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Explore the Calabi problem, a fundamental challenge in differential and algebraic geometry first popularized by Eugenio Calabi in his 1954 ICM lecture, through this mathematical seminar. Discover how this problem investigates which compact complex manifolds can admit Kähler-Einstein metrics, a special type of constant-curvature metric that sits at the intersection of geometric analysis and algebraic geometry. Learn about the historical solutions achieved in the 1970s by Yau, Aubin, and Yau for manifolds with flat or negative curvature, which confirmed Calabi's original predictions. Examine the more complex case of positively curved projective manifolds, known as Fano manifolds, where the existence of Kähler-Einstein metrics is not guaranteed and presents ongoing challenges. Understand the surprising connections that have emerged in recent decades between the existence of these special metrics and birational geometry, which have led to significant theoretical advances. Gain insight into current research developments and the state-of-the-art approaches being used to tackle this problem, particularly in three-dimensional cases, as presented by Carolina Araujo from IMPA.
