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In this learning seminar series we will make an introduction to the locally conformally Kähler (LCK) geometry. A LCK metric is a structure on a complex manifold which falls somewhere between a Hermitian metric and a Kähler metric.

Ingredients of the individual seminars are as follows:

LCK 1: Introduction. Lee form. Torsion 1-form.

LCK 2: Weyl connection.

LCK 3: Relation with conformal structures: Hermite-Weyl structures with vanishing distance curvature. (Complex dimension ≥ 3)

LCK 4: Globally conformally Kähler (GCK) manifolds.

LCK 5: Vaisman's conjectures.

LCK 6: Curvature properties.

LCK 7: Blow up.

LCK 8: Adapted cohomology.

LCK 9: Examples. Hopf manifolds. Inoue surfaces. Generalized Thurston's manifold.

We will be using the following resources.

References:

1. S. Dragomir, L. Ornea - Locally conformal Kähler geometry.
   Progress in Mathematics, 155. Birkhäuser Boston, Inc., Boston, MA, 1998.


2. Vaisman, Izu. Some curvature properties of complex surfaces.
   Ann. Mat. Pura Appl. (4) 132 (1982), 1–18 (1983).


3. Vaisman, Izu. On locally and globally conformal Kähler manifolds.
   Trans. Amer. Math. Soc. 262 (1980), no. 2, 533–542.


4. Gauduchon, Paul. La 1-forme de torsion d'une variété hermitienne compacte.
   [Torsion 1-forms of compact Hermitian manifolds] Math. Ann. 267 (1984), no. 4, 495–518.