[Paper Review] Relative Orbifold Donaldson-Thomas Theory and the Degeneration Formula

Zijun Zhou|arXiv (Cornell University)|Apr 9, 2015

Algebraic Geometry and Number Theory38 references3 citations

TL;DR

This paper extends relative Donaldson-Thomas theory and the degeneration formula to 3-dimensional smooth projective Deligne-Mumford stacks (orbifolds), generalizing expanded degenerations and pairs to the orbifold setting. It defines relative DT invariants via stable quotients and proves properness of their moduli, leading to a degeneration formula for orbifold DT invariants that matches the structure of the Gromov-Witten/DT correspondence in the orbifold setting.

ABSTRACT

We generalize the notion of expanded degenerations and pairs for a simple degeneration or smooth pair to the case of smooth Deligne-Mumford stacks. We then define stable quotients on the classifying stacks of expanded degenerations and pairs and prove the properness of their moduli's. On 3-dimensional smooth projective DM stacks this leads to a definition of relative Donaldson-Thomas invariants and the associated degeneration formula.

Motivation & Objective

• To generalize the notion of expanded degenerations and pairs from schemes to smooth Deligne-Mumford stacks.
• To define stable quotients on the classifying stacks of expanded degenerations and pairs, and prove properness of their moduli spaces.
• To construct relative Donaldson-Thomas invariants for 3-dimensional smooth projective DM stacks without assuming the Calabi-Yau condition.
• To establish a degeneration formula for orbifold DT invariants in the relative and numerical settings, particularly for multi-regular curve classes.
• To provide a framework for completing the GW/DT correspondence diagram in the orbifold setting, especially for transverse $A_n$-singularities.

Proposed method

• Generalizes expanded degenerations and pairs to the context of smooth Deligne-Mumford stacks by constructing standard families and their stacks of degenerations.
• Introduces admissible sheaves and stable quotients on these stacks, ensuring properness of the moduli space via boundedness, separatedness, and a numerical criterion for admissibility.
• Constructs a perfect obstruction theory on the Hilbert stack of the central fiber to define the relative DT invariants in the cycle version of the degeneration formula.
• Uses Kunneth decomposition of the diagonal in the cohomology of the Hilbert scheme of an orbifold surface to decompose the degeneration formula.
• Applies the theory to the multi-regular case, where curve classes arise from pullbacks of classes on the coarse moduli space, and defines partition functions with $q$-weighted Euler characteristics.
• Employs equivariant or quiver-theoretic bases (e.g., Nakajima quiver varieties) for the cohomology of the Hilbert scheme of orbifold surfaces to compute the structure constants in the degeneration formula.

Experimental results

Research questions

• RQ1 How can the notion of expanded degenerations and pairs be generalized from schemes to smooth Deligne-Mumford stacks?
• RQ2 What conditions ensure the properness of the moduli space of 1-dimensional stable quotients on orbifold expanded degenerations?
• RQ3 How can relative Donaldson-Thomas invariants be defined for 3-dimensional smooth projective DM stacks without requiring the Calabi-Yau condition?
• RQ4 What is the form of the degeneration formula for orbifold DT invariants, and how does it relate to the GW/DT correspondence in the orbifold setting?
• RQ5 How do multi-regular curve classes behave in the degeneration formula, and what role do they play in the numerical version of the formula?

Key findings

• The moduli space of 1-dimensional stable quotients on expanded degenerations of smooth Deligne-Mumford stacks is proper, established via boundedness, separatedness, and a numerical criterion for admissibility.
• A cycle version of the degeneration formula is proven using a perfect obstruction theory on the Hilbert stack of the central fiber, linking invariants of the total space to relative invariants of the components.
• For multi-regular curve classes, the numerical degeneration formula is established, expressing the partition function of the central fiber as a sum over products of relative partition functions on the two components, weighted by structure constants from the cohomology of the Hilbert scheme of the orbifold divisor.
• The structure constants $g^{kl}$ in the degeneration formula arise from the Kunneth decomposition of the diagonal in $A^*( ext{Hilb}^m(D))$, and are non-trivial in the orbifold case.
• When the orbifold surface $D$ is of ADE type (e.g., $[bC^2/G]$), the cohomology of $ ext{Hilb}(D)$ admits a natural basis via Nakajima quiver varieties, enabling explicit computation of the degeneration formula.
• The framework provides a foundation for completing the GW/DT correspondence diagram in the orbifold setting, particularly for $A_n$-singularities, and supports the crepant resolution conjecture for DT invariants.