On Dolbeault formality and small deformations

Томассини, Адриано; Torelli, Sara

doi:10.1142/s0129167x14501110

Cited by 13 publications

(17 citation statements)

References 11 publications

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“…We proceed in the following way: we begin with the computation of the Dolbeault cohomology of X t since, as we recalled is Section 2, the first step of the Frölicher spectral sequence is isomorphic to the Dolbeault cohomology, namely E p,q 1 (X t ) ≃ H p,q ∂ (X t ) for every (p, q) ∈ Z 2 . By applying [AK12, Theorem 1.3], in [TT14] Tomassini and Torelli found the ∆ ′′ t -harmonic forms of X t ; for every (p, q) ∈ Z, those forms are a basis for H p,q ∂ (X t ) as C vector space. Then, as proved in [CFUG97], E p,q 2 (X t ) can be described as On the other hand, E p,q 2 (X 0 ) is the cohomology group of the complex…”

Section: Examplementioning

confidence: 99%

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On the degeneration of the Frölicher spectral sequence and small deformations

Maschio¹

2020

Complex Manifolds

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We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a C ∞ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a resultà la Kodaira-Spencer for the dimension of the second step of the Frölicher spectral sequence and we prove that, under a certain hypothesis, the degeneration at the second step is an open property under small deformations of the complex structure. We also provide an example in which we show that, if we omit that hypothesis, we lose the openness of the degeneration.

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Section: Examplementioning

confidence: 99%

“…In the last section, starting with the completely solvable Nakamura manifold and taking suitable complex deformations of the complex structure studied in [TT14], we show, by direct computations, that if Assumption 1 is not satisfied, then also the conclusion of Theorem 1 does not hold.…”

Section: Introductionmentioning

confidence: 97%

On the degeneration of the Frölicher spectral sequence and small deformations

Maschio¹

2020

Complex Manifolds

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“…• The same question may be addressed for other geometric flows other than the Chern-Ricci flow, for example the Hermitian curvature flows in [ST11] or in particular the one studied in [Ust17]. • It could be interesting to further investigate Massey triple products and Dolbeault Massey products, see [TTo14,CT15], or other Massey products, in particular on class VII surfaces with b 2 > 0 and on primary Kodaira surfaces.…”

Section: Dolbeault and Bott-chern Geometric Formalitiesmentioning

confidence: 99%

“…Then it is possible to investigate analogous notions for the Dolbeault cohomology. Neisendorfer and Taylor introduced the notion of (strictly) Dolbeault formality and "complex homotopy groups" in [NT78]; for Hermitian manifolds, the notion of (strictly) geometric-Dolbeault formality has been investigated by Tomassini and Torelli in [TTo14].…”

Section: Introductionmentioning

confidence: 99%

“…In this note, we focus on geometric formalities of complex manifolds and its dependence on the Hermitian metric. In [TTo14,TaT17], the authors study the behaviour of Dolbeault formality, respectively geometric-Bott-Chern formality, under small deformations of the complex structure. Here, we keep the complex structure fixed, and we study geometric formalities with respect to Hermitian metrics evolving along a geometric flow.…”

Section: Introductionmentioning

confidence: 99%

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Geometric Formalities Along the Chern-Ricci Flow

Angella

Sferruzza

2020

Complex Anal. Oper. Theory

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In this paper, we study how the notions of geometric formality according to Kotschick and other geometric formalities adapted to the Hermitian setting evolve under the action of the Chern-Ricci flow on class VII surfaces, including Hopf and Inoue surfaces, and on Kodaira surfaces.ker ∂∂ im ∂+im ∂ [Aep65] cohomologies provide further cohomological invariants on complex manifolds.and we study the possible algebra structure on the space of (de Rham, Dolbeault, Bott-Chern, Aeppli) harmonic forms with respect to ω(t) varying t.We study in details geometric formality according to Kotschick for a whole class of surfaces evolving by the Chern-Ricci flow, i.e. compact complex non-Kähler surfaces with Kodaira dimension Kod(X) = −∞ and first Betti number b 1 (X) = 1, known as class VII of the Enriques-Kodaira classification. In particular, we first rule out class VII surfaces with second Betti number b 2 > 0 by applying arguments as in [Kot01]. Then, we exploit the structure of quotients of Lie groups with invariant complex and Hermitian structure on the only class VII surfaces with b 2 = 0, that is Hopf and Inoue surfaces see [Bog82,Kod64,LYZ94,Tel94], in order to reduce the description of harmonic forms and the equation of the Chern-Ricci flow of such surfaces at the level of invariant forms and thus make explicit computations. We obtain the following.Theorem 3.1. On class VII surfaces, the Chern-Ricci flow preserves the notion of geometric formality according to Kotschick starting at initial invariant metrics.We also study the evolution of geometric formality according to Kotschick on other compact complex non-Kähler surfaces that are diffeomorphic to solvmanifolds, e.g. Kodaira surfaces. Since any complex structures on such surfaces is left-invariant, see [Has05, Theorem 1], we focus on invariant forms also in this case.Proposition 3.2. On Kodaira surfaces, geometric formality according to Kotschick is preserved by the Chern-Ricci flow starting at initial invariant metrics.We note that, also in this case, it is possible to rule out primary Kodaira surfaces by the obstructions in [Kot01] or [Has89], and therefore we focus on secondary Kodaira surfaces with initial invariant metrics.Regarding Dolbeault and Bott-Chern geometric formalities evolving by the Chern-Ricci flow, by applying the analogous procedure on Hopf, Inoue, and Kodaira surfaces, we have reached results as follows. We also checked how the algebraic structures of Aeppli cohomology and its harmonic representatives are modified along the Chern-Ricci flow.

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