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Nicolas Addington

Publications and source records attributed to Nicolas Addington.

18 recordsLinked to original sources
arXiv2024

On an example of Aspinwall, Morrison, and Szendr\H{o}i

We study the cohomology of a 1-parameter family Y_t of Calabi-Yau 3-folds introduced by Aspinwall and Morrison, related to the mirror quintic family. Szendr\H{o}i proved that Y_t, Y_{xi t}, ..., Y_{xi^4 t}, where xi is a fifth root of unity, have the same rational Hodge structure but are not isomorphic, and conjectured that they are not birational or even derived equivalent. We confirm this by proving that their integral Hodge structures are different, and discuss how this fits with known Torelli-type theorems and counterexamples.

math.AG↗
arXiv2023

The Lichtenbaum-Quillen dimension of complex varieties

The Lichtenbaum-Quillen conjecture for smooth complex varieties states that algebraic and topological K-theory with finite coefficients become isomorphic in high degrees. We define the "Lichtenbaum-Quillen dimension" of a variety in terms of the point where this happens, show that it is surprisingly computable, and analyze many examples. It gives an obstruction to rationality, but one that turns out to be weaker than unramified cohomology and some related birational invariants defined by Colliot-Th\'el\`ene and Voisin using Bloch-Ogus theory. Because it is compatible with semi-orthogonal decompositions, however, it allows us to prove some new cases of the integral Hodge conjecture using homological projective duality, and to compute the higher algebraic K-theory of the Kuznetsov components of the derived categories of some Fano varieties.

math.AG↗
arXiv2021

Hodge numbers are not derived invariants in positive characteristic

We study a pair of Calabi-Yau threefolds X and M, fibered in non-principally polarized Abelian surfaces and their duals, and an equivalence D^b(X) = D^b(M), building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while pi_1(M) = (Z/3)^2. In characteristic 3, we find that X and M have different Hodge numbers, which would be impossible in characteristic 0. In an appendix, we give a streamlined proof of Abuaf's result that the ring H^*(O) is a derived invariant of complex threefolds and fourfolds. A second appendix by Alexander Petrov gives a family of higher-dimensional examples to show that h^{0,3} is not a derived invariant in any positive characteristic.

math.AG↗
arXiv2020

A categorical sl_2 action on some moduli spaces of sheaves

We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman, Yoshioka, and Nakajima. We show that these sequences can be given the structure of a geometric categorical sl_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops.

math.AG↗
arXiv2020

On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold

For the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane, we show that a natural Abel-Jacobi map from H^4_prim(Y) to H^2_prim(Z) is a Hodge isometry. We describe the full H^2(Z) in terms of the Mukai lattice of the K3 category A of Y. We give numerical conditions for Z to be birational to a moduli space of sheaves on a K3 surface or to Hilb^4(K3). We propose a conjecture on how to use Z to produce equivalences from A to the derived category of a K3 surface.

math.AG↗
arXiv2019

Rational points and derived equivalence

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a pair of hyperkaehler 4-folds over Q. The latter is independently interesting as a new example of a transcendental Brauer-Manin obstruction to the Hasse principle.

math.AG↗
arXiv2018

Twisted Fourier-Mukai partners of Enriques surfaces

Bridgeland and Maciocia showed that a complex Enriques surface X has no Fourier-Mukai partners apart from itself: that is, if D^b(X) = D^b(Y) then X = Y. We extend this to twisted Fourier-Mukai partners: if alpha is the non-trivial element of Br(X) = Z/2 and D^b(X,alpha) = D^b(Y,beta), then X = Y and beta is non-trivial. Our main tools are twisted topological K-theory and twisted Mukai lattices.

math.AG↗
arXiv2017

Some non-special cubic fourfolds

In [1309.1899], Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics "apolar to a Veronese surface" is not a Noether-Lefschetz divisor. We give an independent proof of this by exhibiting an explicit cubic fourfold X in the divisor and using point counting methods over finite fields to show X is Noether-Lefschetz general. We also show that two other divisors considered in [ibid.] are not Noether-Lefschetz divisors.

math.AG↗
arXiv2016

Cubic fourfolds fibered in sextic del Pezzo surfaces

We exhibit new examples of rational cubic fourfolds, parametrized by a countably infinite union of codimension-two subvarieties in the moduli space. Our examples are fibered in sextic del Pezzo surfaces over the projective plane; they are rational whenever the fibration has a rational section.

