Symplectic Inflation in 4D Topology

• Symplectic inflation is a deformation technique that increases the symplectic area along embedded surfaces by adding Poincaré dual classes.
• It uses local models and controlled iterative modifications to address symplectic embedding, packing, and moduli space problems.
• The method adapts to both smooth and singular scenarios, enabling precise manipulation of configurations in four-dimensional symplectic topology.

Symplectic inflation is a foundational deformation technique in four-dimensional symplectic topology that enables precise manipulation of symplectic forms via local modifications along embedded surfaces or more general singular curves. It is central to the paper of symplectic embedding problems, symplectic cone computations, and the geometry of moduli spaces of symplectic manifolds. The core procedure relies on increasing the symplectic area assigned to a specified homology class, typically through controlled modifications cohomologically of the form $[\omega_t] = [\omega_0] + t\,\mathrm{PD}([S])$, where $S$ is a symplectic surface or collection thereof.

1. Fundamental Definitions and Geometric Principles

Let $(M,\omega_0)$ denote a closed symplectic four-manifold, and let $C \subset M$ be an embedded symplectic surface representing a homology class $[C]=S \in H_2(M;\mathbb{Z})$ with $\omega_0(S)>0$. Symplectic inflation is the process of deforming $\omega_0$ through symplectic forms $\omega_t$ in such a way that the symplectic area of $C$—and more generally, the pairing of the symplectic class with any prescribed homology class—increases. In topological terms, this is described by the path $\omega_t = \omega_0 + t\,\mathrm{PD}(S)$ for $t\geq 0$, where $\mathrm{PD}(S)$ is the Poincaré dual in $H^2(M;\mathbb{R})$. Geometrically, inflation is achieved either by performing a symplectic sum (related to Gompf's pairwise sum construction) or by introducing closed 2-forms representing $\mathrm{PD}(S)$ supported in tubular neighborhoods of the inflation loci (McDuff, 2013).

Inflation can be localized along embedded surfaces with trivial or nontrivial normal bundles, and it admits generalizations to collections of transversally intersecting, $\omega_0$-orthogonal surfaces via the Li–Usher technique [J. Symplectic Geom. 4 (2006), 71–91].

2. Inflation Along Single and Multiple Surfaces

The classical Lalonde–McDuff inflation applies when $C$ is symplectically embedded with nonnegative self-intersection, admitting inflation for arbitrarily large $t\geq 0$. In such cases, the process is unconstrained except by global cohomological positivity ($[\omega_t]^2 > 0$). If $C \cdot C < 0$, the inflation is limited by $t < \omega_0([C]) / |C \cdot C|$, reflecting the requirement that the area of $C$ remains positive (Chakravarthy et al., 28 Mar 2024).

The Li–Usher generalization enables simultaneous inflation along a configuration $S = \{C_i\}_{i=1}^L$ of embedded symplectic surfaces, intersecting transversely and $\omega_0$-orthogonally. A choice of positive multiplicities $m_i>0$ yields a combined inflation class $A = \sum m_i [C_i]$ such that $A \cdot [C_i] > 0$ for all $i$. The inflation is performed iteratively along each $C_i$, using individual steps $\varepsilon < \omega_0([C_i]) / |C_i \cdot C_i|$ for negative curves, with the process repeated sufficiently many times to achieve any desired total weight $K$. Cohomologically, this results in deformation to $[\omega_K] = [\omega_0] + K\,\mathrm{PD}(A)$ (McDuff, 2013).

For singular inflation, where the locus is not smooth or Fredholm regular, one inflates along unions of nodal or normal-crossing divisors, extending cohomological deformations to cases involving non-generic or singular curves (McDuff et al., 2013).

3. Inflation Relative to Singular Divisors

In many embedding and packing problems, one requires inflation relative to a singular symplectic divisor $S$, which may be a union of transversally intersecting embedded curves of arbitrary genera and self-intersections. An almost complex structure $J$ is said to be $(S,N)$-adapted if it makes all components $J$-holomorphic and is integrable on a fibered neighborhood $N(S)$. Inflation in this context relies on the existence of $S$-good homology classes—those satisfying positivity and Gromov invariant conditions, and suitable intersection constraints with the components of $S$.

Absolute singular inflation produces a family of symplectic forms $[\omega_{k,A}] = [\omega] + k\,\mathrm{PD}(A)$, all nondegenerate on $S$, for $A$ an $S$-good class (McDuff et al., 2013). This is further extended by the relative singular inflation theorem, which expresses isotopies between cohomologous symplectic forms that remain nondegenerate on $S$ as 2-parameter families of relative inflation, crucial for establishing deformation-to-isotopy results and the calculation of the relative symplectic cone.

