arXiv subjects

Brent Pym

Publications and source records attributed to Brent Pym.

At least 19 recordsLinked to original sources

arXiv2026

Poisson K-stability and the semiclassical Yau--Tian--Donaldson correspondence

We introduce a notion of K-polystability for compact K\"ahler holomorphic Poisson manifolds. On the one hand, this notion of stability is well-adapted to constructions of moduli spaces. For instance, when the underlying manifold is K-polystable with reductive reduced automorphism group, Poisson K-stability is equivalent to geometric invariant theoretic stability in the space of Poisson bivectors, but there also exist K-unstable varieties that become stable after incorporating a Poisson structure. On the other hand, the Poisson K-stability condition interacts well with generalized K\"aher metrics -- the background geometry of (2,2) supersymmetric string theory. In particular, we conjecture that Poisson K-polystability characterizes the existence of constant scalar curvature symplectic generalized K\"ahler structures with a sufficiently small Poisson tensor -- a natural extension of the Yau--Tian--Donaldson (YTD) conjecture. Our main result is a proof of the existence part of this ``semiclassical YTD conjecture'' for Poisson structures on K\"ahler--Einstein Fano manifolds, using infinite-dimensional momentum map techniques. In this way, we obtain the existence of many new examples of symplectic generalized K\"ahler structure of constant scalar curvature, and prove the conjecture completely in the case of the projective plane.

math.DG↗

arXiv2026

Bondal's conjecture in dimension five

Bondal's conjecture in Poisson geometry gives lower bounds on the degeneracy loci of Poisson Fano manifolds, where the rank of the Poisson structure drops. By work of several authors, it was previously known to hold for Fano manifolds of dimension at most four. We give the first proof of this conjecture for Fano manifolds of dimension five, and partial results for Fano manifolds of all odd dimensions. The proof uses: (i) an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, extending a previous result of the first author; (ii) the "modular residues" of Poisson structures introduced by Gualtieri and the third author; and (iii) a cohomological constraint on invariant subvarieties for Pfaff fields, extending earlier results of Esteves--Kleiman to the case in which the Pfaff distribution on the subvariety admits a closed strongly directed positive current.

math.AG↗

arXiv2026

Weighted blowups and 3d Poisson desingularizations

We establish existence of functorial orbifold reductions of singularities for Poisson subvarieties in smooth Poisson threefolds. Namely, we show that with enough weighted blowups, one can reduce the singularities of such Poisson subvarieties to certain simple, explicit, local normal forms: Du Val surface singularities where the Poisson structure is locally Jacobian, and plane curves lying in the vanishing locus of a particular linear Poisson structure. The proof combines Abramovich--Temkin--W{\l}odarczyk and McQuillan's recent approach to resolution of singularities for varieties via weighted blowups with some new normal forms for three-dimensional Poisson brackets derived via Poisson cohomology. Along the way, we describe necessary and sufficient conditions for a polyvector field to lift to the weighted blowup of an orbifold along a suborbifold, generalizing criteria of Polishchuk for unweighted blowups of Poisson structures on smooth varieties.

math.AG↗

arXiv2024

Creating quantum projective spaces by deforming q-symmetric algebras

We construct a large collection of "quantum projective spaces", in the form of Koszul, Calabi-Yau algebras with the Hilbert series of a polynomial ring. We do so by starting with the toric ones (the q-symmetric algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. We then prove that these algebras are identified with the canonical quantizations of corresponding families of quadratic Poisson structures, in the sense of Kontsevich. In this way, we obtain the first broad class of quadratic Poisson structures for which his quantization can be computed explicitly, and shown to converge, as he conjectured in 2001.

math.QA↗

arXiv2024

Numerically flat foliations and holomorphic Poisson geometry

We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact K\"ahler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that such foliations induce a decomposition of the tangent bundle of the ambient manifold, have leaves uniformized by Euclidean spaces, and have torsion canonical bundle. Additionally, we prove that smooth two-dimensional foliations with numerically trivial canonical bundle on projective manifolds are either isotrivial fibrations or have numerically flat tangent bundles. This in turn implies a global Weinstein splitting theorem for rank-two Poisson structures on projective manifolds. We also derive new Hodge-theoretic conditions for the existence of zeros of Poisson structures on compact K\"ahler manifolds.

