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For a steady-state equivariant dynamical system, an axial subgroup is a stabilizer subgroup whose real fixed-point subspace of a group action has dimension one. Requiring an actual stabilizer subgroup matters: a smaller subgroup can have the same fixed line without being the full symmetry of any point on that line. The fixed line reduces the steady-state bifurcation equation to one real amplitude.

In an equivariant Hopf bifurcation, a complex axial isotropy subgroup is instead a stabilizer subgroup for the spatial symmetry group combined with the common phase circle action, with one-dimensional complex fixed-point subspace of a group action. Distinct conjugacy classes of such subgroups can describe distinct types of bifurcating branches.

A symplectic blowup replaces a small symplectic ball around the point by an exceptional divisor ${\mathbb{CP}}^{n-1}$, for ${\dim }_{\mathbb{R}}X=2n$. One can construct its form by symplectic reduction. In a Darboux chart, consider ${\mathbb{C}}^n\times \mathbb{C}$ with the circle actionOn ${\mu }^{-1}(R^2/2)$ the action is free because $z\ne 0$. Its quotient therefore carries a smooth symplectic form. Where $w\ne 0$, choose the representative with $w$ positive real; the reduced form is the original form on $|z|>R$. Where $w=0$, the quotient is the Hopf fibration quotient of $|z|=R$, namely ${\mathbb{CP}}^{n-1}$. The local quotient is the tautological complex line bundle near its zero section, hence the local blowup. Glue this model to the exterior of the ball to obtain a symplectic form on the blowup.

The area of a projective line in the exceptional divisor is $\epsilon =\pi R^2$. For a closed $X$, changing this size changes the symplectic volume:Consequently the blowup is not determined up to symplectomorphism without a size choice. The underlying smooth blowup is fixed, but the symplectic construction has a parameter, as in symplectic blowup size changes volume.

For the blowdown map $\pi$ and exceptional divisor $E$, the First Chern class isIn real dimension four this becomes ${\pi }^{\ast }{c}_1(TX)-\mathrm{PD}[E]$, the first Chern class formula for a symplectic blowup.

The tangent bundle of the standard four-torus is a trivial complex rank-two bundle, so its symplectic canonical class is zero. For a generic fiber $F$ of a symplectic Lefschetz pencil, the symplectic adjunction formula gives $2g-2=F^2$. Two generic fibers intersect precisely at the base points, each with local intersection number one in the pencil model $[z_1:z_2]$. HenceThis is base-point count for a Lefschetz pencil on a symplectic four-torus.