Remarks on $J$-tame inflation

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 $[\omega_t]=[\omega]+tPD(Z)$, for $0\leq t$ 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 isotop $\omega$ within its cohomology class to a new form that still tames $J$ and which is compatible with $J$ along the submanifold $Z$.

arXiv version