Thursday 28 May 2026 at 4:30 pm, as part of the “Mathematics Colloquium of Milano” series, Claire Voisin (IMJ-PRG, Paris) will give a talk entitled Points and zero-cycles of algebraic varieties.
The event is aimed at a general mathematical audience, in particular PhD students and Post-docs and will take place in Room U4-01 (U4 Building | Piazza della Scienza 4, Milano). An apero will follow.
Abstract: An algebraic variety X is defined by polynomial equations in an ambient space. The defining equations have coefficients in a field K. When K is not algebraically closed, there might be no points of X defined over K, but there are always points defined over finite extensions of K, if X is non-empty. The study of these points is of interest both for geometry and arithmetic. I will introduce the Chow group of zero-cycles of X, which is built on these points and gives a partial way of computing them, and its universal Chow group. The Chow group CH0(X) of zero-cycles onX contains very important information. If the field K is the field of complex numbers, one knows that it is related to holomorphic forms on X (Mumford). Over non-algebraically closed fields, the universal Chow group of zero-cycles is related to rationality properties of X. I will discuss the notion of Mumford finite vs. infinite dimensionality of CH0 and, on the arithmetic side, the recently developed notion of boundedness vs. unboundedness of CH0. I will also present related boundedness vs. unboundedness statements about points on Fano hypersurfaces.
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Points and zero-cycles of algebraic varieties is the 1st of four meetings of the colloquia cycle Mathematics Colloquium of Milano - Breakthroughs and Perspectives organised by the Departments of Mathematics of the University of Milano, Polytechnic of Milano, University of Milano-Bicocca and Bocconi University, in which internationally renowned mathematicians present and discuss the most important ideas and breakthroughs in contemporary mathematics and its applications.
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