Article Dans Une Revue The Journal of Geometric Analysis Année : 2024

Twisted Adiabatic Limit for Complex Structures

Résumé

Given a complex manifold X and a smooth positive function η thereon, we perturb the standard differential operator d = ∂ + ∂ acting on differential forms to a first-order differential operator D η whose principal part is η∂ + ∂. The role of the zero-th order part is to force the integrability property D 2 η = 0 that leads to a cohomology isomorphic to the de Rham cohomology of X, while the components of types (0, 1) and (1, 0) of D η induce cohomologies isomorphic to the Dolbeault and conjugate-Dolbeault cohomologies. We compute Bochner-Kodaira-Nakano-type formulae for the Laplacians induced by these operators and a given Hermitian metric on X. The computations throw up curvature-like operators of order one that can be made (semi-)positive under appropriate assumptions on the function η. As applications, we obtain vanishing results for certain harmonic spaces on complete, non-compact, manifolds and for the Dolbeault cohomology of compact complex manifolds that carry certain types of functions η. This study continues and generalises the one of the operators d h = h∂ + ∂ that we introduced and investigated recently for a positive constant h that was then let to converge to 0 and, more generally, for constants h ∈ C. The operators d h had, in turn, been adapted to complex structures from the well-known adiabatic limit construction for Riemannian foliations. Allowing now for possibly non-constant functions η creates positivity in the curvature-like operator that stands one in good stead for various kinds of applications.

Problem 1.1. Let r ≥ 2 be an integer. Find a geometric property (P ) that certain compact complex manifolds X satisfy such that, whenever X 0 has property (P ) and has its Frölicher spectral sequence degenerate at E r , the Frölicher spectral sequence of every fibre X t with t ∈ B close enough to 0 degenerates again at E r .

We refrain from speculating on the nature (metric? cohomological?) of any such property (P ), but we stress the need for it to be readily verifiable on concrete examples of manifolds. An analytic such property, in terms of the decay rate to 0 of the small eigenvalues of certain Laplacians, was given in [Pop19], but that result seems hard to apply in explicit families of manifolds. One of the goals of the present paper is to generalise the main construction of [Pop19] (whose key points we recall in outline in §1.1 for the reader's convenience) in order to make it more flexible.

A solution to Problem 1.1 is expected to play a central role in various contexts, including in the further development of a non-Kähler mirror symmetry theory that started in [Pop18] and has continued with [PSU22], [PSU21a] and [PSU21b].

Reminder of the main facts from [Pop19]

Let X be a complex manifold with dim C X = n.

By adapting to the setting of complex structures the adiabatic limit construction for Riemannian foliations initiated and subsequently studied in e.g. [Wit85] and [MM90], we defined in [Pop19], for constants h > 0, the first-order differential operators

and the zero-th order pointwise operators θ h : Λ p, q T ⋆ X -→ Λ p, q T ⋆ X, u -→ θ h u := h p u,

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Dates et versions

hal-04930900 , version 1 (05-02-2025)

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Dan Popovici. Twisted Adiabatic Limit for Complex Structures. The Journal of Geometric Analysis, 2024, 35 (1), pp.36. ⟨10.1007/s12220-024-01867-4⟩. ⟨hal-04930900⟩
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