2,040 research outputs found
Quantum deformations of projective three-space
We describe the possible noncommutative deformations of complex projective
three-space by exhibiting the Calabi--Yau algebras that serve as their
homogeneous coordinate rings. We prove that the space parametrizing such
deformations has exactly six irreducible components, and we give explicit
presentations for the generic members of each family in terms of generators and
relations. The proof uses deformation quantization to reduce the problem to a
similar classification of unimodular quadratic Poisson structures in four
dimensions, which we extract from Cerveau and Lins Neto's classification of
degree-two foliations on projective space. Corresponding to the ``exceptional''
component in their classification is a quantization of the third symmetric
power of the projective line that supports bimodule quantizations of the
classical Schwarzenberger bundles.Comment: 27 pages, 1 figure, 1 tabl
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
and . 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 are the only log symplectic structures on
projective four-space whose singular points are all elliptic.Comment: 33 pages, comments welcom
Poisson modules and degeneracy loci
In this paper, we study the interplay between modules and sub-objects in
holomorphic Poisson geometry. In particular, we define a new notion of
"residue" for a Poisson module, analogous to the Poincar\'e residue of a
meromorphic volume form. Of particular interest is the interaction between the
residues of the canonical line bundle of a Poisson manifold and its degeneracy
loci---where the rank of the Poisson structure drops. As an application, we
provide new evidence in favour of Bondal's conjecture that the rank \leq 2k
locus of a Fano Poisson manifold always has dimension \geq 2k+1. In particular,
we show that the conjecture holds for Fano fourfolds. We also apply our
techniques to a family of Poisson structures defined by Fe\u{\i}gin and
Odesski\u{\i}, where the degeneracy loci are given by the secant varieties of
elliptic normal curves.Comment: 33 page
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
Stone-Type Dualities for Separation Logics
Stone-type duality theorems, which relate algebraic and
relational/topological models, are important tools in logic because -- in
addition to elegant abstraction -- they strengthen soundness and completeness
to a categorical equivalence, yielding a framework through which both algebraic
and topological methods can be brought to bear on a logic. We give a systematic
treatment of Stone-type duality for the structures that interpret bunched
logics, starting with the weakest systems, recovering the familiar BI and
Boolean BI (BBI), and extending to both classical and intuitionistic Separation
Logic. We demonstrate the uniformity and modularity of this analysis by
additionally capturing the bunched logics obtained by extending BI and BBI with
modalities and multiplicative connectives corresponding to disjunction,
negation and falsum. This includes the logic of separating modalities (LSM), De
Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics
extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as
corollaries soundness and completeness theorems for the specific Kripke-style
models of these logics as presented in the literature: for DMBI, the
sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene
BI (connecting our work to Concurrent Separation Logic), this is the first time
soundness and completeness theorems have been proved. We thus obtain a
comprehensive semantic account of the multiplicative variants of all standard
propositional connectives in the bunched logic setting. This approach
synthesises a variety of techniques from modal, substructural and categorical
logic and contextualizes the "resource semantics" interpretation underpinning
Separation Logic amongst them
Corrections and Remarks
This document contains corrections to errors discovered to-date in,
and also some remarks upon, both Samin Ishtiaq's thesis, A Relevant
Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL
[IP99] papers, by Ishtiaq and Pym, that follow from it
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