Torsion pairs and 3-fold flops

TL;DR

The paper classifies t-structures on the local derived category of a 3-fold flopping contraction that are intermediate with respect to perverse hearts. It builds a comprehensive lattice of torsion classes via algebraic, geometric, and semi-geometric hearts, using mutation theory for modifying modules and Bridgeland–Chen wall-crossings to connect birational models. The main achievement is a complete description of the heart fan as an affine-root-system–type intersection arrangement Arr(Δ, 𝔍), together with a precise partial order on algebraic intermediates and a brick-based description of bricks in perverse hearts, thereby enabling a full, numerical account of all intermediate t-structures in D^0X. The framework also extends to minimal (and partial) Kleinian resolutions and yields a robust parallel between algebraic mutation data and birational geometry, with immediate consequences for spherical objects and semibricks, and a clear path toward a full understanding of t-structures in derived categories of surfaces and 3-folds.

Abstract

This paper classifies t-structures on the local derived category of a 3-fold flopping contraction, that are intermediate with respect to the heart of perverse coherent sheaves. Equivalently, this describes the complete lattice of torsion classes for the associated modification algebra. The intermediate hearts are (1) categories of coherent sheaves on birational models and tilts thereof in skyscrapers, (2) algebraic t-structures described in the homological minimal model programme, or (3) combinations of the above over appropriate open covers. An analogous classification is also proved for minimal (and partial) resolutions of Kleinian singularities, thus providing a description of all torsion pairs in the module categories of (contracted) affine preprojective algebras. The results have immediate applications to the classification of spherical modules and (semi)bricks, and are first steps towards describing all t-structures and spherical objects in derived categories of surfaces and 3-folds.

Figures (6)

• Figure 1: The heart fan for a minimal resolution of the $\text{cA}_1$ singularity, where the algebraic hearts are enumerated via mutation as above and we write $\Phi_i$ for the functor $\Psi_i^{-1}$ ($i=0,1$). Hyperplanes are induced by the $\widetilde{\text{A}}_1$ root system, where the simple real roots $\alpha_0,\alpha_1$ are identified with the $\mathop{\mathrm{\mathbf{K}}}\nolimits$-theory classes of $\omega_{C}[1]$ and $\mathscr{O}_C(-1)$ respectively.
• Figure 2: The $3$-dimensional heart fan for a $\text{cA}_2$ resolution, sliced along affine hyperplanes (left) and the unit sphere (right). The mutation functors are abbreviated, e.g. by writing $\Psi_{01}$ for $\Psi_1\circ \Psi_0$. Cones in the hyperplane $\{\delta=0\}$ have been labelled by the $3$-fold whose geometric hearts they are associated to, where e.g. $\nu_{212}X$ is the flop of the second curve in $\nu_{12}X$.
• Figure 3: Continuing from \ref{['fig:AffA2']}, the actions of $\mathop{\mathrm{\textnormal{Pic}}}\nolimits X$ and $\mathop{\mathrm{\textnormal{Pic}}}\nolimits(\nu_1 X)$ on $H$ are shown. Here $\left[\!ij\!\right]$ denotes the line bundle on $X$ which has degrees $i$ and $j$ on the two exceptional curves respectively, and we use double brackets ($\llbracket \!ij\! \rrbracket$) for bundles on the flop $\nu_1 X$. Note that since $\left[\!01\!\right]$ and its proper transform $\llbracket \!01\! \rrbracket$ are both trivial on the flopped curve, their actions coincide i.e. $\left\llbracket\!01\!\right\rrbracket \check{}\,\!\!\otimes\!H = \left[\!01\!\right] \check{}\,\!\!\otimes\!H$. The shaded region represents $\sigma$-$\mathop{\mathrm{\textnormal{tilt}}}\nolimits^+(H)$ for $\sigma$ shown in \ref{['fig:AffA2']}.
• Figure 4: The mutation class of $J=\{2,3,5,6,7\}$ inside the $\widetilde{\text{E}}_7$ Dynkin graph $\raisebox{5pt}{\dynkin[label,edge length=6pt,ordering=Dynkin, root radius=1.5pt] E[1]7}$, where subgraphs are indicated by marking off their vertices with a cross (${\dynkin[root radius=2pt] A{X}}$). Arrows indicate simple mutation, the symbol $\nu$ has been omitted for brevity.
• Figure 5: Representative affine slices of the $\mathcal{E}_{7,4}$ arrangement, associated to the Dynkin data $\dynkin[extended, edge length = 5pt, edge/.style={draw=none}, root radius=1.5pt, affine mark ={o}, ordering=Dynkin] E{*XX*XXX}$ ($J=\{2,3,5,6,7\}$) as in \ref{['figure:E74mutation']}. The principal chambers $\mathrm{C}_J^+,\mathrm{C}_J^0,\mathrm{C}_J^-$ are indicated by the symbols +,0,-- respectively.

Theorems & Definitions (176)

• Theorem A: (= \ref{['thmC']}, \ref{['prop:partialperverseinterval']})
• Theorem B: (= \ref{['thm:brickclassification']})
• Theorem C: (= \ref{['cor:heartfanofzeroper']})
• Theorem D: (= \ref{['thm:nefintermediacy']}, \ref{['cor:partialorderoftwists']}, \ref{['thm:nefcomparisons']})
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4: demonetLatticeTheoryTorsion2023
• Lemma 2.5
• proof