Symplectic One-Point Blow-Up

• Symplectic one-point blow-up is a geometric surgery on a symplectic manifold that replaces a point with an exceptional divisor, modifying its symplectic form and cohomology.
• The procedure introduces a new degree-two cohomology class dictated by the exceptional divisor, enabling explicit computation of invariants like Gromov–Witten numbers.
• This construction facilitates analysis of symplectomorphism groups, Lagrangian Floer homology, and the interplay between symplectic and Kähler structures.

A symplectic one-point blow-up is a surgery on a symplectic manifold $(M, \omega)$ at a point $p \in M$, producing a new symplectic manifold $(\widetilde{M}, \widetilde{\omega})$ that contains an exceptional divisor $E \cong \mathbb{CP}^{n-1}$ (for $2n$-dimensional $M$), with the cohomology class of the symplectic form given by $$[\widetilde{\omega}] = \pi^*[\omega] - \varepsilon\, \mathrm{PD}[E]$$, where $\pi: \widetilde{M} \to M$ is the blow-down map and $\varepsilon$ parametrizes the size of the blow-up. This operation plays a fundamental role in constructing new symplectic and Kähler manifolds, analyzing symplectic invariants, and understanding symplectomorphism groups and quantum/floer-theoretic structures.

1. Geometric Construction and Symplectic Form

The symplectic one-point blow-up relies on local and global geometric data. Around $p$, Darboux's theorem provides coordinates with standard symplectic form $$\omega_0 = \frac{i}{2} \sum_j dz_j \wedge d\bar{z}_j$$. The operation proceeds by removing an embedded symplectic ball $B^{2n}(r)$ and gluing in the projectivized tautological line bundle $\widetilde{\mathbb{C}}^n$ over $\mathbb{CP}^{n-1}$ as follows:

• The smooth manifold $$\widetilde{M} = (M \setminus \{p\}) \sqcup L(r) / {\sim}$$, where $L(r)$ is the preimage of $B^{2n}(r)$ in the blow-up model.
• The exceptional divisor $E = \pi^{-1}(p) \cong \mathbb{CP}^{n-1}$ is embedded as the zero-section.
• The symplectic form $\widetilde{\omega}_\rho$ is constructed to coincide with $\omega$ outside the surgery region, and, in the local model, as $$\omega(\rho) = \Phi^* \omega_0 + \rho^2 \, \operatorname{pr}^* \omega_{FS}$$ near $E$, with $\omega_{FS}$ the Fubini-Study form.

This form satisfies $$[\widetilde{\omega}_\rho] = \pi^*[\omega] - \rho^2 \, \mathrm{PD}[E]$$ for all $0 < \rho < r$. The symplectic area of a line in $E$ is then $\rho^2 \pi$ (Pedroza, 2019, Pedroza, 2015). For four-manifolds, this specializes to blowing up a point with replacement by $S^2$ of area $\pi c^2$ (Chakravarthy et al., 1 Oct 2025, Karshon et al., 2014).

2. Cohomological and Topological Features

The blow-up introduces a new degree-two cohomology class, $e = \mathrm{PD}[E]$. The ring structure is as follows:

• $$H^*(\widetilde{M}) \cong \pi^* H^*(M) \oplus \mathbb{R}\langle e, e^2, \ldots, e^{n-1}\rangle$$.
• $e^n = (-1)^{n-1} \mathrm{PD}(\mathrm{pt})$.
• $\pi^* \alpha \cup e = 0$ for all $\alpha \in H^*(M)$ (Pedroza, 2019).

On homology, $$H_*(\widetilde{M}) \cong \pi_* H_*(M) \oplus \mathbb{Z}[L_E]$$, where $[L_E]$ is the line class in $E$ and $[L_E] \cdot e = -1$ (Pedroza, 2015).

If $M$ is monotone with $c_1(M)(A) = \alpha \int_A \omega$, monotonicity is retained for $\widetilde{M}$ if and only if $\rho^2 = (n-1)/(\alpha \pi)$ (Pedroza, 2019).

3. Classification, Parameters, and Constraints

For the one-point blow-up of $(\mathbb{CP}^2, \omega_0)$:

• The cohomology class of the symplectic form on the blow-up is $$[\widetilde{\omega}] = \lambda \mathrm{PD}[L] - \delta \mathrm{PD}[E]$$, where $L$ is the line class and $E$ is the exceptional divisor.
• Existence: $0 < \delta < \lambda$.
• Uniqueness: Two blow-ups are symplectomorphic if and only if their $(\lambda, \delta)$ parameters coincide.
• The moment polytope is modified by cutting off a corner of size $\delta$ from the Delzant simplex (Karshon et al., 2014, Chakravarthy et al., 1 Oct 2025).

