H2020Индивидуална стипендия2021–2023

CanMetCplxGeom · Finding canonical metrics in complex differential geometry

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2021-09-01 → 2023-08-31
Финансиране от ЕС: 203 852 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Сложните геометрични фигури се изследват, за да се открият оптимални мерки за разстояние върху тях, като например при т.нар. Калаби-Яу многообразия. Това помага да се разбере връзката между алгебричната стабилност и съществуването на специфични метрики в геометрията.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Finding canonical metrics in complex differential geometry

A central problem in complex geometry is the search for canonical Kähler metrics, which are optimal notions of distance on a Kähler manifold. This goes back to the Uniformisation theorem in complex dimension one, and is a highly active area of research to the present day. In higher dimensions, Aubin and Yau's resolution of the Calabi conjecture showed that one always obtains such metrics for canonically polarised and Calabi-Yau manifolds. However, in general, higher dimensional complex manifolds may or may not admit a canonical metric. A key goal in complex differential geometry is to understand whether or not a given complex manifold admits a canonical Kähler metric, such as those of constant scalar curvature (cscK), or more generally, extremal metrics. This is a highly non-trivial question, and has surprising links to algebraic geometry. The Yau-Tian-Donaldson (YTD) conjecture is central to the field. This predicts that an algebraic notion of stability should be able to detect whether or not a given Kähler manifold admits a canonical metric. It is similar in spirit to the Hitchin-Kobayashi correspondence for vector bundles, which characterises the existence of Hermite-Einstein metrics on vector bundles via the notion of slope stability. This was established in the 1980's by Uhlenbeck-Yau and Donaldson. A key goal in the project is to produce such metrics. Another is to study how these metrics behave in families. Together with Dervan, I started a research programme on this theme. We introduced an equation, called the Optimal Symplectic Connection equation, for good families of canonical metrics on fibrations, and a notion of stability for fibrations. For projectivised vector bundles, this recovers the notions of Hermite-Einstein metrics and slope stability as in the Hitchin-Kobayashi correspondence for vector bundles. An overall goal in this research programme is to establish a Hitchin-Kobayashi/YTD type conjecture. This requires the development of many differential and algebro-geometric tools.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

This proposal is in the area of complex differential geometry, a prominent field of mathematics. It stands at the intersection of differential and algebraic geometry. The basic objects are manifolds, spaces that locally look like flat space, and vector bundles over them - a collection of vector spaces parametrised by a manifold. In complex differential geometry one seeks optimal notions of distance, so-called canonical metrics. In higher dimensions, canonical metrics may or may not exist. The key question is to determine whether or not a given space has a canonical metric, a very challenging problem. The Yau-Tian-Donaldson conjecture stands at the heart of this problem, and relates the existence of a solution to algebro-geometric notions of stability.The aim of this research proposal is to give several new constructions of canonical metrics for complex manifolds, holomorphic vector bundles and families of such objects. It also seeks to show connections of the existence of these metrics, a solution to a PDE, with purely algebraic notions, for an equation for families of canonical metrics. This will be approached mainly with techniques from perturbative and variational PDE theory and algebraic geometry, but will also use some computational methods and probability theory. The proposal seeks to develop new techniques for well studied equations, and to apply more well known techniques to new equations, to advance the constructions and the theory of canonical metrics in a major way.The action would give a unique opportunity for a reciprocal transfer of knowledge as part of a prominent research group in the field, whose research focus and strengths differ from that of the ER. It would provide the ER with the independence needed to form his own research group in the future, and expand the ER's academic network through new connections. Though currently working in Europe, the ER was previously in North America. The fellowship would allow the ER to remain within the EU.

Оригинален текст от CORDIS (на английски).

Участници

GOETEBORGS UNIVERSITET · GoeteborgКоординаторШвеция