Flops and Poisson Deformations in Symplectic Geometry

• Flops and Poisson deformations are fundamental in symplectic geometry, with flops as crepant birational modifications and deformations altering Poisson structures.
• The interplay is characterized by unique equivariant C*-actions that extend Poisson brackets and linearize line bundles for consistent deformation theory.
• These methods underpin applications in moduli theory and mirror symmetry, as illustrated by explicit examples like the Atiyah flop in conifold singularities.

A symplectic variety is a normal complex algebraic variety $X$ whose smooth locus $X_{\mathrm{sm}}$ admits a holomorphic $2$-form $\omega$ that is nondegenerate on $X_{\mathrm{sm}}$ and extends, via pullback, to any resolution of $X$. Equivalently, the associated Poisson bivector $\theta := \omega^{-1}$ on $X_{\mathrm{sm}}$ extends as a Poisson structure on $X$. Poisson deformations are flat families of Poisson varieties over an Artinian base that recover the original structure at the special fiber. Flops in this context are birational modifications that preserve the symplectic structure, interpreted via Poisson geometry. The interplay between flops and Poisson deformations provides foundational results for the study of symplectic birational geometry. Recent corrections and addenda (Namikawa, 3 Jan 2026) address essential technical details of equivariance and line bundle linearization, reinforcing the theoretical infrastructure underpinning this program.

1. Symplectic Varieties, Poisson Deformations, and Flops

A symplectic variety $(X, \omega)$ is defined by the existence of a nondegenerate holomorphic $2$-form on its smooth part $X_{\mathrm{sm}}$, extendable to any resolution. The associated Poisson bracket $\{\ ,\ \}$ on $\mathcal{O}_X$ is induced by the inverse $\omega^{-1}$. A Poisson deformation of $(X, \omega)$ over an Artinian algebra $(A, m)$ is a flat family $\mathcal{X} \to \operatorname{Spec} A$ with a compatible Poisson bracket on $\mathcal{O}_{\mathcal{X}}$ that restricts to the bracket on $X$ at the closed point.

The minimal model program for symplectic varieties entails projective birational morphisms $f: X \rightarrow Y$ from symplectic $X$, contracting loci of codimension at least two, with "crepant" singularities. A symplectic flop is a crepant birational transform $Y \leftarrow X^+$ from the same center, realized by wall-crossing in the movable cone. The existence and classification of such flops are characterized by the deformation theory of Poisson structures and period maps as established in [Na 2008].

2. Universal Poisson Deformations and $\mathbf{C}^*$-Equivariance

The universal formal Poisson deformation is a formal deformation space pro-representing the Poisson deformation functor $\mathrm{PD}_X$. Under a good $\mathbf{C}^*$-action of positive weight $l$ on $\omega$, the deformation base $R$ and the universal deformation tower $\{X_n, \{\ ,\ \}_n\}$ can be endowed with $\mathbf{C}^*$-actions. The main result establishes that for a conical symplectic variety $(X, \omega)$, there exist unique $\mathbf{C}^*$-actions on the base $S_n = \operatorname{Spec}(R/m^{n+1})$ and on $X_n$ such that the towers and all canonical maps are $\mathbf{C}^*$-equivariant, and for $\sigma \in \mathbf{C}^*$, the induced automorphism $(\varphi_{\sigma})_n$ of $X_n$ satisfies

$$(\varphi_{\sigma})_n: (X_n, \sigma^{-l} \{\ ,\ \}_n) \cong (X_n, \{\ ,\ \}_n)$$

as Poisson isomorphisms.

The proof leverages formal semi-universality (Rim's method), inductively extending $\mathbf{C}^*$-actions by analyzing obstructions in the truncated Lichnerowicz–Poisson complex, which is identified with the de Rham complex on the regular locus. These obstruction groups vanish due to Hodge theory and linear reductivity of tori (i.e., $\mathbf{C}^*$), and uniqueness is established via Hochschild cohomology applied to equivariant automorphism groups. The result ensures that $R$ inherits a natural grading by weights, and the period map

$$\operatorname{Per}: X_n \to H^2(X) \otimes m_R / m_R^2$$

respects the $\mathbf{C}^*$-action, i.e., $$\operatorname{Per}(\sigma \cdot x) = \sigma^l \operatorname{Per}(x)$$ (Namikawa, 3 Jan 2026).

