Calabi–Yau caps, uniruled caps and symplectic fillings

Tian-Jun Li University of Minnesota Cheuk Yu Mak University of Minnesota Kouichi Yasui Hiroshima University

Algebraic Geometry Symplectic Geometry mathscidoc:1705.01002

Proc. London Math. Soc., 114, (3), 159–187, 2017
We introduce symplectic Calabi–Yau caps to obtain new obstructions to exact fillings. In particular, they imply that any exact filling of the standard contact structure on the unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinite family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admitting minimal strong fillings of arbitrarily large b2. Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/ adjunction contact structures to present a unified picture to the existing finiteness results on the topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find new classes of contact 3-manifolds with the finiteness properties and extend Wand’s obstruction of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero.
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  title={Calabi–Yau caps, uniruled caps and symplectic fillings},
  author={Tian-Jun Li, Cheuk Yu Mak, and Kouichi Yasui},
  booktitle={Proc. London Math. Soc.},
Tian-Jun Li, Cheuk Yu Mak, and Kouichi Yasui. Calabi–Yau caps, uniruled caps and symplectic fillings. 2017. Vol. 114. In Proc. London Math. Soc.. pp.159–187.
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