arXiv subjects

Alexander Perry

Publications and source records attributed to Alexander Perry.

At least 19 recordsLinked to original sources

arXiv2026

The period-index conjecture is false for motivic reasons

For any integer $d \geq 3$ and any algebraically closed field $k$ of characteristic $0$ and transcendence degree at least $d-3$, or of characteristic $p>2$ and transcendence degree at least $d-2$, we construct a $d$-dimensional variety over $k$ with a Brauer class of period $2$ and index $2^d$, violating the period-index conjecture; in particular, the period-index conjecture fails in dimension $3$ over $\overline{\mathbf{Q}}$ and $\overline{\mathbf{F}_p(t)}$, and in all dimensions $d \geq 3$ over $\mathbf{C}$ and $\overline{\mathbf{F}_p((t))}$. To bound the index of Brauer classes from below, we employ an obstruction of motivic nature, which requires proving the nonexistence of integral Hodge or Tate classes satisfying a certain equation modulo $2$. Motivated by these counterexamples, we propose a period-index conjecture with corrections at small primes and, as evidence, prove its Hodge-theoretic counterpart.

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arXiv2026

Stability conditions and moduli spaces on projective families

We extend the construction of stability conditions on projective schemes over a field to projective families over an arbitrary base, and prove that they admit proper relative moduli spaces of semistable objects. We also prove a number of complementary results: the existence of mass-Hom bounds for these stability conditions, as conjectured by Halpern-Leistner and Robotis; a comparison with tilt-stability on surfaces and threefolds; a construction of stability conditions on the supported derived category of total spaces of certain vector bundles, including all local Calabi--Yau varieties; and a simple new proof of Bondal and Orlov's reconstruction theorem.

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arXiv2026

Inducing t-structures on semiorthogonal components

Given a triangulated category with a t-structure, we introduce a method for inducing t-structures on its semiorthogonal components, based on the construction of an associated perverse t-structure on the ambient category. As applications, we construct bounded t-structures in many new examples, including: almost all known phantom and quasiphantom categories; the semiorthogonal complement of the structure sheaf on a Fano variety; the residual component of an Enriques surface; the categorical resolution of a nodal cubic curve appearing in an early counterexample to the Jordan-H\"{o}lder property for semiorthogonal decompositions; and Brill-Noether modifications of the derived category of a curve.

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arXiv2026

The semiregularity theorem for equivariant noncommutative varieties

We generalize the classical semiregularity theorem of Buchweitz and Flenner to the setting of noncommutative algebraic geometry, with group actions. This applies in particular to twisted derived categories, in which case it answers a question of Markman and streamlines part of his proof of the Hodge conjecture for abelian fourfolds. Along the way, we prove that for many finite group actions on derived categories of varieties, the invariant category is of geometric origin.

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arXiv2025

Stability conditions on products of curves and Hilbert schemes of surfaces

We prove that stability conditions on the derived category of a product of curves of positive genus are uniquely determined by their central charge and the phase of skyscraper sheaves. As an application, we construct stability conditions on Hilbert schemes of points on certain surfaces, including some K3 surfaces of Kummer type.

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arXiv2025

A Looming of phantoms

Following Krah's method, we construct new examples of phantom categories as semiorthogonal components of the derived categories of two types of rational surfaces: the blowup of the plane at 11 points in general position, and the blowup of the second Hirzebruch surface at 9 points in general position. We also pose conjectures about the existence of phantom subcategories in the derived categories of other rational surfaces, obtained as the blowups of the other Hirzebruch surfaces.

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arXiv2024

The period-index conjecture for abelian threefolds and Donaldson-Thomas theory

We prove the period-index conjecture for unramified Brauer classes on abelian threefolds. To do so, we develop a theory of reduced Donaldson-Thomas invariants for 3-dimensional Calabi-Yau categories, with the feature that the noncommutative variational integral Hodge conjecture holds for classes with nonvanishing invariant. The period-index result is then proved by interpreting it as the algebraicity of a Hodge class on the twisted derived category, and specializing within the Hodge locus to an untwisted abelian threefold with nonvanishing invariant. As a consequence, we also deduce the integral Hodge conjecture for generically twisted abelian threefolds.

