In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in ${\open P}^2\times{\open P}^2$ defined over ${\open C}$ and with Picard number 3. We describe the group of automorphisms ${\cal A}={\rm Aut}(V/{\open C})$ on V. For an ample divisor D and an arbitrary curve C0 on V, we investigate the asymptotic behavior of the quantity $N_{{\cal A}(C_0)}(t)=\#\{C \in {\cal A}(C_0): C\cdot D<t\}$. We show that the limit $\lim_{t\to \infty } {\log N_{{\cal A}(C\,)}(t)\over \log t}=\alpha\fleqno{}$ exists, does not depend on the choice of curve C or ample divisor D, and that .6515<α<.6538.

We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel–Jacobi map for submanifolds.

Some curvature conditions about the geodesics emanating from a submanifold are obtained. These conditions are used to to study the topological and geometric properties of the ambient spaces which admit some minimal submanifolds.

We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.