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Antony Maciocia

Publications and source records attributed to Antony Maciocia.

At least 19 recordsLinked to original sources
arXiv2025

Higher rank DT/PT wall-crossing in Bridgeland stability

We prove that the Gieseker moduli space of stable sheaves on a smooth projective threefold $X$ of Picard rank 1 is separated from the moduli space of PT stable objects by a single wall in the space of Bridgeland stability conditions on $X$, thus realizing the higher rank DT/PT correspondence as a wall-crossing phenomenon in the space of Bridgeland stability conditions. In addition, we also show that only finitely many walls pass through the upper $(β,α)$-plane parametrizing geometric Bridgeland stability conditions on $X$ which destabilize Gieseker stable sheaves, PT stable objects or their duals when $α>α_0$.

math.AG↗
arXiv2023

Fully Triangulated Categories

We modify the axioms of triangulated categories to include both higher triangles and distinguished maps of higher triangles. The distinguished maps are specializations of Neeman's ``good'' maps of $2$-triangles. The axioms both simplify Neeman's axioms in his 1991 paper and generalize them to higher triangles in the way proposed by Balmer et al. We provide a geometric formulation via directed truncated simplices to enable a more concrete approach. The axioms are modelled by homotopy and derived categories. We look at a number of key theorems including the fact that sums of maps of $2$-triangles are distinguished if and only if the maps are distinguished and a strong version of the $3$x$3$ lemma. These illustrate some key proof methods. We also show that maps of faces of distinguished triangles are distinguished. We construct some useful distinguished $5$-triangles.

math.CT↗
arXiv2022

Walls and asymptotics for Bridgeland stability conditions on 3-folds

We consider Bridgeland stability conditions for three-folds conjectured by Bayer-Macrì-Toda in the case of Picard rank one. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections, and showing that they are essentially regular. Next, we prove that walls within a certain region of the upper half plane that parametrizes geometric stability conditions must always intersect the curve given by the vanishing of the slope function and, for a fixed value of s, have a maximum turning point there. We then use all of these facts to prove that Gieseker semistability is equivalent to asymptotic semistability along a class of paths in the upper half plane, and to show how to find large families of walls. We illustrate how to compute all of the walls and describe the Bridgeland moduli spaces for the Chern character (2,0,-1,0) on complex projective 3-space in a suitable region of the upper half plane.

math.AG↗
arXiv2021

Vertical asymptotics for Bridgeland stability conditions on 3-folds

Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomlov-Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to $X$ in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and $ch^β(E)=(-R,0,D,0)$, we prove that there are only a finite number of nested walls in the $(α,s)$-plane. Moreover, when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gieseker semistable sheaves, and when $β=0$ there are no semistable objects in the innermost chamber. In both cases, the only limit semistable objects of the form $E$ or $E[1]$ (where $E$ is a sheaf) that do not get destabilized until the innermost wall are precisely the (shifts of) instanton sheaves.

math.AG↗
arXiv2019

Fourier-Mukai transforms for K3 and elliptic fibrations

Given a non-singular variety with a K3 fibration f : X --> S we construct dual fibrations Y --> S by replacing each fibre X_s of f by a two-dimensional moduli space of stable sheaves on X_s. In certain cases we prove that the resulting scheme Y is a non-singular variety and construct an equivalence of derived categories of coherent sheaves Φ: D(Y) --> D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we show how the equivalences Φidentify certain moduli spaces of stable bundles on elliptic threefolds with Hilbert schemes of curves.

math.AG↗
arXiv2019

Complex surfaces with equivalent derived categories

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves D(X). To do this we find, for each such surface X, the set of surfaces Y for which there exists a Fourier-Mukai transform D(Y) --> D(X).

math.AG↗
arXiv2017

Rank Two Fourier-Mukai Transforms for K3 Surfaces

We study rank two locally-free Fourier-Mukai transforms on K3 surfaces and show that they come in two distinct types according to whether the determinant of a suitable twist of the kernel is positive or not. We show that a necessary and sufficient condition on the existence of Fourier-Mukai transforms of rank 2 between the derived categories of K3 surfaces X and Y with negative twisted determinant is that Y is isomorphic to X and there must exist a line bundle with no cohomology. We use these results to prove that all reflexive K3 surfaces (including the degenerate ones) admit Fourier-Mukai transforms.