math.AG↗

arXiv2024

Semiclassical Hodge theory for log Poisson manifolds

We construct a mixed Hodge structure on the topological K-theory of smooth Poisson varieties, depending weakly on a choice of compactification. We establish a package of tools for calculations with these structures, such as functoriality statements, projective bundle formulae, Gysin sequences and Torelli properties. We show that for varieties with trivial A-hat class, the corresponding period maps for families can be written as exponential maps for bundles of tori, which we call the "quantum parameters". As justification for the terminology, we show that in many interesting examples, the quantum parameters of a Poisson variety coincide with the parameters appearing in its known deformation quantizations. In particular, we give a detailed implementation of an argument of Kontsevich, to prove that his canonical quantization formula, when applied to Poisson tori, yields noncommutative tori with parameter "$q = e^\hbar$".

math.AG↗

arXiv2024

Logarithmic morphisms, tangential basepoints, and little disks

We develop the theory of "virtual morphisms" in logarithmic algebraic geometry, introduced by Howell. It allows one to give algebro-geometric meaning to various useful maps of topological spaces that do not correspond to morphisms of (log) schemes in the classical sense, while retaining functoriality of key constructions. In particular, we explain how virtual morphisms provide a natural categorical home for Deligne's theory of tangential basepoints: the latter are simply the virtual morphisms from a point. We also extend Howell's results on the functoriality of Betti and de Rham cohomology. Using this framework, we lift the topological operad of little $2$-disks to an operad in log schemes over the integers, whose virtual points are isomorphism classes of stable marked curves of genus zero equipped with a tangential basepoint. The gluing of such curves along marked points is performed using virtual morphisms that transport tangential basepoints around the curves. This builds on Vaintrob's analogous construction for framed little disks, for which the classical notion of morphism in logarithmic geometry sufficed. In this way, we obtain a direct algebro-geometric proof of the formality of the little disks operad, following the strategy envisioned by Beilinson. Furthermore, Bar-Natan's parenthesized braids naturally appear as the fundamental groupoids of our moduli spaces, with all virtual basepoints defined over the integers.

math.AG↗

arXiv2023

Regularized integrals and manifolds with log corners

We introduce a natural geometric framework for the study of logarithmically divergent integrals on manifolds with corners and algebraic varieties, using the techniques of logarithmic geometry. Key to the construction is a new notion of morphism in logarithmic geometry itself, introduced by Howell, which allows us to interpret the ubiquitous rule of thumb ''$\lim_{\epsilon\to 0} \log \epsilon := 0$'' as the restriction to a submanifold. Via a version of de Rham's theorem with logarithmic divergences, we obtain a functorial characterization of the classical theory of ``regularized integration'': it is the unique way to extend the ordinary integral to the logarithmically divergent context while respecting the basic laws of calculus (change of variables, Fubini's theorem, and Stokes' formula.)

math.DG↗

arXiv2023

Resurgence of a de Sitter Glauber-Sudarshan State: Nodal Diagrams and Borel Resummation

We show that an explicit construction of a four-dimensional de Sitter space may be performed using a diagrammatic approach via nodal diagrams emanating from the path integral representation of the Glauber-Sudarshan state. Sum of these diagrams typically leads to an asymptotic series of Gevrey kind which can then be Borel resummed, thus reproducing the non-perturbative structure of the system. Our analysis shows that four-dimensional de Sitter space is not only possible in string theory overcoming the no-go and the swampland criteria -- albeit as a Glauber-Sudarshan state -- but it may also be non-perturbatively stable within a controlled temporal domain. Somewhat consistently, the Borel resummation of the Gevrey series provides strong hint towards the positivity of the cosmological constant.

hep-th↗

arXiv2022

Holonomic Poisson geometry of Hilbert schemes

We undertake a detailed study of the geometry of Bottacin's Poisson structures on Hilbert schemes of points in Poisson surfaces, i.e. smooth complex surfaces equipped with an effective anticanonical divisor. We focus on three themes that, while logically independent, are linked by the interplay between (characteristic) symplectic leaves and deformation theory. Firstly, we construct the symplectic groupoids of the Hilbert schemes and develop the classification of their symplectic leaves, using the methods of derived symplectic geometry. Secondly, we establish local normal forms for the Poisson brackets, and combine them with a toric degeneration argument to verify that Hilbert schemes satisfy our recent conjecture characterizing holonomic Poisson manifolds in terms of the geometry of the modular vector field. Finally, using constructible sheaf methods, we compute the space of first-order Poisson deformations when the anti-canonical divisor is reduced and has only quasi-homogeneous singularities. (The latter is automatic if the surface is projective.) Along the way, we find a tight connection between the Poisson geometry of the Hilbert schemes and the finite-dimensional Lie algebras of affine transformations, which is mediated by syzygies. In particular, we find that the Hilbert scheme has a natural subvariety that serves as a global counterpart of the nilpotent cone, and we prove that the Lie algebras of affine transformations have holonomic dual spaces -- the first such series of Lie algebras to be discovered.

