Geometry and complex manifolds

Complex manifolds are spaces that locally resemble ordinary Euclidean space but carry a richer geometric structure, and much of modern research concerns which of these spaces admit a *Kähler metric* — a particularly well-behaved notion of distance and curvature that ties together differential geometry, complex analysis, and algebraic geometry. A central problem, originating in conjectures of Calabi and resolved in landmark cases by Yau and others, asks when a manifold admits a metric of constant or prescribed scalar curvature, with the answer turning out to depend on subtle algebraic conditions called K-stability. The complex Monge-Ampère equation — a highly nonlinear PDE — lies at the heart of constructing such metrics, and tools like Ricci flow are used to deform metrics toward canonical ones, most notably in the study of Fano manifolds and Calabi-Yau manifolds, the latter of which appear as the geometric backbone of string-theoretic models in physics. Active directions include understanding precisely how K-stability governs the existence of Kähler-Einstein metrics on singular or non-smooth spaces, and extending the analytic techniques developed for smooth manifolds to these more general settings.

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Keywords

Kähler MetricsStabilityScalar CurvatureK-StabilityComplex Monge-Ampère EquationRicci Flow