Final Report Summary - STABILITYMETRICS (Stability in algebraic geometry and its relationship with canonical Kahler metrics)
One of the principle ideas in mathematics is that of the "shape" of an object. For a very concrete example, one can consider three different types of 3 dimensional balls: a spherical football, an elongated rugby ball or a golf ball whose surfaces is bumpy. Of course each of these are very different, as witnessed by their different size and shape. However, from another point of view, they can be seen to be very similar: each can essentially be made by taking a completely spherical ball and then "deforming" it by pushing and pulling parts without making any cuts or tears. Among these three types of ball, the football is particularly special in that it is completely spherical so has more symmetry than the others. In fact, as one can see rather easily, the spherical football has the most symmetric among all the possible deformations that one can create.
In the above illustration, the football is an example of an object that has a particularly nice form of shape. Mathematicians have long wanted to understand this from various points of view, in particular for objects that have higher dimensions. The research in this grant concerns an area of pure mathematics called complex algebraic geometry that is concerned with this problem. More precisely, it concerns the idea of shape not through symmetries but through the idea of distance on an object, and an important conjecture that governs when there might exist a special form of this distance (analogous to the spherical football thought of above). In technical language it centres around a conjectural equivalence between the existence of certain special metrics in Kahler geometry and a notion of stability in projective geometry. The work builds on what is already known and generalises it to other mathematical domains, connecting with other parts of pure mathematics.
During the period of this grant proposal, the researcher has worked on a number of problems in this area. In collaboration with 4 different authors, the researcher has produced 6 articles which hold a number of new ideas, techniques and theorems along with their proofs. These are listed below and the full text can be found through the project website.
The resources from this grant have allowed the researcher increased time for research and to permit the researcher's academic travel (within the United Kingdom, and European countries including France and Sweden) and to invite academic visitors and collaborators to the researcher's home institution.
[1] (J Ross and R P Thomas) Weighted Bergman kernels on orbifolds. Jour. Diff. Geom. 88, p. 87-108, 2011.
[2] (J Ross and R P Thomas) Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kahler metrics Jour. Diff. Geom. 88, p. 109-160, 2011
[3] (J Ross and M Garcia-Fernandez) Limits of balanced metrics on vector bundles and polarised manifolds To appear in the Proceedings of the London Mathematical Society
[4] (J Ross and J Keller) A note on Chow stability of the Projectivisation of Gieseker Stable Bundles To appear in the Journal of Geometric Analysis
[5] (J Ross and D Witt-Nystrom) Analytic test configurations and geodesic rays Preprint arXiv: 1101. 1612. Submitted.
[6] (J Ross and D Witt-Nystrom) Envelopes of positive metrics with prescribed singularities. Preprint arXiv: 1210. 2220. Submitted
In the above illustration, the football is an example of an object that has a particularly nice form of shape. Mathematicians have long wanted to understand this from various points of view, in particular for objects that have higher dimensions. The research in this grant concerns an area of pure mathematics called complex algebraic geometry that is concerned with this problem. More precisely, it concerns the idea of shape not through symmetries but through the idea of distance on an object, and an important conjecture that governs when there might exist a special form of this distance (analogous to the spherical football thought of above). In technical language it centres around a conjectural equivalence between the existence of certain special metrics in Kahler geometry and a notion of stability in projective geometry. The work builds on what is already known and generalises it to other mathematical domains, connecting with other parts of pure mathematics.
During the period of this grant proposal, the researcher has worked on a number of problems in this area. In collaboration with 4 different authors, the researcher has produced 6 articles which hold a number of new ideas, techniques and theorems along with their proofs. These are listed below and the full text can be found through the project website.
The resources from this grant have allowed the researcher increased time for research and to permit the researcher's academic travel (within the United Kingdom, and European countries including France and Sweden) and to invite academic visitors and collaborators to the researcher's home institution.
[1] (J Ross and R P Thomas) Weighted Bergman kernels on orbifolds. Jour. Diff. Geom. 88, p. 87-108, 2011.
[2] (J Ross and R P Thomas) Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kahler metrics Jour. Diff. Geom. 88, p. 109-160, 2011
[3] (J Ross and M Garcia-Fernandez) Limits of balanced metrics on vector bundles and polarised manifolds To appear in the Proceedings of the London Mathematical Society
[4] (J Ross and J Keller) A note on Chow stability of the Projectivisation of Gieseker Stable Bundles To appear in the Journal of Geometric Analysis
[5] (J Ross and D Witt-Nystrom) Analytic test configurations and geodesic rays Preprint arXiv: 1101. 1612. Submitted.
[6] (J Ross and D Witt-Nystrom) Envelopes of positive metrics with prescribed singularities. Preprint arXiv: 1210. 2220. Submitted