Narrow Results

Let X ⊂ ℙ^2m+1 be a projective variety with isolated singularities, complete intersection of a smooth hypersurface of degree k, with a smooth hypersurface F of degree n > k. Denote by NS_m(F) and NS_m(X) the mth Néron–Severi groups. We prove that if rkNS_m(F) = 1, then rkNS_m(X) = 1. Moreover, we prove that if is general and n > max{k, 2m + 1}, then the natural map NS_m(X) ⊗ ℚ → NS_m(F) ⊗ ℚ is surjective. When X is a threefold, we deduce that X is factorial if and only if rkNS₂(F) = 1. This allows us to prove the existence of factorial threefolds with many singularities.

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

A well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi–Yau threefolds.