$L^2$-vanishing theorem and a conjecture of Kollár
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911093406826496 |
|---|---|
| author | Deng, Ya Wang, Botong |
| author_facet | Deng, Ya Wang, Botong |
| contents | In 1995, Kollár conjectured that a smooth complex projective $n$-fold $X$ with generically large fundamental group has Euler characteristic $χ(X, K_X)\geq 0$. In this paper, we prove the conjecture assuming $X$ has linear fundamental group, i.e., there exists a representation $π_1(X)\to {\rm GL}_N(\mathbb{C})$ with finite kernel. We deduce the conjecture by proving a stronger $L^2$ vanishing theorem: for the universal cover $\widetilde{X}$ of such $X$, its $L^2$-Dolbeault cohomology $H_{(2)}^{n,q}(\widetilde{X})=0$ for $q\neq 0$. The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11399 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^2$-vanishing theorem and a conjecture of Kollár Deng, Ya Wang, Botong Algebraic Geometry Complex Variables In 1995, Kollár conjectured that a smooth complex projective $n$-fold $X$ with generically large fundamental group has Euler characteristic $χ(X, K_X)\geq 0$. In this paper, we prove the conjecture assuming $X$ has linear fundamental group, i.e., there exists a representation $π_1(X)\to {\rm GL}_N(\mathbb{C})$ with finite kernel. We deduce the conjecture by proving a stronger $L^2$ vanishing theorem: for the universal cover $\widetilde{X}$ of such $X$, its $L^2$-Dolbeault cohomology $H_{(2)}^{n,q}(\widetilde{X})=0$ for $q\neq 0$. The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods. |
| title | $L^2$-vanishing theorem and a conjecture of Kollár |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2409.11399 |