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On Amoebas and Multidimensional Residues
Stockholm University, Faculty of Science, Department of Mathematics.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers and an introduction. 

In Paper I we calculate the second order derivatives of the Ronkin function of an affine polynomial in three variables. This gives an expression for the real Monge-Ampére measure associated to the hyperplane amoeba. The measure is expressed in terms of complete elliptic integrals and hypergeometric functions. 

In Paper II and III we prove that a certain semi-explicit cohomological residue associated to a Cohen-Macaulay ideal or more generally an ideal of pure dimension, respectively, is annihilated precisely by the given ideal. This is a generalization of the local duality principle for the Grothendieck residue and the cohomological residue of Passare. These results follow from residue calculus, due to Andersson and Wulcan, but the point here is that our proof is more elementary. In particular, it does not rely on the desingularization theorem of Hironaka.

In Paper IV we prove a global uniform Artin-Rees lemma for sections of ample line bundles over smooth projective varieties. We also prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proofs are based on multidimensional residue calculus.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2012. , 20 p.
Keyword [en]
amoeba, multidimensional residue, duality principle, effective, uniform, Artin-Rees, Ronkin function
National Category
Mathematical Analysis Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-82843ISBN: 978-91-7447-617-0 (print)OAI: oai:DiVA.org:su-82843DiVA: diva2:573936
Public defence
2013-01-18, 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At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 3: Manuscript. Paper 4. Manuscript.

Available from: 2012-12-27 Created: 2012-11-28 Last updated: 2013-01-07Bibliographically approved
List of papers
1. An explicit calculation of the Ronkin function
Open this publication in new window or tab >>An explicit calculation of the Ronkin function
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We calculate the second order derivatives of the Ronkin function in the case of an affine linear polynomial in three variables and give an expression of them in terms of complete elliptic integrals and hypergeometric functions. This gives a semi-explicit expression of the associated Monge-Ampère measure, the Ronkin measure.

Keyword
Ronkin function, amoeba
National Category
Mathematical Analysis Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-82846 (URN)
Available from: 2012-11-28 Created: 2012-11-28 Last updated: 2012-12-06Bibliographically approved
2. A local Grothendieck duality theorem for Cohen-Macaulay ideals
Open this publication in new window or tab >>A local Grothendieck duality theorem for Cohen-Macaulay ideals
2012 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 111, no 1, 42-52 p.Article in journal (Refereed) Published
Abstract [en]

We give a new proof of a recent result due to Mats Andersson and Elizabeth Wulcan, generalizing the local Grothendieck duality theorem. It can also be seen as a generalization of a previous result by Mikael Passare. Our method does not require the use of the Hironaka desingularization theorem and it provides a semi-explicit realization of the residue that is annihilated by functions from the given ideal.

Keyword
Grothendieck, residue, duality
National Category
Mathematical Analysis Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-82845 (URN)000313542800003 ()
Available from: 2012-11-28 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
3. A local duality principle for ideals of pure dimension
Open this publication in new window or tab >>A local duality principle for ideals of pure dimension
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We prove that a certain cohomological residue associated to an ideal of pure dimension is annihilated exactly by the ideal. The cohomological residue is quite explicit and generalizes the classical local Grothendieck residue and the cohomological residue of Passare.

Keyword
duality, duality principle, pure dimension, residue
National Category
Mathematical Analysis Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-82847 (URN)
Available from: 2012-11-28 Created: 2012-11-28 Last updated: 2012-12-06Bibliographically approved
4. An effective uniform Artin-Rees lemma
Open this publication in new window or tab >>An effective uniform Artin-Rees lemma
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We prove a global uniform Artin-Rees lemma type theorem for sections of ample line bundles over smooth projective varieties. This result is used to prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proof is based on multidimensional residue calculus.

Keyword
Artin-Rees, uniform, effective, polynomials, holomorphic, section, line bundle
National Category
Mathematical Analysis Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-82848 (URN)
Available from: 2012-11-28 Created: 2012-11-28 Last updated: 2012-12-06Bibliographically approved

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