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The singular-curve adjunction formula in a symplectic manifold of real dimension four gives $c_1(A)=A^2+2-2\delta$ for a simple J-holomorphic sphere, with a nonnegative singularity count $\delta$. Hence the displayed inequality holds. This is the version needed when the sphere need not be embedded.

An anti-symplectic map between symplectic manifolds is a diffeomorphism $f$ satisfying $f^{\ast }{\omega }^{\prime }=-\omega$. For symplectic vector spaces, the analogous definition uses an invertible linear map.

Independent functions $Q_1,\dots ,Q_n$ on a $2n$-dimensional symplectic manifold can locally form the position half of canonical coordinates exactly when $\{Q_i,Q_j\}=0$ throughout a neighbourhood. Their commuting Hamiltonian vector fields span the fibres of $Q$. Flow from a transverse section gives coordinates $(Q,t)$ with $\omega =\sum d{t}_i\wedge d{Q}_i+\beta (Q)$. Write $\beta =d\alpha$ locally and take $P_i=-t_i-{\alpha }_i(Q)$. Then $\omega =\sum d{Q}_i\wedge d{P}_i$.

For a hypersurface $N$ of a symplectic manifold, the restriction of the symplectic form has a one-dimensional kernel, its characteristic line field. On a regular value level $N=H^{-1}(c)$, this line is generated by the Hamiltonian vector field of $H$.

A coisotropic submanifold $N$ satisfies $(T_{p}N{)}^{\omega }\subset T_{p}N$, where the superscript denotes the symplectic orthogonal complement. Every hypersurface of a symplectic manifold is coisotropic.

This subgroup uses Hamiltonians whose support in $[0,1]\times M$ is compact. Their flows are the identity outside a fixed compact set. On closed manifolds this support condition imposes no extra restriction. On an open symplectic manifold, it excludes nontrivial whole-fiber cotangent fiber translations, even when their generating one-form is exact.

An almost complex structure $J$ on a symplectic manifold $(M,\omega )$ is compatible with $\omega$ whenand $g_J(u,v)=\omega (u,Jv)$ is a positive-definite inner product. Every symplectic vector bundle admits a compatible almost complex structure.

The Darboux theorem says that every point of a $2n$-dimensional symplectic manifold has local coordinates in which the symplectic form is ${\sum }_{j}d{x}_j\wedge d{y}_j$. Thus symplectic manifolds have no local invariants beyond dimension.

Denis Auroux studies symplectic geometry, with work linking geometric constructions to topology and algebraic geometry. His Harvard research profile identifies the specialization.^[ref]

His lectures with Ivan Smith develop descriptions of symplectic manifolds through Lefschetz pencils and branched covers of the projective plane. The resulting invariants combine the algebra of monodromy with geometric information from pseudoholomorphic curves. This creates a bridge between concrete maps and the classification of manifolds.^[ref]

For a smooth map between Riemannian manifolds, its Dirichlet energy is half the integral of the squared norm of its differential, using the source and target metrics. When the source is two-dimensional, a conformal change of its metric leaves this energy unchanged. For a J-holomorphic curve into a symplectic manifold with a compatible almost complex structure, the Energy identity for a J-holomorphic curve identifies it with the pulled-back symplectic area. The norm of a differential here is the full tensor norm, summing its squared values on an orthonormal source frame.

The displacement energy of a subset of a symplectic manifold is the infimum of the energy norms of compactly supported Hamiltonian diffeomorphisms moving it off itself. The infimum is $+\infty$ if no such map exists. One must use the same chosen norm convention in both this definition and any energy lower bound. A compact set can have zero displacement energy even when every nonidentity map has positive norm.

In a product symplectic manifold $S^2(a)\times T^{2n-2}$, every compatible almost complex structure admits a J-holomorphic sphere in class $F$ through each point. Here $S^2(a)$ has area $a$, and the torus carries a constant symplectic form. All spherical classes are integer multiples of $F$, so a curve in class $F$ cannot split into positive-area nonconstant spheres. For the product complex structure the fibre through a point is unique and regular. Its one-point count modulo two is one; generic transversality and the compact one-dimensional parameterized moduli spaces preserve this count. Approximation and Gromov compactness for spheres remove the genericity requirement. This gives the area-$a$ curve used in the proof of the Gromov non-squeezing theorem.

For a symplectic blowup of a $2n$-dimensional symplectic manifold, the displayed formula gives its First Chern class. The exceptional divisor has normal bundle $\mathcal{O}(-1)$; the discrepancy coefficient is $n-1$. In dimension four it is one.

