We present a microlocal version of the Riemann–Hilbert correspondence for regular holonomic $\cal D$-modules. We show that a regular holonomic system of microdifferential equations is associated to a perverse sheaf concentrated in degree 0. Moreover, we show that this perverse sheaf can be recovered from the local system it determines on the complementary of its singular locus. We characterize the classes of perverse sheaves and local systems associated to regular holonomic systems of microdifferential equations.

We compute relations of rational equivalence among special codimension 2 cycles on families of Abelian surfaces using elements of a higher Chow group. These relations are similar to those between Heegner points and special divisors obtained by Zagier, Van der Geer and others.

Stark conjectured that given a positive integer h, there are only a finite number of CM-fields L with class number equal to h. We prove this conjecture for fields L of degree [ges ]6 whose normal closure is solvable.

Let M be a manifold of X = $\Bbb C$n, A a small analytic disc attached to M, zo a point of ∂A where A is tangent to M, z1 another point of ∂A where M extends to a germ of manifold M1 with boundary M. We prove that CR functions on M which extend to M1 at z1 also extend at zo to a new manifold M2. The directions M1 and M2 point to, are related by a sort of connection associated to A which is dual to the connection obtained by attaching ‘partial analytic lifts’ of A to the co-normal bundle to M in X.

The connection between these Fano 3-folds and plane quartic curves is explained.

We prove a generalisation of a theorem of Nagata on ruled surface to the case of the fiber bundle E/P → X, associated to a principal G-bundle E. Using this we prove boundedness for the isomorphism classes of semi-stable G-bundles in all characteristics.

We present an analogy of the famous formula that the square of the Pfaffian is equal to the determinant for an alternating matrix for the case where the entries are the generators of the orthogonal Lie algebras. This identity clarifies the relation between the two sets of central elements in the enveloping algebra of the orthogonal Lie algebras. We employ systematically the exterior calculus for the proof.