Geometry and complex manifolds

Kähler geometry sits at the crossroads of complex analysis, differential geometry, and algebraic geometry, studying spaces that carry a rich interplay between their complex structure and a notion of curvature. A central concern is whether a given complex manifold admits a distinguished metric — one whose scalar curvature satisfies a prescribed equation — and a deep conjecture now largely resolved through the Yau–Tian–Donaldson program ties the existence of such metrics on Fano manifolds to an algebraic condition called K-stability. Calabi–Yau manifolds, which admit Ricci-flat metrics, have become indispensable in both pure mathematics and theoretical physics, while the complex Monge–Ampère equation provides the analytic engine for constructing and analyzing these metrics. Active directions include understanding canonical metrics on singular spaces, the behavior of Ricci flow as a tool for geometrizing manifolds, and extending stability theory beyond the smooth setting to broader classes of geometric objects.

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Keywords

Kähler MetricsStabilityScalar CurvatureK-StabilityComplex Monge-Ampère EquationRicci Flow