We give a complete and self-contained exposition of the J-tame inflation lemma: Given any tame almost complex structure J on a symplectic 4-manifold \((M,\omega )\) , and given any compact, embedded, J-holomorphic submanifold Z, it is always possible to construct a deformation of symplectic forms \(\omega _t\) in classes \([\omega _t]=[\omega ]+t \mathrm{{PD}}{Z}\) , for \(0\le t\) less than an upper bound \(0<T\) that only depends on the self-intersection \(Z\cdot Z\) . The original proofs of this fact make the unwarranted assumption that one can find a family of normal planes along Z that is both J invariant and \(\omega \) -orthogonal to TZ—which amounts, in effect, to assuming the compatibility of J and \(\omega \) along Z. We explain how the original constructions can be adapted to avoid this assumption when Z has nonpositive self-intersection, and we discuss the difficulties with this line of argument in general to establish the full inflation when Z has positive self-intersection. We overcome this problem by proving a ‘preparation lemma’, which states that prior to inflation, one can isotope \(\omega \) within its cohomology class to a new form that still tames J and which is compatible with J along the submanifold Z. This preparation lemma can be regarded as an infinitesimal version of the “tamed-to-compatible” conjecture of S. K. Donaldson along an almost-complex submanifold Z.
