The volume of an isolated singularity

Abstract

48 pages. v4: Appendix is new, plus several minor changes made throughout the paper following the referee's suggestions. Final version, to appear in Duke Math JWe introduce a notion of volume of a normal isolated singularity that generalizes Wahl's characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of non-invertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a non-invertible polarized endomorphism is pseudoeffective. Our techniques build on Shokurov's b-divisors. We define the notion of nef Weil b-divisors, and of nef envelopes of b-divisors. We relate the latter to the pull-back of Weil divisors introduced by de Fernex and Hacon. Using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by Takagi, we carry over to the isolated singularity case the intersection theory of nef Weil b-divisors formerly developed by Boucksom, Favre, and Jonsson in the smooth case

Singularities of varieties admitting an endomorphism

“…For the proof of this statement we will use the tools and terminology of [BdFF12]: given a canonical divisor K X on X, there is a unique canonical divisor K Xπ , for each birational model π : X π → X, with the property that π * K Xπ = K X . Thus we obtain a canonical b-divisor K X over X. Boucksom, de Fernex and Favre define the nef envelope Env X (−K X ) of the Weil divisor −K X as the largest nef Weil b-divisor Z that is both relatively nef over X and satisfies Z X ≤ −K X .…”

Entropy and flatness in local algebraic dynamics

“…Now since ϕ is contracting, using Lemma 4 we quickly see that the family tϕ nt pmqRu t N defines the m-adic topology of R. By Lemma 47 it suffices to verify equation (6) for this family of m-primary ideals. We need to show length R R{ϕ n pϕ nt pmqqR¨length R R{ϕ nt pmqR¨¤length R R{ϕ n pmqR¨.…”

Comparing multiplier ideals to test ideals on numerically ℚ‐Gorenstein varieties

“…A priori, one might need to take into account infinitely many resolutions to determine that we have computed J (X, Z). The condition of numerically Q-Gorenstein singularities, introduced and studied in [3,4], is designed to provide a work-around for this issue. The definitions of numerically Q-Cartier and numerically Q-Gorenstein given here follow [4]; they are equivalent to the definitions of numerically Cartier and numerically Gorenstein given in [3].…”

“…In [3,4], the authors introduce and study a new class of singularities, called numerically Q-Gorenstein singularities, which encompasses the class of Q-Gorenstein singularities. It turns out that the theory of multiplier ideals on normal varieties, as introduced in [5], becomes particularly simple on numerically Q-Gorenstein varieties.…”

“…Since every normal surface is numerically Q-Gorenstein (cf. [3,Example 2.29]), the theorem applies, in particular, to all pairs (X, Z) where X is any normal surface and Z is an effective Q-Cartier Q-divisor on X.…”