Searcharxiv⌕ Search

arXiv subjects

Brent Doran

Publications and source records attributed to Brent Doran.

At least 19 recordsLinked to original sources
arXiv2020

Projective completions of graded unipotent quotients

The aim of this paper is to show that classical geometric invariant theory (GIT) has an effective analogue for linear actions of a non-reductive algebraic group $H$ with graded unipotent radical on a projective scheme $X$. Here the linear action of $H$ is required to extend to a semi-direct product $\hat{H} = H \rtimes \mathbb{G}_m$ with a multiplicative one-parameter group which acts on the Lie algebra of the unipotent radical $U$ of $H$ with all weights strictly positive, and which centralises a Levi subgroup $R \cong H/U$ of $H$. We show that $X$ has an $H$-invariant open subscheme (the 'hat-stable locus') which has a geometric quotient by the $H$-action. This geometric quotient has a projective completion which is a categorical quotient (indeed, a good quotient) by $\hat{H}$ of an open subscheme of a blow-up of the product of $X$ with the affine line; with additional blow-ups a projective completion which is itself a geometric quotient can be obtained. Furthermore the hat-stable locus of $X$ and the corresponding open subsets of the blow-ups of the product of $X$ with the affine line can be described effectively using Hilbert-Mumford-like criteria combined with the explicit blow-up constructions. Applications include the construction of moduli spaces of sheaves and Higgs bundles of fixed Harder--Narasimhan type over a fixed nonsingular projective scheme, and of moduli spaces of unstable projective curves of fixed singularity. More recently, cohomology theory for reductive GIT quotients were extended by the first and fourth author to the non-reductive situation studied in this paper, and this was used to prove the Green--Griffiths--Lang and Kobayashi hyperbolicity conjectures for generic projective hypersurfaces with polynomial bounds on their degree.

math.AG↗
arXiv2020

Geometric invariant theory for graded unipotent groups and applications

Let $U$ be a graded unipotent group over the complex numbers, in the sense that it has an extension $\hat{U}$ by the multiplicative group such that the action of the multiplicative group by conjugation on the Lie algebra of $U$ has all its weights strictly positive. Given any action of $U$ on a projective variety $X$ extending to an action of $\hat{U}$ which is linear with respect to an ample line bundle on $X$, then provided that one is willing to replace the line bundle with a tensor power and to twist the linearisation of the action of $\hat{U}$ by a suitable (rational) character, and provided an additional condition is satisfied which is the analogue of the condition in classical GIT that there should be no strictly semistable points for the action, we show that the $\hat{U}$-invariants form a finitely generated graded algebra; moreover the natural morphism from the semistable subset of $X$ to the enveloping quotient is surjective and expresses the enveloping quotient as a geometric quotient of the semistable subset. Applying this result with $X$ replaced by its product with the projective line gives us a projective variety which is a geometric quotient by $\hat{U}$ of an invariant open subset of the product of $X$ with the affine line and contains as an open subset a geometric quotient of a U-invariant open subset of $X$ by the action of $U$. Furthermore these open subsets of $X$ and its product with the affine line can be described using criteria similar to the Hilbert-Mumford criteria in classical GIT.

math.AG↗
arXiv2019

Extraction of higher-order nonlinear electronic response to strong field excitation in solids using high harmonic generation

State-of-the-art experiments employ strong ultrafast optical fields to study the nonlinear response of electrons in solids on an attosecond time-scale. Notably, a recent experiment retrieved a 3rd order nonlinear susceptibility by comparing the nonlinear response induced by a strong laser field to a linear response induced by the otherwise identical weak field. In parallel, experiments have demonstrated high harmonic generation (HHG) in solids, a highly nonlinear process that until recently had only been observed in gases. The highly nonlinear nature of HHG has the potential to extract even higher order nonlinear susceptibility terms, and thereby characterize the entire response of the electronic system to strong field excitation. However, up till now, such characterization has been elusive due to a lack of direct correspondence between high harmonics and nonlinear susceptibilities. Here, we demonstrate a regime where such correspondence can be clearly made, extracting nonlinear susceptibilities (7th, 9th, and 11th) from sapphire of the same order as the measured high harmonics. The extracted high order susceptibilities show angular-resolved periodicities arising from variation in the band structure with crystal orientation. Nonlinear susceptibilities are key to ultrafast lightwave driven optoelectronics, allowing petahertz scaling manipulation of the signal. Our results open a door to multi-channel signal processing, controlled by laser polarization.

