The log canonical threshold and rational singularities

Publication date: 16 February 2022

Abstract: We show that if

f

is a nonzero, noninvertible function on a smooth complex variety

X

and

J_{f}

is the Jacobian ideal of

f

, then

m l c t (f, J_{f}^{2}) > 1

if and only if the hypersurface defined by

f

has rational singularities. Moreover, if it does not have rational singularities, then

m l c t (f, J_{f}^{2}) = m l c t (f)

. We give two proofs, one relying on arc spaces and one that goes through the inequality

w i d e t i l d e a l p h a (f) g e q m l c t (f, J_{f}^{2})

, where

w i d e t i l d e a l p h a (f)

is the minimal exponent of

f

. In the case of a polynomial over

o v e r l i n e m a t h b f Q

, we also prove an analogue of this latter inequality, with

w i d e t i l d e a l p h a (f)

replaced by the motivic oscillation index

m m o i (f)

. We also show a part of Igusa's strong monodromy conjecture, for poles larger than

- m l c t (f, J_{f}^{2})

. We end with a discussion of lct-maximal ideals: these are ideals

I

with the property that

m l c t (I) < m l c t (J)

for every

J

with

I s u b s e t n e q J

.

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