# Meeting in December 2015

• Jian Xiao: Zariski decomposition for curves. We study Zariski decompositions using Legendre-Fenchel type transforms. In this way, we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves, yielding some fundamental positivity results for curve classes. We also give a refined structure of the movable cone of curves, and compare the volume function for curves with the mobility, showing some relations between enumerative geometry and convex analysis. This talk is based on the joint work with Brian Lehmann.
• Junyan Cao: Kodaira dimension of algebraic fiber spaces over surfaces. Let $f$ be a fibration between two projective manifolds. The Iitaka conjecture states that the Kodaira dimension of total space is not less than the sum of the Kodaira dimension of the generic fiber and that of the base manifold.  We prove a log-version (with klt pair) of the conjecture, under the assumption that the base is of dimension 2. The proof relies mainly on a uniformation theorem for compact kähler orbifolds with trivial first Chern class proved by F. Campana, and also the positivity properties of direct images of relative pluri-canonical bundles. This is a joint work with J.-A. Chen.
• Damian Brotbek: Complete intersection varieties with ample cotangent bundle. This is a joint work with Lionel Darondeau. We prove that any smooth projective variety $M$ contains subvarieties with ample cotangent bundle of any dimension less or equal to half the dimension of $M$. We construct such subvarieties as suitable complete intersection varieties.
• Shane Kelly: —