[Paper Review] Global shifted potentials for moduli stacks of sheaves on Calabi-Yau four-folds I (derived schemes)

Dennis Borisov, Artan Sheshmani+1 more|arXiv (Cornell University)|Aug 1, 2019

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper establishes that any derived $$\mathbb{C}$$-scheme with a $-2$-shifted symplectic structure and a Hausdorff classical point space admits a globally defined Lagrangian distribution as a dg $C^{∞}$-manifold. The key contribution is a strictification result for Lagrangian distributions, paving the way for constructing such distributions on stable loci of derived $m$Quot$-stacks in Part II.

ABSTRACT

It is shown that any derived scheme over $\mathbb{C}$ equipped with a $-2$-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg $C^{\infty}$-manifold. This is in preparation for Part II, where this result is used to construct Lagrangian distributions on stable loci of derived $ m Quot$-stacks. The main tool for proving the theorem in the current paper is a strictification result for Lagrangian distribution.

Motivation & Objective

• To establish the existence of globally defined Lagrangian distributions on derived schemes with $-2$-shifted symplectic structures.
• To ensure the classical point space is Hausdorff, a topological condition necessary for geometric consistency.
• To develop a strictification result for Lagrangian distributions as a foundational tool for subsequent constructions.
• To prepare the groundwork for Part II, where Lagrangian distributions will be constructed on stable loci of derived $m$Quot$-stacks.

Proposed method

• Utilize the theory of derived algebraic geometry over $\mathbb{C}$ to analyze derived schemes with $-2$-shifted symplectic structures.
• Apply a strictification result to transform formal or local Lagrangian data into a globally defined dg $C^{∞}$-manifold structure.
• Employ the Hausdorff condition on the classical point space to ensure topological regularity and compatibility with global constructions.
• Leverage the interplay between shifted symplectic geometry and dg $C^{∞}$-manifolds to define and control Lagrangian distributions.
• Use the derived enhancement of the classical point space to lift geometric structures globally.

Experimental results

Research questions

• RQ1 Under what conditions does a derived scheme with a $-2$-shifted symplectic structure admit a globally defined Lagrangian distribution?
• RQ2 How can one globally realize Lagrangian distributions on derived schemes with Hausdorff classical point spaces?
• RQ3 What role does the strictification of Lagrangian distributions play in enabling global constructions?
• RQ4 How does the $-2$-shifted symplectic structure interact with the dg $C^{∞}$-manifold structure to support Lagrangian geometry?
• RQ5 What structural properties are required to extend local Lagrangian data to a global dg $C^{∞}$-Lagrangian distribution?

Key findings

• Any derived scheme over $\mathbb{C}$ with a $-2$-shifted symplectic structure and a Hausdorff classical point space admits a globally defined Lagrangian distribution as a dg $C^{∞}$-manifold.
• The strictification result for Lagrangian distributions is instrumental in transforming local or formal data into a global geometric structure.
• The global Lagrangian distribution is compatible with the dg $C^{∞}$-manifold framework, ensuring smoothness and derived enhancement.
• The construction is independent of local models, relying instead on global coherence via the strictification mechanism.
• The result provides a foundational toolset for Part II, where it enables the construction of Lagrangian distributions on stable loci of derived $m$Quot$-stacks.