# Half-spherical twists on derived categories of coherent sheaves

## TL;DR

This paper introduces new autoequivalences of derived categories for certain singular fibers of elliptic surfaces and degenerations, linking geometric autoequivalences to mapping class groups via mirror symmetry.

## Contribution

It constructs autoequivalences from spherical objects on fibers, providing novel symmetries for singular varieties and connecting them to half twists and mapping class groups.

## Key findings

- Autoequivalences induced by spherical objects on fibers
- Connection between autoequivalences and half twists on punctured tori
- Description of autoequivalence groups of elliptic surfaces

## Abstract

For a flat morphism $\pi \colon X \to T$ 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.