[Paper Review] $L^2$-theory for the $\bar{\partial}$-operator on complex spaces with isolated singularities

Jean Ruppenthal|arXiv (Cornell University)|Oct 11, 2011

Advanced Mathematical Physics Problems11 references3 citations

TL;DR

This paper refines the $L^2$-theory for the $\bar{\partial}$-operator on complex spaces with isolated singularities by using a resolution of singularities to construct a smooth model for $L^2$-cohomology. It establishes explicit isomorphisms between $L^2$-cohomology of $(0,q)$- and $(n,q)$-forms and cohomology of holomorphic line bundles on the resolution, achieving a canonical realization of $L^2$-cohomology via pull-backs and duality.

ABSTRACT

We present a refined, improved $L^2$-theory for the $\bar{\partial}$-operator for $(0,q)$ and $(n,q)$-forms on Hermitian complex spaces of pure dimension $n$ with isolated singularities. The general philosophy is to use a resolution of singularities to obtain a regular model of the $L^2$-cohomology.

Motivation & Objective

• To develop a refined $L^2$-cohomology theory for the $\bar{\partial}$-operator on Hermitian complex spaces with isolated singularities.
• To construct a smooth, explicit model for $L^2$-cohomology by leveraging resolution of singularities and canonical sheaves.
• To realize $L^2$-cohomology isomorphisms explicitly via pull-backs and push-forwards of differential forms.
• To establish a canonical isomorphism between $L^2$-cohomology of $(0,q)$-forms and cohomology of holomorphic line bundles on the resolution.
• To prove that the $\overline{\partial}_{s}$-complex is a fine resolution of the canonical sheaf $\mathcal{K}_X^s$ under the assumption $D = Z - |Z|$.

Proposed method

• Use a resolution of singularities $\pi: M \to X$ with only normal crossings and exceptional divisor $Z = \pi^{-1}(\operatorname{Sing}X)$, assuming $D = Z - |Z|$.
• Define the canonical sheaf $\mathcal{K}_X^s$ as the kernel of the $\overline{\partial}_s$-operator on $L^2$-$(n,0)$-forms satisfying a Dirichlet boundary condition at $\operatorname{Sing}X$.
• Establish the isomorphism $\mathcal{K}_X^s \cong \pi_*(\mathcal{K}_M \otimes \mathcal{O}(-D))$ under the assumption $D = Z - |Z|$.
• Apply Takegoshi's vanishing theorem to show $R^q\pi_*(\mathcal{K}_M \otimes \mathcal{O}(|Z| - Z)) = 0$ for $q \geq 1$, enabling the use of the Leray spectral sequence.
• Construct a commutative diagram relating $L^2$-cohomology groups via natural pairings and isomorphisms induced by duality and pull-backs.
• Realize cohomology isomorphisms explicitly on the level of forms using harmonic representatives and cut-off functions to preserve compact support.

Experimental results

Research questions

• RQ1 How can $L^2$-cohomology for the $\bar{\partial}$-operator on singular complex spaces be modeled explicitly via resolution of singularities?
• RQ2 Under what conditions does the canonical sheaf $\mathcal{K}_X^s$ admit the representation $\pi_*(\mathcal{K}_M \otimes \mathcal{O}(-Z + |Z|))$?
• RQ3 Can the isomorphisms between $L^2$-cohomology and cohomology of holomorphic line bundles be realized explicitly via pull-backs and duality?
• RQ4 What is the role of the $\overline{\partial}_s$-complex in resolving $\mathcal{K}_X^s$ as a fine sheaf?
• RQ5 How does the assumption $D = Z - |Z|$ enable a canonical realization of $L^2$-cohomology via the resolution $\pi: M \to X$?

Key findings

• The canonical sheaf $\mathcal{K}_X^s$ is isomorphic to $\pi_*(\mathcal{K}_M \otimes \mathcal{O}(-D))$ with $D = Z - |Z|$, providing a coherent, locally free model for $L^2$-cohomology.
• The $\overline{\partial}_s$-complex forms a fine resolution of $\mathcal{K}_X^s$, ensuring the $L^2$-cohomology is well-behaved and computable.
• The $L^2$-cohomology of $(0,q)$-forms on $X$ is canonically isomorphic to the cohomology of $\mathcal{O}(Z - |Z|)$ on the resolution $M$ via the map $i_q$.
• The isomorphism $i_q: H^q_E(\widetilde{\Omega}, \mathcal{O}(Z - |Z|)) \to H^{0,q}_{\max}(\widetilde{\Omega}, L_{Z - |Z|})$ is realized explicitly by pull-back of forms with compact support.
• The natural pairing induces an isomorphism between $H^{n,p}_{\min}(\widetilde{\Omega}, L_{|Z| - Z})^*$ and $H^{0,q}_{\max}(\Omega^*)$, establishing duality in the $L^2$-setting.
• The injectivity of $i_q$ is proven by showing that if $i_q[a] = 0$, then $[a] = 0$ in $H^q_E(\widetilde{\Omega}, \mathcal{O}(Z - |Z|))$, using cut-off functions and compactly supported $\overline{\partial}$-solutions.