Robert Lazarsfeld: Local syzygies of multiplier ideals

The multiplier ideal of a function f is an ideal that measures in a somewhat subtle manner the singularities of f. Thanks largely to the vanishing theorems they satisfy, multiplier ideals have found many applications in recent years to both local and global algebraic geometry, and to commutative algebra. Because of their importance, there has been interest -- especially from the algebraic side of the field -- in the question of which ideals can be realized as multiplier ideals. Multiplier ideals are integrally closed, but up to now have not been known to satisfy any other local properties. I will discuss a theorem with Kyungyong Lee to the effect that multiplier ideals satisfy some unexpected conditions involving higher syzygies. It follows that in dimensions three or higher, multiplier ideals are in fact quite special among all integrally closed ideals.