[Paper Review] Poisson deformations and birational geometry

Yoshinori Namikawa|arXiv (Cornell University)|Feb 27, 2015

Algebraic Geometry and Number Theory19 references24 citations

TL;DR

This paper establishes a deep equivalence between two geometric structures on the second cohomology $ H^2(Y, \mathbb{C}) $ of a crepant resolution $ \pi: Y \to X $ of an affine symplectic variety with a $ \mathbb{C}^* $-action: the chamber decomposition from the ample cones of different resolutions and the chamber decomposition from the locus $ D \subset H^2(Y, \mathbb{C}) $ where Poisson deformations of $ Y $ fail to be affine. The key result is that these two chamber structures coincide, unifying birational geometry and Poisson deformation theory.

ABSTRACT

Let \pi: Y -> X be a crepant projective resolution of an affine symplectic variety X with a good C^*-action. We interpret the second cohomology H^2(Y, C) in two ways. First, H^2(Y, C) is the Picard group of Y tensorised with C. By the ample cones of different crepant resolutions of X, there is a natural chamber structure in H^2(Y, C). The second interpretation of H^2(Y, C) is the base space of the universal Poisson deformation $\mathcal Y$ of Y. Let D \subset H^2(Y, C) be the locus where the corresponding Poisson varieties are not affine. Then D is the union of finite number of hyperplanes, which gives a chamber structure in H^2(Y, C). These two chamber structures coincide.

Motivation & Objective

• To understand the geometric structure of the second cohomology $ H^2(Y, \mathbb{C}) $ of a crepant resolution $ Y \to X $ of an affine symplectic variety $ X $.
• To relate the chamber decomposition arising from the ample cones of different crepant resolutions of $ X $ to a Poisson-geometric structure on $ H^2(Y, \mathbb{C}) $.
• To show that the locus $ D \subset H^2(Y, \mathbb{C}) $, where Poisson deformations of $ Y $ are not affine, defines a chamber structure that matches the birational chamber structure.

Proposed method

• Interpret $ H^2(Y, \mathbb{C}) $ as the complexified Picard group $ \operatorname{Pic}(Y) \otimes \mathbb{C} $, linking it to line bundle positivity and ample cones.
• Use the existence of a good $ \mathbb{C}^* $-action on $ X $ to ensure the resolution $ Y $ has well-behaved geometric and cohomological properties.
• Construct the universal Poisson deformation $ \mathcal{Y} $ of $ Y $, with base space $ H^2(Y, \mathbb{C}) $, using deformation theory of Poisson structures.
• Define the locus $ D \subset H^2(Y, \mathbb{C}) $ as the set of parameters for which the corresponding Poisson variety is not affine.
• Show that $ D $ is a finite union of hyperplanes, thereby defining a chamber structure in $ H^2(Y, \mathbb{C}) $.
• Prove that this Poisson-geometric chamber structure coincides with the birational chamber structure from the ample cones of different crepant resolutions.

Experimental results

Research questions

• RQ1 How do the chamber decompositions arising from the ample cones of different crepant resolutions of $ X $ relate to Poisson deformation theory on $ Y $?
• RQ2 What is the geometric nature of the locus $ D \subset H^2(Y, \mathbb{C}) $ where Poisson deformations of $ Y $ fail to be affine?
• RQ3 Can the chamber structure from birational geometry be recovered from the Poisson deformation base space $ H^2(Y, \mathbb{C}) $?
• RQ4 Is there a canonical identification between the birational chamber structure and the Poisson chamber structure in $ H^2(Y, \mathbb{C}) $?

Key findings

• The second cohomology $ H^2(Y, \mathbb{C}) $ carries a natural chamber decomposition from the ample cones of different crepant resolutions of $ X $, reflecting birational geometry.
• The space $ H^2(Y, \mathbb{C}) $ also parametrizes the universal Poisson deformation $ \mathcal{Y} $ of $ Y $, making it a moduli space for Poisson structures.
• The locus $ D \subset H^2(Y, \mathbb{C}) $, where the Poisson deformations are not affine, is a finite union of hyperplanes, defining a chamber structure.
• This Poisson-geometric chamber structure is identical to the chamber structure arising from the ample cones of different crepant resolutions.
• The coincidence of these two chamber structures establishes a deep link between birational geometry and Poisson deformation theory in the context of symplectic resolutions.
• The result holds under the assumption that $ X $ is an affine symplectic variety with a good $ \mathbb{C}^* $-action, ensuring the necessary geometric control.