Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds

Abstract

The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-1)$-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d-critical loci, by introducing notions such as `d-critical flips', `d-critical flops', etc. They are not birational maps of the underlying spaces, but rather should be understood as virtual birational maps. We show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry. Among them, we show that wall-crossing diagrams of Pandharipande-Thomas (PT) stable pair moduli spaces, which are relevant in showing the rationality of PT generating series, form a d-critical minimal model program.

72 pages

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Moduli stacks of semistable sheaves and representations of Ext-quivers
• Derived Algebraic Geometry and Deformation Quantization
• Homotopy L-infinity spaces
• Gopakumar-Vafa invariants and wall-crossing
• Birational geometry and derived categories

Cited by in corpus (7)

• Curve counting via stable objects in derived categories of Calabi-Yau 4-folds
• On the Chow theory of projectivizations
• Semiorthogonal decompositions of stable pair moduli spaces via d-critical flips
• Derived categories of Quot schemes of locally free quotients
• Derived categories of Quot schemes of locally free quotients via categorified Hall products
• Categorical Donaldson-Thomas theory for local surfaces: $\mathbb{Z}/2$-periodic version
• On window theorem for categorical Donaldson-Thomas theories on local surfaces and its applications