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J. P. Pridham

Publications and source records attributed to J. P. Pridham.

At least 19 recordsLinked to original sources
arXiv2026

Deformation quantisation of exact shifted symplectic structures, with an application to vanishing cycles

We extend the author's and CPTVV's correspondence between shifted symplectic and Poisson structures to establish a correspondence between exact shifted symplectic structures and non-degenerate shifted Poisson structures with formal derivation, a concept generalising constructions by De Wilde and Lecomte. Our formulation is sufficiently general to encompass derived algebraic, analytic and $\mathcal{C}^{\infty}$ stacks, as well as Lagrangians and non-commutative generalisations. We also show that non-degenerate shifted Poisson structures with formal derivation carry unique self-dual deformation quantisations in any setting where the latter can be formulated. One application is that for (not necessarily exact) $0$-shifted symplectic structures in analytic and $\mathcal{C}^{\infty}$ settings, it follows that the author's earlier parametrisations of quantisations are in fact independent of any choice of associator, and generalise Fedosov's parametrisation of quantisations for classical manifolds. Our main application is to complex $(-1)$-shifted symplectic structures, showing that our unique quantisation of the canonical exact structure, a sheaf of twisted $BD_0$-algebras with derivation, gives rise to BBDJS's perverse sheaf of vanishing cycles, equipped with its monodromy operator.

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arXiv2025

Derived deformation functors, Koszul duality, and Maurer-Cartan spaces

We summarise the chain of comparisons showing Hinich's derived Maurer-Cartan functor gives an equivalence between differential graded Lie algebras and derived Schlessinger functors on Artinian differential graded-commutative algebras. We include some motivating deformation problems and analogues for more general Koszul dual pairs of operads.

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arXiv2025

Derived derivations govern contraderived deformations of dg algebras over dg (pr)operads

We show that Hinich's simplicial nerve of the differential graded Lie algebra (DGLA) of derived derivations of a dg algebra $A$ over a dg properad $\mathcal{P}$ is equivalent to the space of deformations of $A$ as a $\mathcal{P}_{\infty}$-algebra in Positselski's contraderived dg category. This resolves Hinich's counterexamples to the general existence of derived deformations. It also generalises his results when $A$ is homologically bounded below, since contraderived deformations are then precisely derived deformations.

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arXiv2025

Non-commutative derived analytic moduli functors

We develop a formulation for non-commutative derived analytic geometry built from differential graded (dg) algebras equipped with free entire functional calculus (FEFC), relating them to simplicial FEFC algebras and to locally multiplicatively convex complete topological dg algebras. The theory is optimally suited for accommodating analytic morphisms between functors of algebraic origin, and we establish forms of Riemann-Hilbert equivalence in this setting. We also investigate classes of topological dg algebras for which moduli functors of analytic origin tend to behave well, and relate their homotopy theory to that of FEFC algebras. Applications include the construction of derived non-commutative analytic moduli stacks of pro-étale local systems and non-commutative derived twistor moduli functors, both equipped with shifted analytic bisymplectic structures, and hence shifted analytic double Poisson structures.

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arXiv2025

Shifted bisymplectic and double Poisson structures on non-commutative derived prestacks

We introduce the notions of shifted bisymplectic and shifted double Poisson structures on differential graded associative algebras, and more generally on non-commutative derived moduli functors with well-behaved cotangent complexes. For smooth algebras concentrated in degree $0$, these structures recover the classical notions of bisymplectic and double Poisson structures, but in general they involve an infinite hierarchy of higher homotopical data, ensuring that they are invariant under quasi-isomorphism. The structures induce shifted symplectic and shifted Poisson structures on the underlying commutative derived moduli functors, and also on underlying representation functors. We show that there are canonical equivalences between the spaces of shifted bisymplectic structures and of non-degenerate $n$-shifted double Poisson structures. We also give canonical shifted bisymplectic and bi-Lagrangian structures on various derived non-commutative moduli functors of modules over Calabi--Yau dg categories. Unlike their commutative counterparts, these structures enjoy a formal integration property, which we exploit to show that Calabi--Yau and pre-Calabi--Yau structures on a dg algebra correspond respectively to bisymplectic and double Poisson structures on its quotient prestacks by the adjoint $\mathbb{G}_m$-action.

