Detecting canonical metrics in complex geometry

An important area of research in complex geometry is the search for canonical metrics, which can be seen as the study of optimal shapes.

The proposed project seeks to develop new constructions of these metrics both for old and new equations and provide deep links between their existence and criteria from algebraic geometry.

It will feature a blend of mathematical techniques: many belong to differential geometry and geometric analysis, but also algebraic tools from algebraic geometry are central to solve the objectives of the research plan.

The proposed goals will be undertaken through individual work as well as collaborations in small groups, featuring old and new collaborators that are both within and outside the department of the host institution. The study of canonical metrics has been a central object of study in geometry since the 19th century through to today.

In dimension 1, one can always find such metrics, but in higher dimension they do not always exist. It is in general very difficult to determine whether such a metric exists.

Having constructions ensuring their existence is crucial and the proposal seeks to do that both from an analytic and algebraic perspective.

Further, the constructions in the proposal will reveal subtle links between the existence of these metrics and algebraic geometry.

The proposal will develop an understanding of how such metrics behave in families and give an analytic approach to the study of Bridgeland stability conditions.