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

An introduction to the subject can be found at:

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

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 at least 3)

LCK 4: Globally conformally Kähler (GCK) manifolds. Vaisman's conjectures. Curvature properties.

LCK 5: Blow up. Adapted cohomology.

LCK 6: Examples. Hopf manifolds. Inoue surfaces. Generalized Thurston's manifold.