

2000  PhD, Versailles University, France  2001  2002  Postdoc, University of Cologne  2002   Maître de conférences, University of Paris VI, France  2007   Professor (W2, Bonn Junior Fellow), University of Bonn  2010  Habilitation, University of Paris VI, France 


I work at the intersection of representation theory and algebraic geometry. I have been working on the singularities of Schubert varieties and the existence of small resolutions which allow to compute KahzdanLuzstig polynomials. I have also been working on enumerative geometry of homogeneous spaces and especially quantum cohomology. On this subject I computed the quantum cohomology of new varieties and discovered some unexpected symmetries.
I have also been interested in the geometry of low codimension subvarieties in homogeneous spaces (see [8] and [7]).


ANR (French National Research Agency) project: “New Symmetries in GromovWitten Theory”
Member


Former Research Area F In [1], we study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points GromovWitten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In [2], we prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, once localized at the quantum parameter, has a nontrivial involution mapping Schubert classes to multiples of Schubert classes. This can be stated as a strange duality property for the GromovWitten invariants, which turn out to be very symmetric.
In [3], we prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q=1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in a previous paper. We deduce VafaIntriligator type formulas for the GromovWitten invariants.
For a variety with an open orbit, we define its boundary as the complement of the open orbit. The action sheaf is the subsheaf of the tangent sheaf made of vector fields tangent to . We prove in [4], for a large family of smooth spherical varieties, the vanishing of the cohomology groups for , extending results of F. Bien and M. Brion.
In [5], we show that for any minuscule or cominuscule homogeneous space X, the GromovWitten varieties of degree d curves passing through three general points of X are rational or empty for any d. Applying techniques of A. Buch and L. Mihalcea to constructions of the authors together with L. Manivel, we deduce that the equivariant Ktheoretic three points GromovWitten invariants are equal to classical equivariant Ktheoretic invariants on auxiliary spaces.
In [6], we decompose the fibers of the Springer resolution for the odd nilcone of the Lie superalgebra into locally closed subsets. We use this decomposition to prove that almost all fibers are connected. However, in contrast with the classical Springer fibers, we prove that the fibers can be disconnected and non equidimensional.
In [7], we prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in a homogeneous space. Using generalizations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in where is any isotropic Grassmannian. We also deduce simple connectedness properties for subvarieties of . 


[ 1] P. E. Chaput, L. Manivel, N. Perrin
Quantum cohomology of minuscule homogeneous spaces Transform. Groups , 13: (1): 4789 2008 DOI: 10.1007/s0003100890015[ 2] PierreEmmanuel Chaput, Laurent Manivel, Nicolas Perrin
Quantum cohomology of minuscule homogeneous spaces. II. Hidden symmetries Int. Math. Res. Not. IMRN (22): Art. ID rnm107, 29 2007 DOI: 10.1093/imrn/rnm107[3] P. E. Chaput, L. Manivel, N. Perrin
Quantum cohomology of minuscule homogeneous spaces III. Semisimplicity and consequences Canad. J. Math. , 62: (6): 12461263 2010 DOI: 10.4153/CJM20100509 [ 4] Boris Pasquier, Nicolas Perrin
Local rigidity of quasiregular varieties Math. Z. , 265: (3): 589600 2010 DOI: 10.1007/s002090090531x[ 8] Nicolas Perrin
Small codimension smooth subvarieties in evendimensional homogeneous spaces with Picard group \Bbb Z C. R. Math. Acad. Sci. Paris , 345: (3): 155160 2007 DOI: 10.1016/j.crma.2007.06.012



2007  Schubert calculus and Schubert geometry, Banff, AB, Canada  2009  Combinatorial, Enumerative and Toric geometry, Mathematical Sciences Research Institute (MSRI), Berkeley, CA, USA  2009  RomanianGerman symposium on moduli spaces in Geometry and Physics, Sibiu (Hermannstadt), Romania  2010  Algebraic Groups, Oberwolfach, Germany  2010  Classical Algebraic Geometry, Oberwolfach, Germany  2010  Complex Algebraic Geometry, Paris, France  2010  Complex Geometry, Vector bundles, Algebraic varieties with a group action, Algebraic Cycles, Hyderabad, India 


Piotr Achinger (2015): “K(pi, 1) Spaces in Algebraic Geometry”,
now EPDI Postdoc, Banach Center, Warsaw, Poland


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