physics.optics↗
arXiv2017

Graded linearisations

When the action of a reductive group on a projective variety has a suitable linearisation, Mumford's geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how Mumford's GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action induces a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumford's GIT are moduli spaces of stable curves and of (semi)stable bundles over a fixed nonsingular curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves or unstable bundles (with suitable fixed discrete invariants in each case, related to their singularities or Harder--Narasimhan type).

math.AG↗
arXiv2016

Constructing quotients of algebraic varieties by linear algebraic group actions

In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumford's geometric invariant theory (GIT). The article surveys some recent work on geometric invariant theory and quotients of varieties by linear algebraic group actions, as well as background material on linear algebraic groups, Mumford's GIT and some of the challenges that the non-reductive setting presents. The earlier work of two of the authors in the setting of unipotent group actions is extended to deal with actions of any linear algebraic group. Given the data of a linearisation for an action of a linear algebraic group H on an irreducible variety $X$, an open subset of stable points $X^s$ is defined which admits a geometric quotient variety $X^s/H$. We construct projective completions of the quotient $X^s/H$ by considering a suitable extension of the group action to an action of a reductive group on a reductive envelope and using Mumford's GIT. In good cases one can also compute the stable locus $X^s$ in terms of stability (in the sense of Mumford for reductive groups) for the reductive envelope.

math.AG↗
arXiv2016

A simplicial approach to effective divisors in $\overline{M}_{0,n}$

We study the Cox ring and monoid of effective divisor classes of $\overline{M}_{0,n} = Bl\mathbb{P}^{n-3}$, over a ring R. We provide a bijection between elements of the Cox ring, not divisible by any exceptional divisor section, and pure-dimensional singular simplicial complexes on {1,...,n-1} with nonzero weights in R satisfying a zero-tension condition. This leads to a combinatorial criterion, satisfied by many triangulations of closed manifolds, for a divisor class to be among the minimal generators for the effective monoid. For classes obtained as the strict transform of quadrics, we present a complete classification of minimal generators, generalizing to all n the well-known Keel-Vermeire classes for n=6. We use this classification to construct new divisors with interesting properties for all n > 6.

math.AG↗
arXiv2015

Algebraic symplectic analogues of additive quotients

Motivated by the study of hyperkahler structures in moduli problems and hyperkahler implosion, we initiate the study of non-reductive hyperkahler and algebraic symplectic quotients with an eye towards those naturally tied to projective geometry, like cotangent bundles of blow-ups of linear arrangements of projective space. In the absence of a Kempf-Ness theorem for non-reductive quotients, we focus on constructing algebraic symplectic analogues of additive quotients of affine spaces, and obtain hyperkahler structures on large open subsets of these analogues by comparison with reductive analogues. We show that the additive analogue naturally arises as the central fibre in a one-parameter family of isotrivial but non-symplectomorphic varieties coming from the variation of the level set of the moment map. Interesting phenomena only possible in the non-reductive theory, like non-finite generation of rings, already arise in easy examples, but do not substantially complicate the geometry.

math.AG↗
arXiv2015

Smooth models of motivic spheres

We study the representability of motivic spheres by smooth varieties. We show that certain explicit "split" quadric hypersurfaces have the $\mathbb A^1$-homotopy type of motivic spheres over the integers and that the $\mathbb A^1$-homotopy types of other motivic spheres do not contain smooth schemes as representatives. We then study some applications of these representability/non-representability results to the construction of new exotic $\mathbb A^1$-contractible smooth schemes. Then, we study vector bundles on even dimensional "split" quadric hypersurfaces by developing an algebro-geometric variant of the classical construction of vector bundles on spheres via clutching functions.

math.KT↗
arXiv2014

Symplectic and hyperkahler implosion

We review the quiver descriptions of symplectic and hyperkähler implosion in the case of SU(n) actions. We give quiver descriptions of symplectic implosion for other classical groups, and discuss some of the issues involved in obtaining a similar description for hyperkähler implosion.

