A Twisted Adiabatic Limit Approach to Vanishing Theorems for Complex Line Bundles
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914830694219776 |
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| author | Popovici, Dan |
| author_facet | Popovici, Dan |
| contents | Given an $n$-dimensional compact complex Hermitian manifold $X$, a $C^\infty$ complex line bundle $L$ equipped with a connection $D$ whose $(0,\,1)$-component $D''$ squares to zero and a real-valued function $η$ on $X$, we prove that the $D''$-cohomology group of $L$ of any bidegree $(p,\,q)$ such that either $(p>q \hspace{1ex}\mbox{and}\hspace{1ex} p+q\geq n+1)$ or $(p<q \hspace{1ex}\mbox{and}\hspace{1ex} p+q\leq n-1)$ vanishes when two extra hypotheses are made. The first hypothesis requires a certain real-valued, not necessarily closed, $(1,\,1)$-form depending on $p,\,q$, on the curvature of $D$ and on a $(1,\,1)$-form induced by $η$ to be positive definite. The second hypothesis requires the norm of $\partialη$ to be small relative to $|η|$. This theorem, for which we also give a number of variants, is proved by generalising our very recent twisted adiabatic limit construction for complex structures to connections on complex line bundles. This twisting of $D$ induces first-order differential operators acting on the $L$-valued forms, for which we obtain commutation relations involving their formal adjoints, and two twisted Laplacians for which we obtain a comparison formula reminiscent of the classical Bochner-Kodaira-Nakano identity. The main features of our results are that $X$ need not be Kähler, $L$ need not be holomorphic and the types of $C^\infty$ functions that $X$ supports play a key role in our hypotheses, thus capturing some of their links with the geometry of manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Twisted Adiabatic Limit Approach to Vanishing Theorems for Complex Line Bundles Popovici, Dan Differential Geometry Algebraic Geometry Complex Variables Given an $n$-dimensional compact complex Hermitian manifold $X$, a $C^\infty$ complex line bundle $L$ equipped with a connection $D$ whose $(0,\,1)$-component $D''$ squares to zero and a real-valued function $η$ on $X$, we prove that the $D''$-cohomology group of $L$ of any bidegree $(p,\,q)$ such that either $(p>q \hspace{1ex}\mbox{and}\hspace{1ex} p+q\geq n+1)$ or $(p<q \hspace{1ex}\mbox{and}\hspace{1ex} p+q\leq n-1)$ vanishes when two extra hypotheses are made. The first hypothesis requires a certain real-valued, not necessarily closed, $(1,\,1)$-form depending on $p,\,q$, on the curvature of $D$ and on a $(1,\,1)$-form induced by $η$ to be positive definite. The second hypothesis requires the norm of $\partialη$ to be small relative to $|η|$. This theorem, for which we also give a number of variants, is proved by generalising our very recent twisted adiabatic limit construction for complex structures to connections on complex line bundles. This twisting of $D$ induces first-order differential operators acting on the $L$-valued forms, for which we obtain commutation relations involving their formal adjoints, and two twisted Laplacians for which we obtain a comparison formula reminiscent of the classical Bochner-Kodaira-Nakano identity. The main features of our results are that $X$ need not be Kähler, $L$ need not be holomorphic and the types of $C^\infty$ functions that $X$ supports play a key role in our hypotheses, thus capturing some of their links with the geometry of manifolds. |
| title | A Twisted Adiabatic Limit Approach to Vanishing Theorems for Complex Line Bundles |
| topic | Differential Geometry Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2406.06286 |