This work is devoted to the study of the foundations of quantum K-theory, a K-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary K-ring K(X) of a smooth projective variety X, analogous to the relation between quantum cohomology and ordinary cohomology. This new quantum product also gives a new class of Frobenius manifolds.

We present a new method to solve certain ar {\partial} -equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a ar {\partial} -lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at ar {\partial} -level, as well as certain injectivity theorem on compact Kahler manifolds.

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space P^n, in which case Bir(X) is the Cremona group of rank n, or when X⊂P^{n+1} is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank n⩾3 is not generated by linear and Jonquières elements.

We study the set of connected components of certain unions of affine Deligne-Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic $\s $-conjugacy classes. As an application, we verify the Axioms in\cite {HR} for certain PEL type Shimura varieties. We also show that for any nonbasic $\s $-conjugacy class in a residually split group, the set of connected components is" controlled" by the set of straight elements associated to the $\s $-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with\cite {Zhou}, this allows one to verify in the residually split case, the description of the mod- p isogeny classes on Shimura varieties conjectured by Langland and Rapoport. Along the way, we determine the Picard group of the Witt vector affine Grassmannian of\cite {BS} and\cite {Zhu} which is of independent interest.

We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.