BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Carolina Araujo (Impa) DTSTART:20230426T173000Z DTEND:20230426T183000Z DTSTAMP:20260423T021651Z UID:BRAG/68 DESCRIPTION:Title: Th e Calabi Problem for Fano Threefolds\nby Carolina Araujo (Impa) as par t of Brazilian algebraic geometry seminar\n\n\nAbstract\nThe Calabi Proble m is a formidable problem in the confluence of differential and algebraic geometry. It asks which compact complex manifolds admit a Kähler-Einstein metric. A necessary condition for the existence of such a metric is that the canonical class of the manifold has a definite sign. For manifolds wit h zero or positive canonical class\, the Calabi problem was solved by Yau and Aubin/Yau in the 1970s. They confirmed Calabi's prediction\, showing t hat these manifolds always admit a Kähler-Einstein metric. On the other h and\, for projective manifolds with negative canonical class\, called “F ano manifolds”\, the problem is much more subtle: Fano manifolds may or may not admit a Kähler-Einstein metric. The Calabi problem for Fano manif olds has attracted much attention in the last decades\, resulting in the f amous Yau-Tian-Donaldson conjecture. The conjecture\, which is now a theor em\, states that a Fano manifold admits a Kähler-Einstein metric if and o nly if it satisfies a sophisticated algebro-geometric condition\, called “K-polystability”. In the last few years\, tools from birational geome try have been used with great success to investigate K-polystability. In t his talk\, I will present an overview of the Calabi problem\, the recent c onnections with birational geometry\, and the current state of the art in dimension 3.\n LOCATION:https://researchseminars.org/talk/BRAG/68/ END:VEVENT END:VCALENDAR