[Paper Review] The Unifying Double Complex on Supermanifolds

Sergio L. Cacciatori, Simone Noja+1 more|arXiv (Cornell University)|Apr 22, 2020

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper constructs a unifying double complex on supermanifolds via a triple tensor product of sheaves, revealing that the de Rham complex of differential forms and the Spencer complex of integral forms both arise from its spectral sequences. The key result is that cohomology of differential and integral forms on real and complex supermanifolds are isomorphic and compute the de Rham cohomology of the reduced manifold, while the Hodge-to-de Rham spectral sequence fails to degenerate at page one for Kähler reduced supermanifolds.

ABSTRACT

We unify the notions of differential and integral forms on real, complex and algebraic supermanifolds. We do this by constructing a double complex resulting from a triple tensor product of sheaves, whose associated spectral sequences give the de Rham complex of differential forms and the Spencer complex of integral forms at page one. For real and complex supermanifolds both the spectral sequences converge at page two to the locally constant sheaf. We use this fact to show that the cohomology of differential forms is isomorphic to the cohomology of integral forms, and they both compute the de Rham cohomology of the reduced manifold. Furthermore, we show that, in contrast with the case of ordinary complex manifolds, the Hodge-to-de Rham (or Frolicher) spectral sequence of supermanifolds with Kahler reduced manifold does not converge in general at page one.

Motivation & Objective

• To unify differential and integral forms on real, complex, and algebraic supermanifolds using a common algebraic structure.
• To resolve the lack of a unified framework for cohomological invariants in supergeometry.
• To clarify the relationship between differential form cohomology and integral form cohomology on supermanifolds.
• To investigate the behavior of the Hodge-to-de Rham spectral sequence in the supermanifold setting, particularly for Kähler reduced manifolds.

Proposed method

• Construct a double complex from the triple tensor product of sheaves of differential forms, integral forms, and constants.
• Analyze the associated spectral sequences to extract the de Rham complex and the Spencer complex at page one.
• Use convergence of spectral sequences to relate cohomology of differential and integral forms to the locally constant sheaf on real and complex supermanifolds.
• Apply the convergence result to prove isomorphism between differential and integral form cohomologies.
• Demonstrate that the Hodge-to-de Rham spectral sequence does not converge at page one for supermanifolds with Kähler reduced manifold, contrasting with ordinary complex manifolds.
• Leverage sheaf-theoretic and spectral sequence techniques to establish cohomological equivalences.

Experimental results

Research questions

• RQ1 How can differential and integral forms on supermanifolds be unified under a single algebraic framework?
• RQ2 What is the relationship between the cohomology of differential forms and the cohomology of integral forms on supermanifolds?
• RQ3 Does the Hodge-to-de Rham spectral sequence degenerate at page one for supermanifolds with Kähler reduced manifold, as it does for ordinary complex manifolds?
• RQ4 To what extent do the cohomologies of differential and integral forms compute the de Rham cohomology of the reduced manifold?
• RQ5 What structural properties of the double complex ensure convergence of spectral sequences to the locally constant sheaf on real and complex supermanifolds?

Key findings

• The cohomology of differential forms on real and complex supermanifolds is isomorphic to the cohomology of integral forms.
• Both the differential and integral form cohomologies compute the de Rham cohomology of the reduced manifold.
• The spectral sequences associated with the double complex converge at page two to the locally constant sheaf on real and complex supermanifolds.
• The Hodge-to-de Rham spectral sequence for supermanifolds with Kähler reduced manifold does not converge at page one, in contrast to the classical case of ordinary complex manifolds.
• The unifying double complex is constructed via a triple tensor product of sheaves, providing a systematic framework for superform cohomology.
• The double complex structure reveals a deep duality between differential and integral forms in supergeometry.