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 denote an equivariant embedding of a connected reductive group G over an algebraically closed field k. Let B denote a Borel subgroup of G and let Z denote a B B-orbit closure in X. When the characteristic of k is positive and X is projective we prove that Z is globally F-regular. As a consequence, Z is normal and CohenMacaulay for arbitrary X and arbitrary characteristics. Moreover, in characteristic zero it follows that Z has rational singularities. This extends earlier results by the second author and M. Brion.

We describe the closure of the strata of Abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne–Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.