Let \(X\) be a smooth complex projective variety and \(D(X)\) be the bounded derived category of coherent sheaves on \(X\). In the case when \(X\) is a Calabi-Yau 3-fold, \(D(X)\) represents the category of \(D\)-branes of type \(B\) and it is conjectured to be equivalent to the derived Fukaya category on a mirror manifold [M. Kontsevich, Proc. ICM ’94, Vol. I. Basel: Birkhäuser, 120–139 (1995; Zbl 0846.53021)]. On the mirror side there are symplectic automomorphisms obtained by taking Dehn twists along Lagrangian spheres [P. Seidel, Bull. Soc. Math. France 128, No. 1, 103–149 (2000; Zbl 0992.53059)]. The notions of spherical objects and the associated spherical twists were introduced by P. Seidel and R. Thomas [Duke Math. J. 108, No. 1, 37–108 (2001; Zbl 1092.14025)], in order to realize Dehn twists under the mirror symmetry.
In this article the author introduces a new class of autoequivalences of derived categories of coherent sheaves on smooth projective varieties, which generalizes the notion of spherical twists. These autoequivalences are associated to a certain class of objects, which are not spherical in general and are interpreted by the author as a “fat” generalization of those. Let \(R\) be a noetherian, artinian local \(\mathbb C\)-algebra. The notion of \(R\)-spherical objects is introduced and the author imitates the construction of spherical twists to present the corresponding autoequivalences. Typically they are obtained by deforming the structure sheaves of \((0, -2)\)-curves on 3-folds, or by deforming \(\mathbb P\)-objects introduced by D. Huybrechts and R. Thomas [Math. Res. Lett. 13, No. 1, 87–98 (2006; Zbl 1094.14012)].