On the restriction formula

TL;DR

This paper investigates conditions under which the restriction formula for multiplier ideal sheaves and complex singularity exponents becomes an equality, demonstrating that equality holds outside a measure zero set in a projective space.

Contribution

It proves that the restriction formula equality holds for a generic choice of submanifold outside a measure zero set in a projective space.

Findings

Equality in the restriction formula holds outside a measure zero set.

The result applies to multiplier ideal sheaves and complex singularity exponents.

The approach uses genericity in a projective space setting.

Abstract

Let $\varphi$ be a quasi-psh function on a complex manifold $X$ and let $S\subset X$ be a complex submanifold. Then the multiplier ideal sheaves $\mathcal{I}(\varphi|_S)\subset\mathcal{I}(\varphi)|_{S}$ and the complex singularity exponents $c_{x}\left(\varphi|_{S}\right)\leqslant c_{x}(\varphi)$ by Ohsawa-Takegoshi $L^{2}$ extension theorem. An interesting question is to know whether it is possible to get equalities in the above formulas. In the present article, we show that the answer is positive when $S$ is chosen outside a measure zero set in a suitable projective space.