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.

In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are examples of threefolds that are both supersingular and rationally chain connected.

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The celebrated Mirror Theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of its mirror dual orbifold. In this article, we establish a mirror-dual statement. Namely, the B model of the Fermat quintic threefold is shown to be equivalent to the A model of its mirror, and hence establishes the mirror symmetry as a true duality.

This an expository article on Givental's axiomatic Gromov--Witten theory and some of its applications.