, 14B05 (secondary) Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact Kähler manifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic line bundle on $X$ equipped with a smooth metric $h_F = e^{-φ_F}$. Via the use of the adjoint ideal sheaves (constructed from $φ_F$ and $D$) and the associated residue morphisms, sections of $K_D \otimes \left. F\right|_D$ on $D$ (as well as those of $K_X \otimes D \otimes F$ on $X$) can be related to the $F$-valued holomorphic top-forms on each lc center of $(X,D)$ by an inductive use of a certain residue exact sequence derived from the adjoint ideal sheaves. The theory of harmonic integrals is valid on each lc center (which is compact Kähler), so this provides a pathway to apply the techniques in harmonic theory to the possibly singular Kähler space $D$. To illustrate the use of such apparatus in problems concerning lc pairs, we prove a Kollár-type injectivity theorem for the cohomology on $D$ when $F$ is semi-positive. This in turn also solves the conjecture by Fujino on the injectivity theorem for the compact Kähler lc pair $(X,D)$, providing an alternative proof of a recent result by Cao and Păun. |
| title | An injectivity theorem on snc compact Kähler spaces: an application of the theory of harmonic integrals on log-canonical centers via adjoint ideal sheaves |
| topic | Complex Variables Algebraic Geometry 32J25 (primary), 32Q15, 14B05 (secondary) |
| url | https://arxiv.org/abs/2307.12025 |