A note on spherical functors
We provide a new and very short proof of the fact that a spherical functor between certain triangulated categories induces an autoequivalence.
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arXiv subjects
Ciaran Meachan
Publications and source records attributed to Ciaran Meachan.
Frobenius and spherical codomains and neighbourhoods
Given an exact functor between triangulated categories which admits both adjoints and whose cotwist is either zero or an autoequivalence, we show how to associate a unique full triangulated subcategory of the codomain on which the functor becomes either Frobenius or spherical, respectively. We illustrate our construction with examples coming from projective bundles and smooth blowups. This work generalises results about spherical subcategories obtained by Martin Kalck, David Ploog and the first author.
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Hochschild cohomology and deformations of $\mathbb{P}$-functors
Given a split $\mathbb{P}$-functor $F:\mathcal{D}^b(X) \to \mathcal{D}^b(Y)$ between smooth projective varieties, we provide necessary and sufficient conditions, in terms of the Hochschild cohomology of $X$, for it to become spherical on the total space of a deformation of $Y$, and explain how the spherical twist becomes the $\mathbb{P}$-twist on the special fibre. These results generalise the object case, that is when $X$ is a point, which was studied previously by Huybrechts and Thomas, and we show how they apply to the $\mathbb{P}$-functor associated to the Hilbert scheme of points on a K3 surface. In the appendix we review and reorganise some technical results due to Toda, relating to the interaction of Atiyah classes, the HKR-isomorphism, and the characteristic morphism.
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Universal functors on symmetric quotient stacks of Abelian varieties
We consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $\mathbb{P}$-functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists are related to the well-known Horja twist. In dimension one, our universal functors are fully faithful, giving rise to a semiorthogonal decomposition for the symmetric quotient stack of an elliptic curve (which we compare to the one discovered by Polishchuk--Van den Bergh), and they lift to spherical functors on the canonical cover, inducing twists which descend to give new derived autoequivalences here as well.
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Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational
We provide a criterion for when Hilbert schemes of points on K3 surfaces are birational. In particular, this allows us to generate a plethora of examples of non-birational Hilbert schemes which are derived equivalent.
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Birational Geometry of Singular Moduli Spaces of O'Grady Type
Following Bayer and Macrì, we study the birational geometry of singular moduli spaces $M$ of sheaves on a K3 surface $X$ which admit symplectic resolutions. More precisely, we use the Bayer-Macrì map from the space of Bridgeland stability conditions $\mathrm{Stab}(X)$ to the cone of movable divisors on $M$ to relate wall-crossing in $\mathrm{Stab}(X)$ to birational transformations of $M$. We give a complete classification of walls in $\mathrm{Stab}(X)$ and show that every birational model of $M$ obtained by performing a finite sequence of flops from $M$ appears as a moduli space of Bridgeland semistable objects on $X$. An essential ingredient of our proof is an isometry between the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice of $X$ and the Néron-Severi lattice of $M$ which generalises results of Yoshioka, as well as Perego and Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of $M$ is deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifold found by O'Grady.
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Spherical functors on the Kummer surface
We find two natural spherical functors associated to the Kummer surface and analyse how their induced twists fit with Bridgeland's conjecture on the derived autoequivalence group of a complex algebraic K3 surface.
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Derived autoequivalences of generalised Kummer varieties
In this article, we construct new derived autoequivalences of generalised Kummer varieties. Together with Huybrechts-Thomas twists around $\mathbb{P}^n$-objects, these are the only known examples of such symmetries.
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Rank One Bridgeland Stable Moduli Spaces on A Principally Polarized Abelian Surface
We compute moduli spaces of Bridgeland stable objects on an irreducible principally polarized complex abelian surface corresponding to twisted ideal sheaves. We use Fourier-Mukai techniques to extend the ideas of Arcara and Bertram to express wall-crossings as Mukai flops and show that the moduli spaces are projective.
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