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.