Suppose a symplectic Lie group action on a connected symplectic manifold has global component Hamiltonians and a common fixed point $p_0$. Choose their constants linearly so ${\mu }_{\xi }(p_0)=0$. The difference $c(\xi ,\eta )=\{{\mu }_{\xi },{\mu }_{\eta }\}-{\mu }_{[\xi ,\eta ]}$ has zero Hamiltonian vector field because fundamental fields and Hamiltonian fields have compatible Lie-bracket conventions; it is therefore constant. At $p_0$, all fundamental fields vanish, so both terms in $c$ vanish. Thus $c=0$ everywhere and the Poisson brackets of the components reproduce the group Lie algebra. For a connected group this also yields coadjoint equivariance. A linear unitary action on a vector space has the origin as such a fixed point.

Geometric quantization associates quantum states and observables with a symplectic manifold. A first step uses a complex line bundle with a connection on a vector bundle whose curvature is proportional to the symplectic form divided by $\hslash$; its existence imposes an integrality condition. A real polarization is an integrable half-dimensional distribution with Lagrangian submanifolds as leaves. Requiring sections to be covariantly constant along the leaves reduces the variables to a configuration-space description. Restricting the curvature to a Lagrangian leaf gives zero, permitting local flat sections; global sections can require trivial holonomy. Exact Lagrangian graphs also describe local semiclassical phases through their generating functions.

For a compact symplectic manifold, smooth convergence of tame almost complex structures and a uniform area bound give a subsequence of J-holomorphic spheres converging, after reparametrization, to a finite tree of J-holomorphic spheres, with convergence smooth away from finitely many bubbling points. The total homology class and symplectic area are preserved. Constant components may be added for stability. In a spherical class which cannot split into two positive-area spherical classes, there is only one nonconstant component.

This group consists of time-one maps of Hamiltonian isotopies whose flows are defined throughout the specified time interval. With ${\iota }_{X_H}\omega =-dH$, composition of isotopies generated by $H_t,K_t$ has generator $H_t+K_t\circ {\varphi }_t^{-1}$, where ${\varphi }_t$ is the first isotopy. The inverse path has generator $-H_t\circ {\varphi }_t$. These formulas prove closure under composition and inversion. On a closed symplectic manifold, every smooth time-dependent Hamiltonian has a complete flow over a bounded time interval.

A Hamiltonian diffeomorphism is the time-one map of a Hamiltonian isotopy beginning at the identity on a symplectic manifold. The isotopy is generated by a smooth time-dependent Hamiltonian function $H_t$. Its flow must exist for the whole interval; on an open manifold a common group convention requires compact support throughout that interval.

In the convention ${\iota }_{X_{H_t}}\omega =-d{H}_t$, the infinitesimal preservation law isThus every Hamiltonian diffeomorphism is a symplectomorphism. The conclusion follows from closedness of the symplectic form, not from a special choice of coordinates.

The identity is generated by the zero Hamiltonian. Inverses and compositions are again Hamiltonian endpoints, forming the group of Hamiltonian diffeomorphisms. A generating function path and its endpoint are different data: different isotopies can have the same time-one map. The Hamiltonian condition imposes more structure than only preserving the symplectic form.

A Hamiltonian group action is a Lie group action on a symplectic manifold with an equivariant moment map $\mu :M\to {\mathfrak{g}}^{\ast }$. In the sign convention ${\iota }_{X_H}\omega =dH$, each component satisfies $d⟨\mu ,X⟩={\iota }_{X_M}\omega$, where $X_M$ is the fundamental vector field. The opposite Hamiltonian vector field convention reverses the sign.

A Hamiltonian isotopy is a smooth path of symplectomorphisms ${\varphi }_t$, with ${\varphi }_0=\mathrm{id}$, generated by a smooth time-dependent Hamiltonian function. In the convention ${\iota }_{X_{H_t}}\omega =-d{H}_t$, it satisfiesThe flow must exist throughout the chosen time interval. This is automatic on a closed symplectic manifold; on an open one, uniformly compactly supported Hamiltonians also suffice. The Hamiltonian flow preserves the symplectic form by Cartan's magic formula.

Its time-one map is a Hamiltonian diffeomorphism, an element of the group of Hamiltonian diffeomorphisms. A path and its endpoint are different data: a full rotation of the symplectic plane gives a nonconstant Hamiltonian path whose endpoint is the identity. Distinct paths can therefore represent the same Hamiltonian diffeomorphism.