An invitation to toric degenerations

Abstract

In [GrSi2] we gave a canonical construction of degenerating families of varieties with effective anticanonical bundle. The central fibre X of such a degeneration is a union of toric varieties, glued pairwise torically along toric prime divisors. In particular, the notion of toric strata makes sense on the central fiber. A somewhat complementary feature of our degeneration is their toroidal nature near the 0-dimensional toric strata of X; near these points the degeneration is locally analytically or in the etale topology given by a monomial on an affine toric variety. Thus in this local model the central fiber is a reduced toric divisor. A degeneration with these two properties is called a toric degeneration. The name is probably not well-chosen as it suggests a global toric nature, which is not the case as we will emphasize below. A good example to think of is a degeneration of a quartic surface in P3 to the union of the coordinate hyperplanes. More generally, any Calabi-Yau complete intersection in a toric variety has toric degenerations [Gr2]. Thus the notion of toric degeneration is a very versatile one, conjecturally giving all deformation classes of Calabi-Yau varieties with maximally unipotent boundary points. Our construction has a number of remarkable features. It generalizes the construction of a polarized toric variety from an integral polyhedron (momentum polyhedron)