16 research outputs found
Hori-mological projective duality
Kuznetsov has conjectured that Pfaffian varieties should admit
non-commutative crepant resolutions which satisfy his Homological Projective
Duality. We prove half the cases of this conjecture, by interpreting and
proving a duality of non-abelian gauged linear sigma models proposed by Hori.Comment: 55 pages. V2: slightly rewritten to take advantage of the
`non-commutative Bertini theorem' recently proved by the authors and Van den
Bergh. V3: lots of changes in exposition following referees' comments.
Section 5 has been mostly cut because it was boring. To appear in Duke Math.
J. V3: added funder acknowledgemen
The homological projective dual of Sym^2 P(V)
We study the derived category of a complete intersection X of bilinear
divisors in the orbifold Sym^2 P(V). Our results are in the spirit of
Kuznetsov's theory of homological projective duality, and we describe a
homological projective duality relation between Sym^2 P(V) and a category of
modules over a sheaf of Clifford algebras on P(Sym^2 V^vee).
The proof follows a recently developed strategy combining variation of GIT
stability and categories of global matrix factorisations. We begin by
translating D^b(X) into a derived category of factorisations on an LG model,
and then apply VGIT to obtain a birational LG model. Finally, we interpret the
derived factorisation category of the new LG model as a Clifford module
category.
In some cases we can compute this Clifford module category as the derived
category of a variety. As a corollary we get a new proof of a result of Hosono
and Takagi, which says that a certain pair of nonbirational Calabi-Yau 3-folds
have equivalent derived categories.Comment: 54 pages, Ph.D. thesi
A non-commutative Bertini theorem
We prove a version of the classical 'generic smoothness' theorem with smooth
varieties replaced by non-commutative resolutions of singular varieties. This
in particular implies a non-commutative version of the Bertini theorem.Comment: 6 pages. v2: added funder acknowledgement. Published in J.
Noncommutative Geometr
Fano varieties with torsion in the third cohomology group
We construct first examples of Fano varieties with torsion in their third
cohomology group. The examples are constructed as double covers of linear
sections of rank loci of symmetric matrices, and can be seen as
higher-dimensional analogues of the Artin--Mumford threefold. As an
application, we answer a question of Voisin on the coniveau and strong coniveau
filtrations of rationally connected varieties.Comment: 18 page
Donaldson-Thomas theory
Donaldson-Thomas invariants are integers assigned to a smooth, projective threefold X. These integers are defined by a virtual count of points on a Hilbert scheme of curves on X, and are invariant under deformations of X.
We give an introduction to this theory, and explain its relation to the similar curve-counting theory of Gromov-Witten invariants. We present different techniques for calculating Donaldson-Thomas invariants and for generalizing the invariants to nonprojective X. In the last chapter we calculate the Donaldson-Thomas invariants of a threefold X with a map to a surface S having fibres isomorphic to a fixed elliptic curve E
Automorphisms of Hilbert schemes of points on surfaces
We show that every automorphism of the Hilbert scheme of points on a weak
Fano or general type surface is natural, i.e. induced by an automorphism of the
surface, unless the surface is a product of curves and . In the
exceptional case there exists a unique non-natural automorphism. More
generally, we prove that any isomorphism between Hilbert schemes of points on
smooth projective surfaces, where one of the surfaces is weak Fano or of
general type and not equal to the product of curves, is natural. We also show
that every automorphism of the Hilbert scheme of points on
is natural.Comment: 20 pages; added reference to related work of Hayash
Universal Polynomials for Tautological Integrals on Hilbert Schemes
We show that tautological integrals on Hilbert schemes of points can be
written in terms of universal polynomials in Chern numbers. The results hold in
all dimensions, though they strengthen known results even for surfaces by
allowing integrals over arbitrary "geometric" subsets (and their
Chern-Schwartz-MacPherson classes).
We apply this to enumerative questions, proving a generalised G\"ottsche
Conjecture for all singularity types and in all dimensions. So if L is a
sufficiently ample line bundle on a smooth variety X, in a general subsystem
P^d of |L| of appropriate dimension the number of hypersurfaces with given
singularity types is a polynomial in the Chern numbers of (X,L). When X is a
surface, we get similar results for the locus of curves with fixed "BPS
spectrum" in the sense of stable pairs theory.Comment: 44 pages, minor changes and correction
A proof of the Donaldson-Thomas crepant resolution conjecture
We prove the crepant resolution conjecture for Donaldson-Thomas invariants of
hard Lefschetz CY3 orbifolds, formulated by Bryan-Cadman-Young, interpreting
the statement as an equality of rational functions. In order to do so, we show
that the generating series of stable pair invariants on any CY3 orbifold is the
expansion of a rational function. As a corollary, we deduce a symmetry of this
function induced by the derived dualising functor. Our methods also yield a
proof of the orbifold DT/PT correspondence for multi-regular curve classes on
hard Lefschetz CY3 orbifolds.Comment: 66 pages, 1 figure. Comments welcome. v2: references adde