Computations of Multiplier Ideals via Bernstein-sato Polynomials

Multiplier ideals are very important in higher dimensional geometry to study the singularities of ideal sheaves. It reflects the singularities of the ideal sheaves and provides strong vanishing theorem called the Kawamata-Viehweg-Nadel vanishing theorem (see [3]). However, the multiplier ideals are defined via a log resolution of the ideal sheaf and divisors on the resolved space, and it is difficult to calculate them except some cases. In the case of monomial ideals or principal ideals generated by a non-degenerate polynomial, one can construct a log resolution of the ideal in the category of toric varieties, and there is a combinatorial description of multiplier ideals (see [2], [3]). On the other hand, Budur, Mustaţă, and Saito introduced generalized BernsteinSato polynomials (or b-function) of arbitrary varieties in [1], and provide the description of multiplier ideals in terms of b-functions using the theory of the V -filtration of Kashiwara and Malgralge (Theorem 2.6). This description is convenient for computations using the theory of computer algebra. In the case where the ideal is a principal ideal, algorithms for b-function are known using the theory of Gröbner bases in Weyl algebras ([5], [6], [7] and [9]). We will generalize the algorithms to the case of arbitrary ideals (Theorem 3.3 and Theorem 3.5). Our algorithm induced an algorithm for solving membership problem for multiplier ideals thanks to Theorem 2.6. In particular, we obtain an algorithm for the log canonical thresholds of arbitrary ideals. Finally, we present some examples computed with our algorithm.