Quasi-Homogeneity of the Moduli Space of Stable Maps to Homogeneous Spaces

Christoph Mark Bärligea

Documenta mathematica, Tome 23 (2018), pp. 697-745

Résumé

Let G be a connected, simply connected, simple, complex, linear algebraic group. Let P be an arbitrary parabolic subgroup of G. Let X=G/P be the G-homogeneous projective space attached to this situation. Let d∈H2​(X) be a degree. Let M0,3​(X,d) be the (coarse) moduli space of three pointed genus zero stable maps to X of degree d. We prove under reasonable assumptions on d that M0,3​(X,d) is quasi-homogeneous under the action of G. The essential assumption on d is that d is a minimal degree, i.e. that d is a degree which is minimal with the property that qd occurs with non-zero coefficient in the quantum product σu​⋆σv​ of two Schubert classes σu​ and σv​, where ⋆ denotes the product in the (small) quantum cohomology ring QH∗(X) attached to X. We prove our main result on quasi-homogeneity by constructing an explicit morphism which has a dense open G-orbit in M0,3​(X,d). To carry out the construction of this morphism, we develop a combinatorial theory of generalized cascades of orthogonal roots which is interesting in its own right.

Détail

DOI : 10.4171/dm/631

Classification : 14H10, 14H45, 14M15, 14N10
Mots-clés : homogeneous spaces, moduli space of stable maps, quasi-homogeneity, curve neighborhoods, minimal degrees in quantum products