Pierre-Emmanuel Chaput

On the quantum cohomology of adjoint varieties

We study the quantum cohomology of quasi-minuscule and quasi-cominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical principle we give presentations of the quantum cohomology algebras. These algebras are semi-simple for adjoint non coadjoint varieties and some properties of the induced strange duality are shown.

Quantum cohomology of minuscule homogeneous spaces III Semi-simplicity and consequences

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 Vafa-Intriligator type formulas for the Gromov-Witten invariants.

Quantum cohomology of minuscule homogeneous spaces II : Hidden symmetries

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, once localized at the quantum parameter, has a non trivial involution mapping Schubert classes to multiples of Schubert classes. This can be stated as a strange duality property for the Gromov-Witten invariants, which turn out to be very symmetric.

RATIONALITY OF SOME GROMOV–WITTEN VARIETIES AND APPLICATION TO QUANTUM K-THEORY

We show that for any minuscule or cominuscule homogeneous space X, the Gromov-Witten 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 K-theoretic three points Gromov-Witten invariants are equal to classical equivariant K-theoretic invariants on auxilliary spaces.

On the adjoint quotient of Chevalley groups over arbitrary base schemes

For a split semisimple Chevalley group scheme G with Lie algebra g over an arbitrary base scheme S, we consider the quotient of g by the adjoint action of G. We study in detail the structure of g over S. Given a maximal torus T with Lie algebra t and associated Weyl group W , we show that the Chevalley morphism π : t/W → g/G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient g → g/G commutes, or does not commute, with base change on S.

Stability of the Tangent Bundle of G/P in Positive Characteristics

Let G be an almost simple simply-connected affine algebraic group over an algebraically closed field k of characteristic p >  0. If G has type Bn, Cn or F4, we assume that p >  2, and if G has type G2, we assume that p >  3. Let P ⊂ G be a parabolic subgroup. We prove that the tangent bundle of G/P is Frobenius stable with respect to the anticanonical polarization on G/P.