the problem of toroidalization of morphisms: a step forward

Toroidal varieties are algebraic varieties that are locally (for-mally) toric in structure, and toroidal morphisms are those morphisms ofvarieties which are locally determined by toric morphisms. the problem oftoroidalization, proposed rst in [1], is to construct a toroidal lifting of adominant morphism φ : x ! y of algebraic varieties by blowing up non-singular subvarieties in the target and domain. this problem is evidentlyvery dicult, and it has been solved only when y is a curve, or when φ isdominant and x; y are of dimension ⩽ 3 { see [7]. this article provides acomprehensive survey of the toroidalization problem. in addition, we discusssome recent results in toroidalization of locally toroidal morphisms [2], whichis among patching type problems.