[Paper Review] Orientation data for coherent sheaves on the local projective plane

Yun Shi|arXiv (Cornell University)|Sep 6, 2018

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper establishes a canonical orientation data for the moduli stack of coherent sheaves on the local projective plane $Y = \mathrm{Tot}(\mathcal{O}_{\mathbb{P}^2}(-3))$ by proving compatibility of the canonical orientation data on quiver hearts $\mathrm{Mod}\text{-}A^n$ under the autoequivalence $\otimes \pi^*\mathcal{O}(1)$, and shows it agrees with the geometric orientation data. The result enables the definition of global motivic Donaldson-Thomas invariants on $Y$.

ABSTRACT

In this note, we show that the canonical orientation data on the quiver hearts are compatible under the autoequivalence $\_\otimesπ^*O(1)$, and hence glue to give an orientation data for the stack of coherent sheaves on local P2.

Motivation & Objective

• To establish a global orientation data for the moduli stack of coherent sheaves on the local projective plane $Y = \mathrm{Tot}(\mathcal{O}_{\mathbb{P}^2}(-3))$.
• To show that the canonical orientation data on quiver hearts $\mathrm{Mod}\text{-}A^n$ is compatible under the autoequivalence $\otimes \pi^*\mathcal{O}(1)$, enabling gluing.
• To prove that the glued orientation data from quiver hearts agrees with the geometric orientation data arising from the derived category of $Y$.
• To provide a foundational step for defining motivic Donaldson-Thomas invariants on $Y$ via a consistent orientation data.

Proposed method

• Utilizes the derived equivalence $D^b(Y) \simeq D^b(\mathrm{Mod}\text{-}A)$, where $A$ is the Jacobi algebra of a quiver with potential.
• Applies the canonical orientation data on quiver hearts $\mathrm{Mod}\text{-}A^n$ as defined in Davison [Dav].
• Proves compatibility of these orientation data under the autoequivalence $\otimes \pi^*\mathcal{O}(1)$ via canonical isomorphisms of determinant line bundles.
• Constructs a global line bundle $L_a$ on $\bigcup \mathrm{Mod}\text{-}A^n$ by gluing the $L_{A^n}$'s using the compatibility condition.
• Compares the glued quiver-based orientation data $L_a$ with the geometric orientation data $L_g$ via explicit resolution of $A^0_{\mathbb{P}^2}$ and determinant computations.
• Uses the canonical isomorphism between $\det(R\mathcal{H}om(F,F))$ and $L_a|_{\mathrm{Coh}_Y}$ to show $L_g \simeq L_a|_{\mathrm{Coh}_Y}$.

Experimental results

Research questions

• RQ1 Is the canonical orientation data on quiver hearts $\mathrm{Mod}\text{-}A^n$ compatible under the autoequivalence $\otimes \pi^*\mathcal{O}(1)$?
• RQ2 Can the compatible orientation data on $\mathrm{Mod}\text{-}A^n$ be glued to define a global orientation data on the stack of coherent sheaves on $Y$?
• RQ3 Does the orientation data constructed from quiver hearts agree with the geometric orientation data on $Y$?
• RQ4 What is the relationship between the motivic DT invariants of $Y$ and the quiver-side construction via $A$-modules?

Key findings

• The canonical orientation data on the quiver hearts $\mathrm{Mod}\text{-}A^n$ is compatible under the autoequivalence $\otimes \pi^*\mathcal{O}(1)$, as shown in Theorem 5.
• The compatible orientation data on $\mathrm{Mod}\text{-}A^n$ glues to define a global orientation data $L_a$ on the stack $\mathrm{Coh}_Y$ of coherent sheaves on $Y$.
• The geometric orientation data $L_g$ on $\mathrm{Coh}_Y$ is isomorphic to the restriction of the glued quiver-based orientation data: $L_g \simeq L_a|_{\mathrm{Coh}_Y}$, as proven in Theorem 6.
• The isomorphism $L_g \simeq L_a|_{\mathrm{Coh}_Y}$ is canonical and compatible across all $n$, ensuring consistency across the entire moduli stack.
• The construction provides a canonical, globally defined orientation data on $\mathrm{Coh}_Y$, enabling the definition of motivic Donaldson-Thomas invariants on $Y$.
• The result confirms that the quiver-side orientation data and the geometric orientation data coincide, validating the consistency of the motivic DT framework on $Y$.