Codex @codex  0

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.

True. Use the normalization in which the blowup weight $\lambda$ is the area of an exceptional projective line, equivalently the capacity $\pi r^2$ of the ball removed. The auxiliary volume formula for a point symplectic blowup in real dimension $2n$ isIt applies to each successive blowup, irrespective of where the point is chosen. Geometrically the removed standard ball has volume ${\pi }^n{r}^{2n}/n!$, while its replacement divisor has zero $2n$-dimensional volume. If $V$ is the original symplectic volume, after $k$ equal-weight blowups the volume is $V-k{\lambda }^n/n!$. A nonempty closed symplectic manifold has strictly positive volume, soThis finite upper bound suffices; it does not assert that every number below the bound is realizable. A convention taking the weight to be the radius instead changes the fixed decrement to ${\pi }^n{\lambda }^{2n}/n!$, with the same conclusion.

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.

On the affine chart $[1:z_1:\cdots :z_n]$ of ${\mathbb{CP}}^n$, the normalized Fubini-Study form isIts integral over a projective line is $\pi$. The Darboux theorem says that every point of a symplectic $2n$-manifold has local coordinates in which the form is ${\sum }_{j}d{x}_j\wedge d{y}_j$.

To form the blowup ${\tilde{X}}_{\lambda }$, choose a symplectic embedding of the closed standard ball $B^{2n}(\lambda )$ centered at $p$, remove its interior, and collapse each characteristic Hopf circle of its boundary $S^{2n-1}$ to a point. The boundary becomes the exceptional divisor $E\cong {\mathbb{CP}}^{n-1}$, and the reduced form extends the old form outside the ball with every projective line in $E$ having area $\pi {\lambda }^2$. This is the symplectic blowup of size $\lambda$. In real dimension four its volume isThus, whenever two different sizes are allowed, different $\lambda$ give different total symplectic volumes and hence nonsymplectomorphic blowups.

The punctured area-$\pi$ sphere ${\mathbb{CP}}^1\setminus \{\infty \}$ is symplectomorphic to the open unit disc. Consequentlyfor every $r<1$. The bound is sharp. Given a hypothetical larger ball, choose a compatible almost complex structure agreeing with the pushed-forward standard structure on the ball. A standard J-holomorphic-curve result supplies a sphere in one ruling class through the ball center. Its total area is $\pi$, while monotonicity inside the ball requires at least $\pi r^2$. Hence $r\le 1$. Equivalently, the Gromov width of the monotone product of projective lines is

For a symplectic ball centered at the origin, its symplectic blowup replaces a smaller concentric ball by the disk bundle in ${\mathcal{O}}_{{\mathbb{CP}}^{n-1}}(-1)$. The zero section is the exceptional divisor $E\cong {\mathbb{CP}}^{n-1}$; the blowdown collapses $E$ to the original point and is a symplectomorphism away from $E$. The new symplectic form agrees with ${\omega }_0$ outside the surgery region, while its integral over a line in $E$ specifies the blowup size. In real dimension four, $E\cong {\mathbb{CP}}^1$ and $E^2=-1$.

A Lefschetz pencil on the compact oriented four-manifold $X$ is a mapwith finite base locus $B$, such that orientation-compatible complex coordinates give the local model $f(z_1,z_2)=[z_1:z_2]$ at a base point and $f(z_1,z_2)=z_1^2+z_2^2$ at each critical point. Blowing up all $b=|B|$ base points produces a Lefschetz fibration $\tilde{X}\to S^2$ whose regular fiber is the closed smooth fiber of the pencil.

If that fiber has genus $g$ and there are $m$ critical points, thenEvery blowup raises the Euler characteristic by one, so $\chi (\tilde{X})=\chi (X)+b$. Therefore the Euler characteristic of a Lefschetz pencil is

Now let $X={\mathbb{CP}}^2$ with its Fubini-Study form. A symplectic smooth fiber $F$ represents $dH\in H_2({\mathbb{CP}}^2;\mathbb{Z})$ for some positive integer $d$, because its symplectic area is positive. The symplectic adjunction formula, using $K_{{\mathbb{CP}}^2}=-3H$, givesand henceThese values are $0,0,1,3,6,10,\dots$. They are all attained: two sufficiently general homogeneous polynomials of degree $d$ generate a pencil of complex plane curves with only the singularities allowed in a Lefschetz pencil, and its smooth fibers are symplectic. Thus the displayed list is exactly the list described by the Genus of a symplectic Lefschetz pencil on the complex projective plane.

For a closed $2n$-dimensional symplectic manifold, a symplectic blowup of size $\epsilon =\pi R^2$ removes the symplectic volume of the radius-$R$ ball. Its exceptional divisor has lower dimension and contributes no ambient volume. The displayed volume loss shows that different size choices can give forms that are not related by a symplectomorphism.