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Period integrals and other direct images of D-modules
Stockholm University, Faculty of Science, Department of Mathematics.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three papers, each touching on a different aspect of the theory of rings of differential operators and D-modules. In particular, an aim is to provide and make explicit good examples of D-module directimages, which are all but absent in the existing literature.The first paper makes explicit the fact that B-splines (a particular class of piecewise polynomial functions) are solutions to D-module theoretic direct images of a class of D-modules constructed from polytopes.These modules, and their direct images, inherit all the relevant combinatorial structure from the defining polytopes, and as such are extremely well-behaved.The second paper studies the ring of differential operator on a reduced monomial ring (aka. Stanley-Reisner ring), in arbitrary characteristic.The two-sided ideal structure of the ring of differential operators is described in terms of the associated abstract simplicial complex, and several quite different proofs are given.The third paper computes the monodromy of the period integrals of Laurent polynomials about the singular point at the origin. The monodromy is describable in terms of the Newton polytope of the Laurent polynomial, in particular the combinatorial-algebraic operation of mutation plays an important role. Special attention is given to the class of maximally mutable Laurent polynomials, as these are one side of the conjectured correspondance that classifies Fano manifolds via mirror symmetry.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2015. , 32 p.
Keyword [en]
D-modules, Rings of differential operators, Period integrals
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-116790ISBN: 978-91-7649-182-9 (print)OAI: oai:DiVA.org:su-116790DiVA: diva2:808420
Public defence
2015-09-18, sal 14, hus 5 Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Accepted. Paper 2: Manuscript. Paper 3: Manuscript.

Available from: 2015-08-26 Created: 2015-04-27 Last updated: 2016-10-19Bibliographically approved
List of papers
1. B-splines, polytopes and their characteristic D-modules
Open this publication in new window or tab >>B-splines, polytopes and their characteristic D-modules
(English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125Article in journal (Refereed) Accepted
Abstract [en]

Given a polytope σ⊂R m  , its characteristic distribution δ σ   generates a D -module which we call the characteristic D -module of σ  and denote by M σ  . More generally, the characteristic distributions of a cell complex K  with polyhedral cells generate a D -module M K  , which we call the characteristic D -module of the cell complex. We prove various basic properties of M K  , and show that under certain mild topological conditions on K , the D -module theoretic direct image of M K   coincides with the module generated by the B -splines associated to the cells of K  (considered as distributions). We also give techniques for computing D -annihilator ideals of polytopes.

National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-116808 (URN)
Available from: 2015-04-28 Created: 2015-04-28 Last updated: 2017-12-04
2. Two-sided ideals in the ring of differential operators on a Stanley-Reisner ring
Open this publication in new window or tab >>Two-sided ideals in the ring of differential operators on a Stanley-Reisner ring
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let R be a Stanley-Reisner ring (that is, a reduced monomial ring) with coefficients in a domain k, and K its associated simplicial complex. Also let D_k(R) be the ring of k-linear differential operators on R. We give two different descriptions of the two-sided ideal structure of D_k(R) as being in bijection with certain well-known subcomplexes of K; one based on explicit computation in the Weyl algebra, valid in any characteristic, and one valid in characteristic p based on the Frobenius splitting of R. A result of Traves [Tra99] on the D_k(R)-module structure of R is also given a new proof and different interpretation using these techniques.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-116809 (URN)
Available from: 2015-04-28 Created: 2015-04-28 Last updated: 2016-10-19Bibliographically approved
3. Period integrals and mutation
Open this publication in new window or tab >>Period integrals and mutation
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let f  be a Laurent polynomial in two variables, whose Newton polygon strictly contains the origin and whose vertices are primitive lattice points, and let L f   be the minimal-order differential operator that annihilates the period integral of f . We prove several results about f  and L f   in terms of the Newton polygon of f  and the combinatorial operation of *mutation*, in particular we give an in principle complete description of the monodromy of L f   around the origin. Special attention is given to the class of *maximally mutable* Laurent polynomials, which has applications to the conjectured classification of Fano manifolds via mirror symmetry.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-116810 (URN)
Available from: 2015-04-28 Created: 2015-04-28 Last updated: 2016-10-19Bibliographically approved

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