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arXiv2025

Quantisation of derived Poisson structures

We prove that every $0$-shifted Poisson structure on a derived Artin $n$-stack admits a curved $A_{\infty}$ deformation quantisation whenever the stack has perfect cotangent complex; in particular, this applies to LCI schemes, where it gives a DQ algebroid quantisation. Whereas the Kontsevich--Tamarkin approach to quantisation for smooth varieties hinges on invariance of the Hochschild complex under affine transformations, we instead exploit the observation that the Hochschild complex carries an anti-involution, and that such anti-involutive deformations of the complex of polyvectors are essentially unique. We also establish analogous statements for deformation quantisations in $\mathcal{C}^{\infty}$ and analytic settings.

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arXiv2024

Semiregularity as a consequence of Goodwillie's theorem

We realise Buchweitz and Flenner's semiregularity map (and hence a fortiori Bloch's semiregularity map) for a smooth variety $X$ as the tangent of a generalised Abel--Jacobi map on the derived moduli stack of perfect complexes on $X$. The target of this map is an analogue of Deligne cohomology defined in terms of cyclic homology, and Goodwillie's theorem on nilpotent ideals ensures that it has the desired tangent space (a truncated de Rham complex). Immediate consequences are the semiregularity conjectures: that the semiregularity maps annihilate all obstructions, and that if $X$ is deformed, semiregularity measures the failure of the Chern character to remain a Hodge class. This gives rise to reduced obstruction theories of the type featuring in the study of reduced Gromov--Witten and Pandharipande--Thomas invariants. We also give generalisations allowing $X$ to be singular, and even a derived stack.

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arXiv2024

Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and Galois representations

We develop a characterisation of non-Archimedean derived analytic geometry based on dg enhancements of dagger algebras. This allows us to formulate derived analytic moduli functors for many types of pro-étale sheaves, and to construct shifted symplectic structures on them by transgression using arithmetic duality theorems. In order to handle duality functors involving Tate twists, we introduce weighted shifted symplectic structures on formal weighted moduli stacks, with the usual moduli stacks given by taking $\mathbb{G}_m$-invariants. In particular, this establishes the existence of shifted symplectic and Lagrangian structures on derived moduli stacks of $\ell$-adic constructible complexes on smooth varieties via Poincaré duality, and on derived moduli stacks of $\ell$-adic Galois representations via Tate and Poitou--Tate duality; the latter proves a conjecture of Minhyong Kim. Unweighted shifted symplectic and Lagrangian structures are also established for $\ell$-adic Galois representations of cyclotomic fields $K(μ_{\ell^{\infty}})$, subject to additional constraints related to Iwasawa theory; these derived moduli stacks yield a non-abelian analogue of Selmer complexes, with $0$-shifted symplectic structures related to generalised Cassels--Tate pairings.

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arXiv2024

Notes characterising higher and derived stacks concretely

This is an informal summary of the main concepts in arXiv:0905.4044, based on notes of various seminars. It gives constructions of higher and derived stacks without recourse to the extensive theory developed by Toen, Vezzosi and Lurie. Explicitly, higher stacks are described in terms of simplicial diagrams of affine schemes, which are analogous to atlases for manifolds. We also describe quasi-coherent sheaves and complexes on such objects.

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arXiv2024

A note on étale atlases for Artin stacks and Lie groupoids, Poisson structures and quantisation

We explain how any Artin stack $\mathfrak{X}$ over $\mathbb{Q}$ extends to a functor on non-negatively graded commutative cochain algebras, which we think of as functions on Lie algebroids or stacky affine schemes. There is a notion of étale morphisms for these CDGAs, and Artin stacks admit étale atlases by stacky affines, giving rise to a small étale site of stacky affines over $\mathfrak{X}$. This site has the same quasi-coherent sheaves as $\mathfrak{X}$ and leads to efficient formulations of shifted Poisson structures, differential operators and quantisations for Artin stacks. There are generalisations to higher and derived stacks. We also describe analogues for differentiable and analytic stacks; in particular, a Lie groupoid naturally gives a functor on NQ-manifolds which we can use to transfer structures. In those settings, local diffeomorphisms and biholomorphisms are the analogues of étale morphisms. This note mostly elaborates constructions scattered across several of the author's papers, but with an emphasis on the functor of points perspective. New results include consistency checks showing that the induced notions of structures such as vector bundles or torsors on a stacky affine scheme coincide with familiar definitions in terms of flat connections.

