On the metric structure of non-Kähler complex surfaces

Normal CR structures on compact 3-manifolds

“…Remark. It turns out then that the Chern class of the S 1 -bundle M → Σ is always positive: this is because we chose T to be positive (see [1], Section 3, for a detailed explanation); in particular, we obtain a positive Chern class for the Hopf fibration S 3 → S 2 , apparently contradictory to the negative Chern class of the tautological bundle O(−1) on CP 1 (C 2 {0} → CP 1 ); this is because the canonical metric on S 3 is a Sasakian structure with the opposite orientation. Remark.…”

“…All normal CR compact 3-manifolds are covered by circle bundles over a Riemann surface [7], [8], [1], see also next Section, and, if this circle bundle is not the Hopf fibration S 3 → S 2 , then all Sasakian structures are, up to a finite quotient, regular ( [1], see also next Section). If M is covered by S 3 , then any Sasakian structure on M is a deformation of a regular one [1], see next Section. Therefore the study of these particular Sasakian structures is essential to the understanding of compact normal CR 3-manifolds.…”

“…In dimension 3, the normal CR structures are always deformations of a flat one (Theorem 1, see also [1]), and the key point is that, for a CR Reeb vector field T , they admit compatible Sasakian metrics, for which T is 1 Killing (see Section 2 for details); these metrics are closely related to locally conformally Kähler metrics with parallel Lee form, natural analogs of Kähler structures on non-symplectic complex manifolds [19].…”

“…Topologically, every compact normal CR (or Sasakian) 3-manifold is a Seifert fibration (Proposition 5, see also [8], [7] and [1]), but it turns out that the Sasakian structures themselves can be explicitly described on these manifolds: Theorem 1, Proposition 5 (these are extended versions of the results announced in [1]); In particular, if a compact Sasakian 3-manifold is not covered by S 3 (or non-spherical), its Sasakian structure is always regular, i.e. it is a (finite quotient of a) circle bundle over a Riemann surface and its metric arises from a Kaluza-Klein construction (Corollary 2), which leads to an elementary description of any Sasakian metric on these manifolds, either directly, or as deformations of a CR flat one.…”

Hermitian spin surfaces with small eigenvalues of the Dolbeault operator

“…It was shown by Friedrich [8] [14,5] and a result of Tsukada [26]. But Recall that a Hermitian connection on a Hermitian manifold is a connection with respect to which both the metric h and the complex structure I are parallel.…”