[Paper Review] Hodge theory and cohomology with compact supports

Edward L. Bueler, Igor Prokhorenkov|arXiv (Cornell University)|Apr 28, 1998

Geometry and complex manifolds22 references9 citations

TL;DR

This paper establishes a Hodge theory for noncompact, topologically tame manifolds by constructing a Hodge--Witten--Bismut Laplacian Δμ associated with a measure dμ of rapid growth at infinity. It proves an isomorphism between de Rham cohomology with compact supports and the kernel of Δμ, using an 'extension by zero' property for forms on manifolds with cylindrical ends.

ABSTRACT

This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_μ$ associated to a measure $dμ$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to $\lap_μ$ which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing gaussian growth measures.

Motivation & Objective

• To develop a Hodge theory for noncompact, topologically tame manifolds where standard Hodge theory fails due to non-compactness.
• To address the lack of a Hodge decomposition for de Rham cohomology with compact supports on noncompact manifolds.
• To construct a Laplacian Δμ associated with a measure dμ that grows rapidly at infinity, enabling a well-defined Hodge theory.
• To prove that the kernel of Δμ is isomorphic to the de Rham cohomology with compact supports.
• To establish an 'extension by zero' property for differential forms on manifolds with cylindrical ends under Gaussian growth measures.

Proposed method

• Define a weighted L2 inner product on differential forms using a measure dμ with sufficiently rapid growth at infinity.
• Construct the Hodge--Witten--Bismut Laplacian Δμ as the formal adjoint of the de Rham differential with respect to dμ.
• Prove that the space of Δμ-invariant forms satisfies an 'extension by zero' property on manifolds with cylindrical ends.
• Use the geometry of cylindrical ends to control decay and growth of forms, ensuring the extension property holds.
• Establish spectral properties of Δμ on the space of compactly supported forms, leading to the isomorphism with cohomology.
• Leverage the structure of the manifold at infinity (cylindrical ends) to ensure the measure dμ induces a well-behaved Laplacian.

Experimental results

Research questions

• RQ1 Can a Hodge theory be developed for noncompact manifolds where standard Hodge theory fails?
• RQ2 Is there a Laplacian operator Δμ on a noncompact manifold that realizes the de Rham cohomology with compact supports as its kernel?
• RQ3 Under what conditions on the measure dμ does the 'extension by zero' property hold for differential forms on cylindrical ends?
• RQ4 How does the growth rate of dμ at infinity affect the spectral properties of Δμ?
• RQ5 What geometric conditions on the manifold (e.g., cylindrical ends) are necessary for the isomorphism between H^k_c(M) and ker(Δμ) to hold?

Key findings

• The de Rham cohomology with compact supports H^k_c(M) is isomorphic to the kernel of the Hodge--Witten--Bismut Laplacian Δμ on the manifold M.
• The isomorphism holds for noncompact, topologically tame manifolds with cylindrical ends when the measure dμ has Gaussian-type growth at infinity.
• The space of Δμ-invariant forms satisfies an 'extension by zero' property, meaning forms in ker(Δμ) extend smoothly by zero across the cylindrical ends.
• The construction relies on the interplay between the geometry of the manifold at infinity and the growth of the measure dμ.
• The result provides a Hodge-theoretic realization of compactly supported cohomology via the kernel of a well-defined elliptic operator.
• The isomorphism is canonical and independent of the choice of metric, depending only on the measure dμ with sufficient growth.