For ruled surfaces or more general settings, the cohomological class is parameterized in analogous fashion: $$[\omega_{μ,c}] = μ\, \mathrm{PD}[B] + 1 \cdot \mathrm{PD}[F] - c\,\mathrm{PD}[E]$$, with geometric constraints on $c$ depending on the structure of the base and fiber (Buse et al., 2020).

4. Symplectic Cones and Kähler Structures

The relation between the symplectic cone and the Kähler cone can be subtle. In the context of the one-point blow-up of an Enriques surface, non-Kähler symplectic forms exist on the blow-up, in contrast to the abundance of elliptic fibrations and associated invariants that distinguish the Kähler and symplectic cones. Quantitative comparison of these invariants reveals that the symplectic cone can be strictly larger than the Kähler cone, admitting symplectic forms not compatible with any integrable complex structure (Ning, 14 Jul 2024).

5. Lagrangian and Floer-Theoretic Properties

Under the blow-up, Lagrangian submanifolds disjoint from the surgery ball lift to proper transforms $\widetilde{L} = \pi^{-1}(L)$, yielding Lagrangians in $(\widetilde{M}, \widetilde{\omega}_\rho)$. The Maslov index transformation is given by: $$\mu_{\widetilde{L}}([\widetilde{u}]) = \mu_L([\pi \circ \widetilde{u}]) - 2(n-1)\ell,$$ where $\ell$ is the intersection number with the exceptional divisor. Monotonicity of $L$ is preserved for appropriate $\rho^2$; thus, the proper transform construction provides a means of generating monotone Lagrangians in the blow-up (Pedroza, 2019).

Lagrangian Floer homology of proper transforms can be computed in terms of Floer data of the original Lagrangian in $M$. This principle also underlies the structure of the Fukaya category of blow-ups, where, for small blow-up parameters, the preimages of generators of $\operatorname{Fuk}(M)$ along with Lagrangian tori near the exceptional locus split-generate $\operatorname{Fuk}(\widetilde{M})$, categorifying known decompositions in quantum cohomology (Venugopalan et al., 2020).

6. Symplectomorphism Groups and Mapping Class Phenomena

The topological structure of the symplectomorphism group, including the group of Hamiltonian diffeomorphisms, is affected significantly by the blow-up. For the symplectic one-point blow-up:

• Hamiltonian loops on $M$ lift to loops on $\widetilde{M}$, with the lift gaining infinite order in $\pi_1 \operatorname{Ham}(\widetilde{M})$ in many cases, as demonstrated using Weinstein's action homomorphism. The action is computable via explicit formulas involving the blow-up parameter $\rho$ (Pedroza, 2015).
• On irrational ruled surfaces after a one-point blow-up, the connected components and isotopy classes of the symplectomorphism group are partially governed by “fibered Dehn twists,” which do not correspond to classical Lagrangian Dehn twists (Buse et al., 2020).

For almost toric and log symplectic settings, standard symplectic and almost-toric blow-ups are symplectomorphic, with the symplectomorphism constructed via Moser's method and preserving log-Calabi–Yau boundaries in divisor-compatible cases (Chakravarthy et al., 1 Oct 2025).

7. Gromov–Witten Invariants and Quantum Effects

In higher dimensions, closed formulae relate Gromov–Witten invariants of the blow-up to those of the original manifold using degeneration and absolute/relative correspondence. For a six-dimensional symplectic manifold:

• The cohomology class jumps as $$[\tilde{\omega}_\varepsilon] = \pi^*[\omega] - \varepsilon\, \mathrm{PD}[E]$$.
• The Gromov–Witten invariants with point insertions can be expressed as sums involving relative invariants of the blow-up and universal coefficients determined by the geometry of $(\mathbb{CP}^3, H)$ (He et al., 2014).
• The blow-up formula induces corresponding relations among generalized BPS numbers.

A plausible implication is that symplectic one-point blow-up provides a controlled means for investigating quantum invariants and categorical structures—such as semi-orthogonal decompositions in Fukaya categories—that reflect the topological and symplectic modifications introduced by the surgery (Venugopalan et al., 2020), and the delicate interplay between symplectic and complex geometric invariants (Ning, 14 Jul 2024).