Half-spherical twists on derived categories of coherent sheaves

TL;DR

The paper develops half-spherical twists as a relative refinement of spherical twists in a flat family $\pi:X\to T$, showing that a half-spherical object $\mathcal{E}\in D^b(X_0)$ yields an autoequivalence on $D^b(X_0)$ via restriction of the relative spherical twist $T_{i_*\mathcal{E}}$. Central to the construction is representing $T_{i_*\mathcal{E}}$ as a relative Fourier–Mukai transform and then restricting to the fiber to obtain the autoequivalence $H_{\mathcal{E}}$ on $D^b(X_0)$. The work specializes to elliptic surfaces, where reducible Kodaira fibers give rise to half-spherical twists that, under HMS, correspond to half twists on the associated $n$-punctured torus $T_n$; this provides a concrete bridge between derived symmetries and mapping class groups. The authors then describe the global autoequivalence structure of certain elliptic surfaces by relating the subgroup generated by twists from $(-2)$-curves to products of mapping class groups of punctured tori, yielding explicit exact sequences and constraints on possible autoequivalences. Overall, the methods fuse relative FM transforms, HMS, and mapping class group actions to illuminate derived symmetries in singular fibers and enrich the understanding of autoequivalence groups in elliptic geometries.

Abstract

For a flat morphism $π\colon X \to T$ between smooth quasi-projective varieties and its fiber $X_0$, we prove that spherical objects on $D^b(X)$ pushed-forward from $D^b(X_0)$ induce autoequivalences of $D^b(X_0)$ itself. Our construction provides new derived symmetries for some singular varieties, which include singular fibers of elliptic surfaces (commonly referred to as Kodaira fibers) and type $III$ degenerations of K3 surfaces. In the case of Kodaira fibers of type $I_n$, we also show the induced autoequivalences of $D^b(X_0)$ correspond to the half twists on the $n$-punctured $2$-torus via homological mirror symmetry. As an application, we describe the autoequivalence groups of elliptic surfaces in terms of mapping class groups of punctured tori.

Figures (8)

• Figure 1: Curves along which the half twists generate $\mathop{\mathrm{\mathrm{Im}}}\nolimits(B \xrightarrow{p_j \circ \psi} \mathop{\mathrm{\mathrm{MCG}}}\nolimits(T_{n_j}))$. The big rectangle illustrates the torus $T_{n_j}$. The top and bottom edges (resp. the left and right edges) are identified, and the white circles represent the punctures.
• Figure 2: The Dehn twist $t_\gamma$ along a simple loop $\gamma$.
• Figure 3: The half twist $h_\gamma$ along a simple arc $\gamma$.
• Figure 4: Kodaira fibers of type $\@slowromancap i@_1$, $\@slowromancap i@_2$, $\@slowromancap i@_3$, and $\@slowromancap i@_4$, respectively.
• Figure 5: The curves $\gamma_{{\mathcal{O}}_{F_n}}, \gamma_{{\mathcal{O}}_{x_i}}$ and $\gamma_{{\mathcal{O}}_{G_i}(-1)}$ on the $n$-punctured torus $T_n$.

Theorems & Definitions (112)

• Proposition 1.1: Proposition \ref{['prop:restriction-to-fiber']}
• Theorem 1.2: Proposition \ref{['prop:twist-functor-is-relative-fm']}, Definition \ref{['def:half-spherical-twist']}, and Corollary \ref{['cor:compatibility-of-half-spherical-twists-and-spherical-twists']}
• Theorem 1.3: Theorem \ref{['thm:half-spherical-twist-and-half-twist']}
• Theorem 1.4: MR3568337
• Theorem 1.5: Theorem \ref{['thm:description-of-B']}
• Theorem 2.1: MR4704076 or stacks-project
• Remark 2.2
• Proposition 2.3: Kunneth formula
• proof
• Theorem 2.4: Grothendieck duality, stacks-project