math.AG↗
arXiv2014

On two rationality conjectures for cubic fourfolds

Motivated by the question of rationality of cubic fourfolds, we show that a cubic X has an associated K3 surface in the sense of Hassett if and only if the variety F of lines on X is birational to a moduli space of sheaves on a K3 surface, but that having F birational to Hilb^2(K3) is more restrictive. We compare the loci in the moduli space of cubics where each condition is satisfied.

math.AG↗
arXiv2014

The Pfaffian-Grassmannian equivalence revisited

We give a new proof of the 'Pfaffian-Grassmannian' derived equivalence between certain pairs of non-birational Calabi-Yau threefolds. Our proof follows the physical constructions of Hori and Tong, and we factor the equivalence into three steps by passing through some intermediate categories of (global) matrix factorizations. The first step is global Knoerrer periodicity, the second comes from a birational map between Landau-Ginzburg B-models, and for the third we develop some new techniques.

math.AG↗
arXiv2013

The Brauer group is not a derived invariant

In this short note we observe that the recent examples of derived-equivalent Calabi-Yau 3-folds with different fundamental groups also have different Brauer groups, using a little topological K-theory.

math.AG↗
arXiv2013

Categories of Massless D-Branes and del Pezzo Surfaces

In analogy with the physical concept of a massless D-brane, we define a notion of "Q-masslessness" for objects in the derived category. This is defined in terms of monodromy around singularities in the stringy Kahler moduli space and is relatively easy to study using spherical functors. We consider several examples in which del Pezzo surfaces and other rational surfaces in Calabi-Yau threefolds are contracted. For precisely the del Pezzo surfaces that can be written as hypersurfaces in weighted P3, the category of Q-massless objects is a "fractional Calabi-Yau" category of graded matrix factorizations.

hep-th↗
arXiv2012

D-brane probes, branched double covers, and noncommutative resolutions

This paper describes D-brane probes of theories arising in abelian gauged linear sigma models (GLSMs) describing branched double covers and noncommutative resolutions thereof, via nonperturbative effects rather than as the critical locus of a superpotential. As these theories can be described as IR limits of Landau-Ginzburg models, technically this paper is an exercise in utilizing (sheafy) matrix factorizations. For Landau-Ginzburg models which are believed to flow in the IR to smooth branched double covers, our D-brane probes recover the structure of the branched double cover (and flat nontrivial B fields), verifying previous results. In addition to smooth branched double covers, the same class of Landau-Ginzburg models is also believed to sometimes flow to `noncommutative resolutions' of singular spaces. These noncommutative resolutions are abstract conformal field theories without a global geometric description, but D-brane probes perceive them as non-Kahler small resolutions of a singular Calabi-Yau. We conjecture that such non-Kahler small resolutions are typical in D-brane probes of such theories.

hep-th↗
arXiv2011

New derived symmetries of some hyperk\"ahler varieties

We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb^2 and the variety of lines, we use the theory of spherical functors; to deal with Hilb^n for n > 2 we develop a theory of P-functors. We conjecture that the same construction yields an autoequivalence for any moduli space of sheaves on a K3 surface. In an appendix we give a cohomology and base change criterion which is well-known to experts, but not well-documented.

math.AG↗
arXiv2009

The Derived Category of the Intersection of Four Quadrics

The derived category of a general complete intersection of four quadrics in P^{2n-1} has a semi-orthogonal decomposition < O(-2n+9), ..., O(-1), O, D >, where D is the derived category of twisted sheaves on a certain non-algebraic complex 3-fold coming from a moduli problem. In particular, when n=4 we obtain a (twisted) derived equivalence of Calabi-Yau 3-folds predicted by Gross. This differs from Kuznetsov's result in that our construction is geometric and avoids non-commutative varieties.

math.AG↗
arXiv2009

Spinor sheaves on singular quadrics

We define reflexive sheaves on a singular quadric Q that generalize the spinor bundles on smooth quadrics, using matrix factorizations of the equation of Q. We study the first properties of these spinor sheaves, give a Horrocks-type criterion, and show that they are semi-stable, and indeed stable in some cases.

math.AG↗