3. Linearization of Line Bundles on Formal $\mathbf{C}^*$-Schemes

For projective $\mathbf{C}^*$-equivariant birational morphisms $\hat{f} : \hat{X} \to \hat{Y}$ of complete local $\mathbf{C}$-schemes with $$\hat{f}_* \mathcal{O}_{\hat{X}} = \mathcal{O}_{\hat{Y}}$$, the linearization of line bundles is crucial for constructing equivariant ample bundles required in moduli theory and birational geometry. A line bundle $\hat{L}$ on $\hat{X}$ satisfying

$\hat{a}^* \hat{L} \cong \mathrm{pr}_2^* \hat{L}$

(with $\hat{a}$ the action map) admits a genuine $\mathbf{C}^*$-linearization.

The proof proceeds by considering mod $m^{n+1}$ approximations and analyzing the 2-cocycle measuring failure of linearity in

$$\operatorname{Aut}(L_n; id|_{L_{n-1}}) \cong id + m^n / m^{n+1},$$

which is a rational $\mathbf{C}^*$-module. Using the vanishing of $H^2(\mathbf{C}^*, V)$ for rational $\mathbf{C}^*$-modules, the 2-cocycle is killed inductively, and the Grothendieck existence theorem lifts the linearization to the formal limit.

In the deformation-theoretic context, this result fills a gap in the construction of $\mathbf{C}^*$-linearized ample line bundles on crepant partial resolutions, which is essential for applying global GAGA and for period map comparisons (Namikawa, 3 Jan 2026).

4. Technical Lemmas and Proof Methods

Key lemmas and propositions articulated in (Namikawa, 3 Jan 2026) include:

• Extension of partial $\mathbf{C}^*$-actions (Lemma 2.2): If a Poisson morphism $$\varphi_{n-1}: \mathbf{C}^* \times X_{n-1} \to X_{n-1}$$ lifts the bracket equivariantly, then it can be extended to $X_n$.
• Uniqueness (Proposition 2.3): Any two $\mathbf{C}^*$-actions on the same formal Poisson deformation with fixed base action are equivariantly conjugate by a unique Poisson automorphism $h$ with $h_0 = id$.
• Equivariant universality (Corollary 2.4): The universal formal Poisson family is universal among $\mathbf{C}^*$-equivariant Artinian Poisson deformations.

Proof techniques reduce the analysis to the regular locus using Sumihiro's theorem, translate Poisson structures to de Rham complexes via the Lichnerowicz isomorphism, and exploit the vanishing of higher group cohomology for tori (Milne Prop. 15.16) to resolve extension and linearization problems.

A tabulation of the key proof ingredients can be given as follows:

Ingredient	Role in Proof	Mechanism/Reference
Lichnerowicz–Poisson complex	Controls obstructions	Identified with de Rham on smooth locus
Group cohomology of tori	Shows vanishing of obstructions	$H^i(\mathbf{C}^*, V) = 0$ for $i > 0$
Hochschild cohomology/automorphisms	Uniqueness of structures	Controls equivariant automorphism group

5. Applications and Examples

An explicit application is provided by the conifold $$X = \operatorname{Spec} \mathbf{C}[x, y, u, v]/(xy - uv)$$, a three-dimensional singularity carrying a standard $\mathbf{C}^*$-action of weight 1. Its $1$-dimensional Poisson deformation family is graded of weight 2. The corrected equivariant universal Poisson family realizes the familiar Atiyah flop, exhibiting explicit grading structure consistent with the wall-crossing formalism (Namikawa, 3 Jan 2026).

The existence and uniqueness of graded universal formal deformations, together with the ability to linearize line bundles, ensure that period maps and wall-crossing formulas retain the requisite grading properties for moduli and mirror symmetry considerations.

6. Mathematical Significance and Open Directions

The corrections in (Namikawa, 3 Jan 2026) resolve critical technical flaws in the equivariance and linearization analysis of [Na 2008], reinforcing the theoretical framework for the symplectic minimal model program via Poisson deformation. The existence of unique graded universal Poisson deformations and line bundle linearizations is indispensable for explicit calculations of period maps, wall-crossing phenomena, and the construction of projective symplectic flops.

Ongoing and prospective research directions include:

• Generalization to other reductive groups: Extending the equivariant theory from tori to more general reductive groups $G$ necessitates understanding rigidity and obstruction theory, particularly the vanishing of $H^i(G, V)$ for $i>0$.
• Analytic vs. algebraic settings: While the present results pertain to the formal (algebraic) context, the analytic parallel involving Kuranishi spaces and convergent Poisson brackets remains open.
• Behavior of invariants in families: With a graded universal Poisson family, there is an expectation of induced torus actions on derived categories, moduli of perverse sheaves, and Donaldson–Thomas invariants associated to flops. Examining these invariants under variation in families is an active domain.

The foundational role of group cohomology and the Lichnerowicz–Poisson complex is highlighted in controlling equivariant extensions and obstructions. A plausible implication is that similar methods may prove effective for equivariant deformation problems in broader geometric contexts.