Beijing Institute of Mathematical Sciences and Applications

This course develops logarithmic and toroidal methods for studying degenerations, fibrations, and birational modifications of algebraic varieties. Beginning with toric varieties and the combinatorics of monoids, the course introduces log schemes, log regularity, log smoothness, and toroidal embeddings. It then develops alterations, toroidal modifications, saturated base change, and stable reduction for nodal curves, emphasising how geometric problems can be transformed into controlled local toric models and combinatorial data. The final part explores selected applications to modern birational geometry. Topics include the Shokurov-McKernan conjecture, toroidal models, irrationality of degenerations of Fano varieties, and boundedness of Stein degree of divisors on log Calabi-Yau fibrations. Throughout, emphasis is placed on the organising ideas, representative arguments, and interaction between local logarithmic structures and global birational geometry.