4. Analytical Details and Technical Constructions

The inflation process employs explicit local models. In a tubular neighborhood of a symplectic curve $C$ with self-intersection $n = C \cdot C$, the symplectic form can be written as $\omega = \pi^*(\omega_C) + d(r^2 \alpha)$, with $d\alpha = -n \pi^*(\omega_C)$. Setting $\omega_k = \omega + k\,d(f(r^2)\alpha)$ with $f$ a cutoff function, one obtains a cohomological class $[\omega_k] = [\omega] + k\,\mathrm{PD}(C)$, symplectic for $k$ in the range dictated by the sign of $n$ (McDuff et al., 2013, Chakravarthy et al., 28 Mar 2024).

For $J$-holomorphic curves $Z$ with tame almost complex structures $J$ (i.e., $\omega(v,Jv)>0$ everywhere), the $J$-tame inflation lemma guarantees the existence of a path of symplectic forms $\omega_t$ taming $J$, with $[\omega_t] = [\omega] + t\,\mathrm{PD}([Z])$ for $0 \leq t < T$. If $Z \cdot Z \geq 0$, arbitrarily large inflation is possible; if $Z \cdot Z < 0$, the bound is $T = \omega(Z)/(-Z \cdot Z)$. Achieving full inflation for positive self-intersection curves requires preparing the pair $(\omega, J)$ to be compatible along $Z$, realizable via a local isotopy constructed using a Whitney extension (Chakravarthy et al., 28 Mar 2024).

When inflating along collections of nodal curves or normal-crossing divisors, the existence of embedded $J$-holomorphic representatives for the relevant classes is often obtained via Gromov invariants and reduction techniques, followed by local smoothing ("amalgamation") (McDuff et al., 2013).

5. Applications: Embedding and Packing Problems

Symplectic inflation has been pivotal in resolving symplectic embedding and packing problems, notably embeddings of ellipsoids and packings of balls in four-manifolds. For example, embedding one ellipsoid into another can be reduced to inflating along a configuration of spheres (arising from weighted blow-ups and exceptional divisors), with the inflation process used to construct a symplectic form whose cohomology class matches the desired scaling ratio, enabling the construction of the requisite embedding (McDuff, 2013).

Manifold-level inflation techniques based on global Liouville forms have enabled constructive proofs of maximal packings in $\mathbb{CP}^2$ by six, seven, or eight balls, relying on explicit configurations of singular curves and local crosses at prescribed singularities, then using Liouville flows to push symplectic balls into the manifold. These explicit constructions surpass earlier blow-up based approaches by eliminating intermediary spaces and making the embedding process fully explicit (Opshtein, 2011).

Singular inflation is instrumental in the existence and isotopy classification of Lagrangian spheres and the computation of the relative symplectic cone, as it enables the control of deformations while remaining nondegenerate on prescribed divisors. In particular, the structure of the relative cone $$\mathrm{Cone}_\omega(M,S) = \{ a \in H^2(M;\mathbb{R}) : a^2>0, a(E)>0\, \forall E, a(S_i)>0\, \forall i \}$$ follows immediately from the flexibility of singular inflation along all components of $S$ (McDuff et al., 2013).

6. Limitations, Open Directions, and Technical Considerations

Several open questions remain regarding the scope of symplectic inflation. The realization of symplectic curves with prescribed singularities, necessary for many inflationary arguments, depends on Donaldson-type techniques; in higher dimensions or with more complex singularity types, existence remains unresolved (Opshtein, 2011). The possibility of deriving Gromov’s non-squeezing theorem purely from approximately holomorphic curve constructions is also open.

Inflation in the presence of positive self-intersection submanifolds requires careful compatibility-preparation arguments, as naive constructions may fail to preserve tameness. The preparation lemma solves this obstruction by isotoping the symplectic form to be compatible with the almost complex structure along the inflation submanifold (Chakravarthy et al., 28 Mar 2024).

Finally, the classification of possible packings by arbitrary shapes via inflation (especially in irrational or singularly polarized settings) remains largely incomplete, but extensions to irrational cohomology classes and to unions of curves (polarizations with nodes) are available (Opshtein, 2011).

7. Historical Context and Core Literature

Symplectic inflation originated with the establishment of the Lalonde–McDuff technique and its refinements by Gompf’s sum constructions and Li–Usher’s generalization to configurations of surfaces (McDuff, 2013). The theoretical landscape has been substantially developed via singular inflation (McDuff–Opshtein), manifold-level Liouville inflation methods (Opshtein), and $J$-tame versions removing regularity assumptions (Chakravarthy et al., 28 Mar 2024, McDuff et al., 2013, Opshtein, 2011).

Key developments include:

Reference	Main Contribution	arXiv id
McDuff, "Symplectic embeddings..." (erratum)	Corrected procedure via Li–Usher inflation	(McDuff, 2013)
Opshtein, "Symplectic packings..."	Manifold-level, Liouville-based inflation, maximal packings	(Opshtein, 2011)
McDuff–Opshtein, "Nongeneric J-holomorphic curves..."	Singular inflation, relative cones, embedding applications	(McDuff et al., 2013)
“Remarks on $J$-tame inflation”	$J$-tame inflation for arbitrary tamed structures	(Chakravarthy et al., 28 Mar 2024)

These references collectively underpin modern symplectic inflation and its extensive applications throughout four-dimensional symplectic topology.