We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an ℓ-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number.

We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Lê, Ginsburg and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara and Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.

Let f be a complex polynomial mapping. We relate the behaviour of f ’at infinity‘ to the characteristic cycle associated to the projective closures of fibres of f. We obtain a condition on the characteristic cycle which is equivalent to a condition on the asymptotic behaviour of some of the minors of the Jacobian matrix of f. This condition generalizes the condition in the hypersurface case known as Malgrange‘s condition. The relation between this condition and the behavior of the characteristic cycle is a partial generalization of Parusinski‘s result in the hypersurface case. We show that the new condition implies the C$^∞$-triviality of f.