The paper is Part III of our ongoing project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle, which reduces a statement in Gromov--Witten theory on non-split bundles to the case of split bundles.

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models. In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the Kähler moduli space for ordinary flops of splitting type. In Part III, together with F. Qu, we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.

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Let X be an equivariant embedding of a connected reductive group X over an algebraically closed field X of positive characteristic. Let X denote a Borel subgroup of X . A X -Schubert variety in X is a subvariety of the form $\diag (G)\cdot V $, where X is a X -orbit closure in X . In the case where X is the wonderful compactification of a group of adjoint type, the X -Schubert varieties are the closures of Lusztig's X -stable pieces. We prove that X admits a Frobenius splitting which is compatible with all X -Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any X -Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that X -Schubert varieties have nice singularities we present an example of a non-normal X -Schubert variety in the wonderful compactification of a group of type X . Finally we also extend the Frobenius splitting results to the more general class of X -Schubert varieties.