Noncommutative deformations and flops

Abstract

We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence associate to every such curve a noncommutative deformation algebra. This new invariant extends and unifies known invariants for flopping curves in 3-folds, such as the width of Reid, and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra is finite dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us to make explicit calculations of these deformations for certain (-3,1)-curves. We then show how our new invariant also controls the homological algebra of flops. For any flopping curve in a projective 3-fold with only Gorenstein terminal singularities, we construct an autoequivalence of the derived category of the 3-fold by twisting around a universal family over the noncommutative deformation algebra, and prove that this autoequivalence is an inverse of Bridgeland's flop-flop functor. This demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 3-fold, and thus the Bridgeland stability manifold.

Minor changes. Final version, to appear in Duke Math Journal. 50 pages

References in corpus (4)

• Calabi-Yau algebras
• Frobenius categories, Gorenstein algebras and rational surface singularities
• Flops and Clusters in the Homological Minimal Model Program
• Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras

Cited by in corpus (37)

• On multi-pointed non-commutative deformations and Calabi-Yau threefolds
• Discreteness of silting objects and t-structures in triangulated categories
• Derived autoequivalences of generalised Kummer varieties
• Genus zero Gopakumar-Vafa invariants from open strings
• Flops and Clusters in the Homological Minimal Model Program
• Mutation of frozen Jacobian algebras
• On the Finiteness of the Derived Equivalence Classes of some Stable Endomorphism Rings
• Gopakumar-Vafa invariants do not determine flops
• Mukai flops and P-twists
• Faithful Actions from Hyperplane Arrangements
• Singularity categories via the derived quotient
• Potential algebras with few generators
• ${\mathbb{P}}^n$-functors
• Cluster categories and rational curves
• The derived deformation theory of a point
• Contraction algebra and singularity of three-dimensional flopping contraction
• The Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence
• Non-commutative width and Gopakumar-Vafa invariants
• GV and GW invariants via the enhanced movable cone
• Leavitt path algebras, $B_{\infty }$-algebras and Keller's conjecture for singular Hochschild cohomology
• Non-commutative thickening of moduli spaces of stable sheaves
• Universal functors on symmetric quotient stacks of Abelian varieties
• Classification of contraction algebras and pre-Lie algebras associated to braces and trusses
• Donaldson-Thomas invariants of length 2 flops
• The category of a partitioned fan
• On non-commutative formal deformations of coherent sheaves on an algebraic variety
• Local Normal Forms of Noncommutative Functions
• Spherical functors and the flop-flop autoequivalence
• Categorifying non-commutative deformations
• Perverse schobers on Riemann surfaces: constructions and examples
• Golod-Shafarevich type theorems and potential algebras
• Non-commutative virtual structure sheaves
• Adjoints to a Fourier-Mukai transform
• Contraction algebra and invariants of singularities
• Persistence of unknottedness of clean Lagrangian intersections
• Flops and Hilbert schemes of space curve singularities
• Local forms for the double $A_n$ quiver