Exotic structures and adjunction inequality

Theorem 1.1. ([E]) Let X = B ∪ (1-handles) ∪ (2-handles) be four-dimensional handlebody with one 0-handle and no 3or 4-handles. Then • The standard symplectic structure on B can be extended over 1-handles so that manifold X1 = B 4 ∪ (1-handles) is a compact Stein domain. • If each 2-handle is attached to ∂X1 along a Legendrian knot with framing one less then Thurston-Bennequin framing of this knot, then the symplectic form and complex structure on X1 can be extended over 2-handles to a symplectic form on X, so that X become compact Stein domain.