Mathematics

Given a complex projective manifold, it is very difficult todetect in the Kähler cone which classes admit a Kähler metric withconstant scalar curvature. Fix a Kähler class $\alpha$ with such aspecial Kähler metric, one can ask if it is easier to describeexplicitly a neighborhood V of $\alpha$, such that for all $\beta$ inV, the Kähler class $\beta$ admits a Kähler metric with constantscalar curvature. Of course, one can ask what is the largest possibleneighborhood V ? Under certain conditions, a partial answer to thesequestions can be given by studying Donaldson's J-flow that I willpresent. This flow also appears in mirror symmetry. I will explainthat if the behavior of this flow is still unclear, its finitedimensional analogue can be well understood thanks to geometricquantization. This is joint works with R. Dervan and Y. Hashimoto.