25.20 Locally conformally Kähler geometry2C3E73AAD887

2C3E73AAD887

In locally conformally Kähler geometry, the Lee form constrains both the rigidity of metrics and the volume forms that can be prescribed. The conjecture on pseudoeffectivity of the Lee class, 23.2.12, is formulated in terms of Gauduchon degrees by Ornea and Verbitsky [arXiv version, Conjecture 4.14 and Remark 4.16, OV25Balanced]; its consequence for balanced metrics is proposition 23.2.11. Their theorem on the volume forms of Vaisman metrics [arXiv version, Theorem 4.1 and Lemmas 4.2–4.4, OV24Vaisman] requires invariance under both the Lee and the anti-Lee vector fields. The proof of theorem 23.2.29 uses theorem 14.2.1: uniqueness gives a basic potential, transverse integration by Stokes' theorem fixes the scalar, and the Lee form of the new metric is parallel because it is closed and its dual vector field is Killing. The uniqueness argument keeps the scale of the candidate metric in (23.2.34) as an unknown until integration determines it.

The least-eigenvalue criterion 23.2.15 is proved by applying smooth Stokes to the volume polynomial of 23.2.14. Nonvanishing of the ordinary Bott–Chern class already follows from the conformal Gauduchon argument and the Lee degree (23.2.8). The Hopf calculations in 23.2.16, 23.2.17 distinguish constancy of the least eigenvalue from constancy of the whole spectrum, and distinguish first pointwise degeneration from collapse of total volume. These are explicit compact metrics in every dimension at least two. The criterion gives a smooth semipositive representative under its spectral hypothesis; pseudoeffectivity without that hypothesis remains the conjecture above.

The classification of radial metrics in 23.2.35 follows from explicit identities for the Hopf forms and a periodic scalar equation. Its positive periodic solutions are strictly ordered by the constant of integration, and their Lee periods fill the negative half-line. The positive twisted potential is constructed in the gauge given by the Lee form of the metric itself. Nonconstant radial volume forms force the metric to be non-Vaisman, and uniqueness of the metric is asserted only within the radial family considered there.

The proof of theorem 23.2.47 keeps the divergence terms in the Chern–Ricci-flat specialization of the argument of Barbaro and Otiman [v2, Section 4, equations (2)–(4), pp. 12–13, BO25LCK]. Their Theorem A gives the Gauduchon rigidity conclusion; retaining the scalar-curvature term in the integrated Bianchi identity gives the divergence identity and the equivalence with holomorphicity of the Lee field. The quotient and the two metrics in example 23.2.53 are computed directly: the holomorphic volume trivializes the canonical bundle, the flat gauge has nonzero Lee divergence, and compact pluriharmonic rigidity prevents a simultaneous conformal gauge. The conjecture for manifolds, 23.2.54, is stated in [Introduction, BO25LCK]. The Kodaira example admits a Vaisman metric on its given complex structure, so its conformal obstruction does not refute that conjecture.

The dual description of the Lee–Gauduchon cone and its bimeromorphic invariance in subsection 23.2.55 are due to Ornea and Verbitsky [Theorems 3.11 and 4.5, OV25LeeGauduchon]. The distinction between vanishing of the Lee–Gauduchon class and the strongly Gauduchon condition in subsection 23.2.67 follows [Remarks 5.6–5.8, OV25LeeGauduchon], and the criterion for the existence of strongly Gauduchon metrics used there is due to Popovici [Proposition 1.2, Pop13SG].