math.AG↗

arXiv2021

A local Torelli theorem for log symplectic manifolds

We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing diagrams, and amenable to algorithmic classification. Applying the theory to four-dimensional projective space, we obtain a total of 40 irreducible components of the moduli space, most of which are new. Our main technique is a detailed analysis of the relevant deformation complex (the Poisson cohomology) as an object of the constructible derived category.

math.AG↗

arXiv2021

A global Weinstein splitting theorem for holomorphic Poisson manifolds

We prove that if a compact Kähler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite étale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold. As a step in the proof, we establish a special case of Beauville's conjecture on the structure of compact Kähler manifolds with split tangent bundle.

math.AG↗

arXiv2018

Multiple zeta values in deformation quantization

Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula.

math.QA↗

arXiv2017

Holonomic Poisson manifolds and deformations of elliptic algebras

We introduce a natural nondegeneracy condition for Poisson structures, called holonomicity, which is closely related to the notion of a log symplectic form. Holonomic Poisson manifolds are privileged by the fact that their deformation spaces are as finite-dimensional as one could ever hope: the corresponding derived deformation complex is a perverse sheaf. We develop some basic structural features of these manifolds, highlighting the role played by the divergence of Hamiltonian vector fields. As an application, we establish the deformation-invariance of certain families of Poisson manifolds defined by Feigin and Odesskii, along with the "elliptic algebras" that quantize them.

math.AG↗

arXiv2017

Constructions and classifications of projective Poisson varieties

This paper is intended both an introduction to the algebraic geometry of holomorphic Poisson brackets, and as a survey of results on the classification of projective Poisson manifolds that have been obtained in the past twenty years. It is based on the lecture series delivered by the author at the Poisson 2016 Summer School in Geneva. The paper begins with a detailed treatment of Poisson surfaces, including adjunction, ruled surfaces and blowups, and leading to a statement of the full birational classification. We then describe several constructions of Poisson threefolds, outlining the classification in the regular case, and the case of rank-one Fano threefolds (such as projective space). Following a brief introduction to the notion of Poisson subspaces, we discuss Bondal's conjecture on the dimensions of degeneracy loci on Poisson Fano manifolds. We close with a discussion of log symplectic manifolds with simple normal crossings degeneracy divisor, including a new proof of the classification in the case of rank-one Fano manifolds.

math.AG↗

arXiv2016

Shifted symplectic Lie algebroids

Shifted symplectic Lie and $L_\infty$ algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify zero-, one- and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric "higher structures", such as Courant algebroids twisted by $Ω^2$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well known features of higher structures (such as twists, Pontryagin classes, and tensor products). The proofs are valid in the $C^\infty$, holomorphic and algebraic settings, and are based on a number of technical results on the homotopy theory of $L_\infty$ algebroids and their differential forms, which may be of independent interest.

math.DG↗

arXiv2015

Motivic Donaldson--Thomas invariants of some quantized threefolds

This paper is motivated by the question of how motivic Donaldson--Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi--Yau threefolds, defined by quivers with homogeneous potentials. These families give deformation quantizations of affine three-space, the resolved conifold, and the resolution of the transversal $A_n$-singularity. It turns out that their invariants are generically constant, but jump at special values of the deformation parameter, such as roots of unity. The corresponding generating series are written in closed form, as plethystic exponentials of simple rational functions. While our results are limited by the standard dimensional reduction techniques that we employ, they nevertheless allow us to conjecture formulae for more interesting cases, such as the elliptic Sklyanin algebras.

math.AG↗

arXiv2015

Elliptic singularities on log symplectic manifolds and Feigin--Odesskii Poisson brackets

A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities $\tilde{E}_6,\tilde{E}_7$ and $\tilde{E}_8$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii's Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic.

math.AG↗