math.AG↗
arXiv2015

Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds II

We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian three-fold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov-Gieseker type inequality. This is done by showing any Fourier-Mukai transform gives an equivalence of abelian categories which are double tilts of coherent sheaves.

math.AG↗
arXiv2015

Pre-triangulated categories are triangulated

We prove a stronger version of the octahedral axiom in a pre-triangulated category. The proof uses a new lemma about exact sequences in pointed additive categories which is based on a weak converse of the snake lemma.

math.CT↗
arXiv2015

Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds

We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their conjectural generalized Bogomolov-Gieseker inequality for certain tilt stable objects. We do this by proving that a suitable Fourier-Mukai transform preserves the heart of a particular conjectural stability condition. We also show that the only reflexive sheaves with zero first and second Chern classes are the flat line bundles.

math.AG↗
arXiv2014

Critical k-Very Ampleness for Abelian Surfaces

Let $(S,L)$ be a polarized abelian surface of Picard rank one and let $ϕ$ be the function which takes each ample line bundle $L'$ to the least integer $k$ such that $L'$ is $k$-very ample but not $(k+1)$-very ample. We use Bridgeland's stability conditions and Fourier-Mukai techniques to give a closed formula for $ϕ(L^n)$ as a function of $n$ showing that it is linear in $n$ for $n>1$. As a byproduct, we calculate the walls in the Bridgeland stability space for certain Chern characters.

math.AG↗
arXiv2013

Computing the Walls Associated to Bridgeland Stability Conditions on Projective Surfaces

We derive constraints on the existence of walls for Bridgeland stability conditions for general projective surfaces. We show that in suitable planes of stability conditions the walls are bounded and derive conditions for when the number of walls is globally finite. In examples, we show how to use the explicit conditions to locate walls and sometimes to show that there are no walls at all.

math.AG↗
arXiv2012

A Note on Commuting Reflection Functors for Calabi-Yau d-folds

We study sets of commuting reflection functors in the derived category of sheaves on Calabi-Yau varieties. We show that such a collection is determined by a set of mutually orthogonal spherical objects. We also show that when the spherical objects are locally-free sheaves then the kernel of the composite transform parametrizes properly torsion-free with zero-dimensional singularity sets and conversely that such a kernel gives rise to a collection of mutually orthogonal spherical vector bundles. We do this using a more detailed analysis of the reason why spherical twists give equivalences.

math.AG↗
arXiv2011

A Fourier-Mukai Approach to the Enumerative Geometry of Principally Polarized Abelian Surfaces

We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface (T,l). Using Fourier-Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of T and show that no smooth genus 5 curve on such a surface can contain a g^1_3. We also describe explicitly the singular divisors in the linear system |2l|.

math.AG↗
arXiv2002

A Fourier-Mukai approach to spectral data for instantons

We study U(r) instantons on elliptic surfaces with a section and show that they are in one-one correspondence with spectral data consisting of a curve in the dual elliptic surface and a line bundle on that curve. We use relative Fourier-Mukai transforms to analyse their properties and, in the case of the K3 and abelian surfaces, we show that the moduli space of instantons has a natural Lagrangian fibration with respect to the canonical complex symplectic structure.

math.AG↗
arXiv1998

Fourier-Mukai transforms for quotient varieties

We study Fourier-Mukai transforms for smooth projective varieties whose canonical bundles have finite order, and relate them to equivariant transforms on certain finite covering spaces. Our results lead to new equivalences of derived categories for Enriques and bielliptic surfaces.

math.AG↗
arXiv1997

Generalized Fourier-Mukai Transforms

The paper sets out a generalized framework for Fourier-Mukai transforms and illustrates their use via vector bundle transforms. A Fourier-Mukai transform is, roughly, an isomorphism of derived categories of (sheaves) on smooth varieties X and Y. We show that these can only exist if the first Chern class of the varieties vanishes and, in the case of vector bundle transforms, will exist if and only if there is a bi-universal bundle on XxY which is "strongly simple" in a suitable sense. Some applications are given to abelian varieties extending the work of Mukai.

alg-geom↗