Holomorphic forms of geometrically formal Kaehler manifolds

Jean-Francois Grosjean and Paul-Andi Nagy


We investigate harmonic forms of geometrically formal metrics, which are defined as those having the
exterior product of any two harmonic forms still harmonic.
We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal
K"ahler metric are parallel w.r.t. the Levi-Civita connection. In the general Riemannian case a formal metric with maximal second Betti number is
shown to be flat . Finally we prove that a six-dimensional manifold with $b_1 neq 1, b_2 ge 3$ and not having the cohomology algebra of $mathbb{T}^3 times S^3$ carries a symplectic structure as soon as it admits a formal metric.

Harmonic forms, K"ahler manifold, almost K"ahler structure

Math Review Classification
Primary 53C15, 53C24, 53C55

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19 pages

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