Hodge theory and complex algebraic geometry

By Claire Voisin, Leila Schneps. 2007

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Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal (2010, methodology): “Then, via the Abel-Jacobi map AJ Y , the group of codimension 2 -cycles homologous to 0 on Y modulo rational equivalence is up to torsion isomorphic to the intermediate Jacobian J ( Y ) of Y , This follows from the diagonal decomposition principle [3] (see also [30, II,10.2.1]).”
Aeppli Cohomology Classes Associated with Gauduchon Metrics on Compact Complex Manifolds (2013, unknown): “Since ∂ω n − 2 = ( n − 2) ω n − 3 ∧ ∂ω , the last identity above gives ω n − 3 ∧ ∂ω = 0 . is an isomorphism (see e.g., [19, Lemma 6.20, p. 146]—no assumption on the Hermitian metric ω is needed .”
Curvature of vector bundles associated to holomorphic fibrations (2005, background): “lomorphic section of L⊗ KXt over a ﬁber can be extended to a holomorphic section of L⊗ KXs, for snear t. When L is trivial this follows from the Kähler assumption, by invariance of Hodge numbers, see [23]. When Lis semipositive it follows from a variant of the Ohsawa-Takegoshi extension theorem, tha”