Compact convergence, deformation of the $L^2$-$\overline{\partial}$-complex and canonical $K$-homology classes
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909224937717760 |
|---|---|
| author | Bei, Francesco |
| author_facet | Bei, Francesco |
| contents | Let $(X,γ)$ be a compact, irreducible Hermitian complex space of complex dimension $m$ and with $\mathrm{dim}(\mathrm{sing}(X))=0$. Let $(F,τ)\rightarrow X$ be a Hermitian holomorphic vector bundle over $X$ and let us denote with $\overlineð_{F,m,\mathrm{abs}}$ the rolled-up operator of the maximal $L^2$-$\overline{\partial}$ complex of $F$-valued $(m,\bullet)$-forms. Let $π:M\rightarrow X$ be a resolution of singularities, $g$ a metric on $M$, $E:=π^*F$ and $ρ:=π^*τ$. In this paper, under quite general assumptions on $τ$, we prove the following equality of analytic $K$-homology classes $[\overlineð_{F,m,\mathrm{abs}}]=π_*[\overlineð_{E,m}]$, with $\overlineð_{E,m}$ the rolled-up operator of the $L^2$-$\overline{\partial}$ complex of $E$-valued $(m,\bullet)$-forms on $M$. Our proof is based on functional analytic techniques developed in \cite{KuSh} and provides an explicit homotopy between the even unbounded Fredholm modules induced by $\overlineð_{F,m,\mathrm{abs}}$ and $\overlineð_{E,m}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12667 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Compact convergence, deformation of the $L^2$-$\overline{\partial}$-complex and canonical $K$-homology classes Bei, Francesco Differential Geometry Complex Variables K-Theory and Homology Let $(X,γ)$ be a compact, irreducible Hermitian complex space of complex dimension $m$ and with $\mathrm{dim}(\mathrm{sing}(X))=0$. Let $(F,τ)\rightarrow X$ be a Hermitian holomorphic vector bundle over $X$ and let us denote with $\overlineð_{F,m,\mathrm{abs}}$ the rolled-up operator of the maximal $L^2$-$\overline{\partial}$ complex of $F$-valued $(m,\bullet)$-forms. Let $π:M\rightarrow X$ be a resolution of singularities, $g$ a metric on $M$, $E:=π^*F$ and $ρ:=π^*τ$. In this paper, under quite general assumptions on $τ$, we prove the following equality of analytic $K$-homology classes $[\overlineð_{F,m,\mathrm{abs}}]=π_*[\overlineð_{E,m}]$, with $\overlineð_{E,m}$ the rolled-up operator of the $L^2$-$\overline{\partial}$ complex of $E$-valued $(m,\bullet)$-forms on $M$. Our proof is based on functional analytic techniques developed in \cite{KuSh} and provides an explicit homotopy between the even unbounded Fredholm modules induced by $\overlineð_{F,m,\mathrm{abs}}$ and $\overlineð_{E,m}$. |
| title | Compact convergence, deformation of the $L^2$-$\overline{\partial}$-complex and canonical $K$-homology classes |
| topic | Differential Geometry Complex Variables K-Theory and Homology |
| url | https://arxiv.org/abs/2308.12667 |