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arXiv2023

An introduction to derived (algebraic) geometry

Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly in characteristic $0$, but we also see the tweaks which extend most of the content to analytic and differential settings.

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arXiv2022

Derived moduli of schemes and sheaves

We describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by derived geometric 1-stacks are derived moduli of polarised projective varieties, derived moduli of vector bundles, and derived moduli of abelian varieties.

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arXiv2021

Quantisation of derived Lagrangians

We investigate quantisations of line bundles $\mathcal{L}$ on derived Lagrangians $X$ over $0$-shifted symplectic derived Artin $N$-stacks $Y$. In our derived setting, a deformation quantisation consists of a curved $A_{\infty}$ deformation of the structure sheaf $\mathcal{O}_{Y}$, equipped with a curved $A_{\infty}$ morphism to the ring of differential operators on $\mathcal{L}$; for line bundles on smooth Lagrangian subvarieties of smooth symplectic algebraic varieties, this simplifies to deforming $(\mathcal{L}, \mathcal{O}_{Y})$ to a DQ module over a DQ algebroid. For each choice of formality isomorphism between the $E_2$ and $P_2$ operads, we construct a map from the space of non-degenerate quantisations to power series with coefficients in relative cohomology groups of the respective de Rham complexes. When $\mathcal{L}$ is a square root of the dualising line bundle, this leads to an equivalence between even power series and certain anti-involutive quantisations, ensuring that the deformation quantisations always exist for such line bundles. This gives rise to a dg category of algebraic Lagrangians, an algebraic Fukaya category of the form envisaged by Behrend and Fantechi. We also sketch a generalisation of these quantisation results to Lagrangians on higher $n$-shifted symplectic derived stacks.

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arXiv2020

Deformation quantisation for $(-2)$-shifted symplectic structures

We formulate a notion of $E_{-1}$ quantisation of $(-2)$-shifted Poisson structures on derived algebraic stacks, depending on a flat right connection on the structure sheaf, as solutions of a quantum master equation. We then parametrise $E_{-1}$ quantisations of $(-2)$-shifted symplectic structures by constructing a map to power series in de Rham cohomology. For derived schemes, we show that these quantisations give rise to classes in Borel--Moore homology, and for a large class of examples we show that the classes are closely related to Borisov--Joyce invariants.

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arXiv2020

Non-commutative derived moduli prestacks

We introduce a formalism for derived moduli functors on differential graded associative algebras, which leads to non-commutative enhancements of derived moduli stacks and naturally gives rise to structures such as Hall algebras. Descent arguments are not available in the non-commutative context, so we establish new methods for constructing various kinds of atlases. The formalism permits the development of the theory of shifted bi-symplectic and shifted double Poisson structures in the companion paper.

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arXiv2019

Unifying derived deformation theories

We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories of DGLAs and SHLAs (L infinity algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toen--Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as Andre--Quillen cohomology.

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arXiv2019

The deformation theory of representations of the fundamental group of a smooth variety

Consider the functor describing deformations of a representation of the fundamental group of a variety X. This paper is chiefly concerned with establishing an analogue in finite characteristic of a result proved by Goldman and Millson for compact Kahler manifolds. By applying the Weil Conjectures instead of Hodge theory, we see that if X is a smooth proper variety defined over a finite field, and we consider deformations of certain continuous l-adic representations of the algebraic fundamental group, then the hull of the deformation functor will be defined by quadratic equations. Moreover, if X is merely smooth, then the hull will be defined by equations of degree at most four.

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