math.SG↗
arXiv2014

Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information

Entangled many-body states are an essential resource for quantum computing and interferometry. Determining the type of entanglement present in a system usually requires access to an exponential number of parameters. We show that in the case of pure multi-particle quantum states, features of the global entanglement can already be extracted from local information alone. This is achieved by associating with any given class of entanglement an entanglement polytope---a geometric object which characterizes the single-particle states compatible with that class. Our results, applicable to systems of arbitrary size and statistics, give rise to local witnesses for global pure-state entanglement, and can be generalized to states affected by low levels of noise.

quant-ph↗
arXiv2014

Eigenvalue Distributions of Reduced Density Matrices

Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

quant-ph↗
arXiv2014

Projective linear configurations via non-reductive actions

We study the iterated blow-up X of projective space along an arbitrary collection of linear subspaces. By replacing the universal torsor with an $\mathbb{A}^1$-homotopy equivalent model, built from $\mathbb{A}^1$-fiber bundles not just algebraic line bundles, we construct an "algebraic uniformization": X is a quotient of affine space by a solvable group action. This provides a clean dictionary, using a single coordinate system, between the algebra and geometry of hypersurfaces: effective divisors are characterized via toric and invariant-theoretic techniques. In particular, the Cox ring is an invariant subring of a Pic(X)-graded polynomial ring and it is an intersection of two explicit finitely generated rings. When all linear subspaces are points, this recovers a theorem of Mukai while also giving it a geometric proof and topological intuition. Consequently, it is algorithmic to describe Cox(X) up to any degree, and when Cox(X) is finitely generated it is algorithmic to verify finite generation and compute an explicit presentation. We consider in detail the special case of $\overline{M}_{0,n}$. Here the algebraic uniformization is defined over the integers. It admits a natural modular interpretation, and it yields a precise sense in which $\overline{M}_{0,n}$ is "one non-linearizable $\mathbb{G}_a$ away" from being a toric variety. The Hu-Keel question becomes a special case of Hilbert's 14th problem for $\mathbb{G}_a$.

math.AG↗
arXiv2012

Computing Multiplicities of Lie Group Representations

For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.

cs.CC↗
arXiv2009

A^1-homotopy groups, excision, and solvable quotients

We study some properties of A^1-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of A^1-homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate A^1-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree A^1-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the "next" non-vanishing A^1-homotopy group (beyond π_1^{A^1}) of a smooth toric variety. From this point of view, A^1-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost "as tractable" (in low degrees) as ordinary homotopy for large classes of interesting varieties.

math.AG↗
arXiv2008

Yang-Mills theory and Tamagawa numbers

Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SL_n. This article surveys this link between Yang-Mills theory and Tamagawa numbers, and explains how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth complex projective curve can be adapted to the setting of A^1-homotopy theory to study the motivic cohomology of these moduli spaces.

math.AG↗
arXiv2007

Vector bundles on contractible smooth schemes

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are that of Spec k. One might hope that furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; this is almost certainly true when the scheme is affine. However, in the non-affine case this is false: we show that (essentially) every smooth A^1-contractible strictly quasi-affine scheme that admits a U-torsor whose total space is affine, for U a unipotent group, possesses a non-trivial vector bundle. Indeed we produce explicit arbitrary dimensional families of non-isomorphic such schemes, with each scheme in the family equipped with "as many" (i.e., arbitrary dimensional moduli of) non-isomorphic vector bundles, of every sufficiently large rank n, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible, but that are not necessarily A^1-contractible.

math.AG↗
arXiv2007

On unipotent quotients and some A^1-contractible smooth schemes

We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principal bundle quotient exists and, together with a cohomological vanishing criterion, to characterize whether or not the resulting quasi-affine quotient scheme is affine. We completely analyze the case of G_a-invariant hypersurfaces in a linear G_a-representation W; here the above characterizations admit simple geometric and algebraic interpretations. As an application, we produce arbitrary dimensional families of non-isomorphic smooth quasi-affine but not affine n-dimensional varieties (n \geq 6) that are contractible in the sense of A^1-homotopy theory. Indeed, existence follows without any computation; yet explicit defining equations for the varieties depend only on knowing some linear G_a- and SL_2- invariants, which, for a sufficiently large class, we provide. Similarly, we produce infinitely many non-isomorphic examples in dimensions 4 and 5. Over C, the analytic spaces underlying these varieties are non-isomorphic, non-Stein, topologically contractible and often diffeomorphic to C^n.

math.AG↗