Geometry and complex manifolds

Complex manifolds are spaces that locally resemble ordinary Euclidean space but are governed by the rules of complex numbers, and much of modern research concerns which of these spaces admit a Kähler metric — a particularly well-behaved geometric structure in which distance, angles, and complex structure fit together harmoniously. A central challenge is the Yau–Tian–Donaldson conjecture, which links the existence of such metrics with canonical curvature properties to an algebraic notion called K-stability, forging a deep bridge between differential geometry and algebraic geometry. Calabi–Yau manifolds, whose Ricci curvature vanishes identically, sit at the heart of this program and carry independent significance in theoretical physics as the geometric ingredient in string compactifications, while Fano manifolds — their positively curved cousins — remain the focus of intense investigation into when Kähler–Einstein metrics exist and how Ricci flow deforms a manifold's geometry toward them. Open directions include understanding the behavior of the complex Monge–Ampère equation at the boundary of well-behaved cases, characterizing singularities that develop along Ricci flow, and extending stability criteria to broader classes of non-smooth or degenerate spaces.

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Keywords

Kähler MetricsStabilityScalar CurvatureK-StabilityComplex Monge-Ampère EquationRicci Flow