Frobenius-Ehresmann structures and Cartan geometries in positive characteristic

Abstract

The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of étale coordinate charts on varieties modeled on homogenous spaces of algebraic groups, which we call Frobenius-Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius-Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann-Weil-Thurston principle for Frobenius-Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius-Ehresmann structure are completely determined by their monodromy crystals. Contents1. Introduction 1 2. Frobenius-Ehresmann structures 11 3. Cartan geometries in positive characteristic 16 4. Indigenous bundles on smooth schemes 34 5. Classification for algebraic curves 54 6. Deformation theory of (dormant) indigenous bundles 57 7. Appendix: Connections and PD stratifications on a G-bundle 79 References 93

Deformation theory of holomorphic Cartan geometries

“…We thank Yasuhiro Wakabayashi who pointed out that the kernel of the exact sequence in the statement of the Theorem 3.4 is not the correct one (see[Wa, Remark 6.4.5]). It should be replaced with H 0 (X, Hom(T X, ad(E G ))).…”