Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

By L. A. Cordero, M. Fernández, A. Gray, L. Ugarte. 2000

168 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0c87b1cd50e26eed7fbf2abc769714a9d0f54a8d

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Families of strong KT structures in six dimensions (2002, background): “and the resulting complex structure J is of ‘nilpotent’ type in the language of [4].”
A correction to “Tamed symplectic forms and strong Kähler with torsion metrics” (2012, background): “Equivalently by [6, 37], a complex structure on a 2n-dimensional nilpotent Lie algebra g is nilpotent if there is a basis {α1, .”
Almost parallel structures (2001, methodology): “Some tools are available to compute the Dolbeault cohomology of an invariant complex structure on N/Γ (Cordero et al., 2000; Console and Fino, 2001; Pedersen and Poon, 2001), but not in the most general situation.”
Hermitian structures on six-dimensional nilmanifolds (2004, background): “Keywords: Hermitian structure, Kähler with torsion structure, balanced metric, locally conformal Kähler structure, generalized Hopf metric, nilmanifold MSC 2000: 53C55; 17B30, 32G05”
Geometry of nilmanifolds with left‐invariant complex structure and deformations in the large (2009, methodology): “The question if the Dolbeault cohomology of nilmanifolds with left-invariant complex structure can be calculated using invariant differential forms has been addressed by Console and Fino in [CF01] and Cordero, Fernandez, Gray and Ugarte in [CFGU00]: consider in the Dolbeault-complex for ΩpMJ the sub”
Dolbeault cohomology for almost complex manifolds (2018, result): “The inclusion A gC →֒ AdR(M)⊗ C is compatible with the bigradings, but in general, even in the integrable case it is not known if it induces an isomorphism on Dolbeault cohomology, although this is the case in several situations (see [CFGU00], [CF01], [FRR19]) and conjectured to be true in the integ”
STABILITY OF ABELIAN COMPLEX STRUCTURES (2005, methodology): “Let us use like in [5] the ascending series {g } with g0 = {0}, g1 = {X ∈ g | [X, g] = 0}, .”
Bott–Chern cohomology of solvmanifolds (2012, background): “(We refer to [71, 54, 13, 3, 25, 22, 59, 62] for definitions and notation.”
Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry (2007, background): “tructures is much more accessible than the general case. In particular, the Dolbeault cohomology of an abelian nilmanifold can be expressed in terms of the corresponding Lie algebra cohomology ([CF], [CFGU]), and the same is true for the deformation space ([MPPS], [V2], [CFP]). This notion is specia”
Techniques of computations of Dolbeault cohomology of solvmanifolds (2011, unknown): “Like Nomizu’s theorem ([8]) for the de Rham cohomology of nilmanifolds, it is known that an isomorphism H∗,∗ ∂̄ (N/Γ′′) ∼= H∗,∗ ∂̄ (n) holds if (N, J,Γ ′′) meet one of the following conditions: (N) The complex manifold N/Γ′′ has the structure of an iterated principal holomorphic torus bundle ([3]).”