Claire Voisin

Collège de France


Election Year: 2016
Primary Section: 11, Mathematics
Membership Type: Foreign Associate

Biosketch

Claire Voisin is an algebraic geometer recognized for her work on Hodge theory and algebraic cycles. She is known particularly for her construction of compact Kahler manifolds not homeomorphic to complex projective manifolds, for her proof of the generic Green conjecture on syzygies of canonical curves, and for her contribution to the stable Luroth problem. Voisin was born in the north suburbs of Paris and grew up there. She entered Ecole Normale Superieure in 1981 and she defended her PhD thesis in 1986 under the supervision of Arnaud Beauville. She then got a permanent position at CNRS, that she kept until 2016 where she became Professor at College de France (Algebraic geometry chair). She has been invited as distinguished visiting professor at IAS (Princeton, 2014-2015) and Senior fellow of ETH (Zurich 2017).  She is a member of the Academie des sciences (Paris), and foreign member of the Accademia Nazionale dei Lincei.

Research Interests

A subject central in Voisin's research is the topology of algebraic varieties. A major tool is the notion of Hodge structure introduced by Griffiths. Voisin used the fact that the Hodge decomposition is compatible with the algebra structure on cohomology, in order to show that some cohomology algebras of compact Kahler manifolds are not the cohomology algebras of smooth projective varieties.

There are two classical results on the geometry of canonical curves: the Noether theorem characterizing hyperelliptic curves and the Petri theorem characterizing trigonal or plane quintic curves. M. Green extrapolated these results by conjecturing that the shape of the minimal resolution of the canonical ring of a curve is explicitly determined by the Clifford index of the curve. Voisin proved the Green conjecture for generic curves.

A classical problem called the stable Lüroth problem asks whether a unirational variety must be stably rational. The answer to this question is "no" in dimension at least 3. This was proved by Artin and Mumford who exhibited a stable birational invariant for smooth projective varieties. Voisin used the decomposition of the diagonal with integral coefficients as a stronger necessary condition for stable rationality. She proved by a simple but powerful degeneration argument that a very general quartic double solid has no cohomological decomposition of the diagonal while its Artin-Mumford is trivial.

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