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arXiv2024

Stability conditions on crepant resolutions of quotients of product varieties

We construct stability conditions on crepant resolutions of certain quotients of product varieties, giving as a special case the first examples of stability conditions on strict Calabi-Yau varieties of arbitrary dimension. Along the way, we prove the crepant resolutions are derived equivalent to the corresponding quotient stacks, verifying an instance of a conjecture of Bondal and Orlov.

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arXiv2024

Derived categories of quartic double fivefolds

We construct singular quartic double fivefolds whose Kuznetsov component admits a crepant categorical resolution of singularities by a twisted Calabi--Yau threefold. We also construct rational specializations of these fivefolds where such a resolution exists without a twist. This confirms an instance of a higher-dimensional version of Kuznetsov's rationality conjecture, and of a noncommutative version of Reid's fantasy on the connectedness of the moduli of Calabi--Yau threefolds.

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arXiv2023

Moduli spaces of stable objects in Enriques categories

We study moduli spaces of stable objects in Enriques categories by exploiting their relation to moduli spaces of stable objects in associated K3 categories. In particular, we settle the nonemptiness problem for moduli spaces of stable objects in the Kuznetsov components of several interesting classes of Fano varieties, and deduce the nonemptiness of fixed loci of certain antisymplectic involutions on modular hyperk\"{a}hler varieties.

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arXiv2022

The period-index problem and Hodge theory

Conditional on the Lefschetz standard conjecture in degree 2, we prove that the index of a Brauer class on a smooth projective variety divides a fixed power of its period, uniformly in smooth families. In the other direction, we reinterpret in more classical terms recent work of Hotchkiss which gives Hodge-theoretic lower bounds on the index of Brauer classes. We also prove versions of our results over arbitrary algebraically closed base fields, and as an application construct qualitatively new counterexamples to the integral Tate conjecture.

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arXiv2022

Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions

We settle the last open case of Kuznetsov's conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel-Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories. Along the way, we obtain a categorical description of the periods of Gushel-Mukai varieties, which we use to resolve a conjecture of Kuznetsov and the second author on the birational categorical Torelli problem, as well as to give a simple proof of a theorem of Debarre and Kuznetsov on the fibers of the period map. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.

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arXiv2021

Serre functors and dimensions of residual categories

We describe in terms of spherical twists the Serre functors of many interesting semiorthogonal components, called residual categories, of the derived categories of projective varieties. In particular, we show the residual categories of Fano complete intersections are fractional Calabi--Yau up to a power of an explicit spherical twist. As applications, we compute the Serre dimensions of residual categories of Fano complete intersections, thereby proving a corrected version of a conjecture of Katzarkov and Kontsevich, and deduce the nonexistence of Serre invariant stability conditions when the degrees of the complete intersection do not all coincide.

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arXiv2020

The integral Hodge conjecture for two-dimensional Calabi-Yau categories

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi-Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves.

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arXiv2019

Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperk\"{a}hler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.

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arXiv2019

Categorical cones and quadratic homological projective duality

We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In particular, our construction provides well-behaved categorical resolutions of singular quadrics, which we use to obtain an explicit quadratic version of the main theorem of homological projective duality. As applications, we prove the duality conjecture for Gushel-Mukai varieties, and produce interesting examples of conifold transitions between noncommutative and honest Calabi-Yau threefolds.

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arXiv2019

Homological projective duality for quadrics

We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it.

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arXiv2019

Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and Piyaratne-Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers. Our main application is the generalization of Mukai's theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington-Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperk\"ahler manifolds of K3 type. Other applications include the deformation-invariance of Donaldson-Thomas invariants counting Bridgeland stable objects on Calabi-Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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