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Jihun Yum
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We introduce the concept of Steinness index related to the Stein neighborhood basis. We then show several results: (1) The existence of Steinness index is equivalent to that of strong Stein neighborhood basis. (2) On the Diederich-Forn{ae}ss worm domains in particular, we present an explicit formula relating the Steinness index to the well-known Diederich-Forn{ae}ss index. (3) The Steinness index is 1 if a smoothly bounded pseudoconvex domain admits finitely many boundary points of infinite type.
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We characterize the Diederich-Fornaess index and the Steinness index in terms of a special 1-form, which we call DAngelo 1-form. We then prove that the Diederich-Fornaess and Steinness indices are invariant under CR-diffeomorphisms by showing CR-invariance of DAngelo 1-forms.
In this paper, we prove the semi-continuity theorem of Diederich-Forn{ae}ss index and Steinness index under a smooth deformation of pseudoconvex domains in Stein manifolds.
We propose the concept of Diederich--Forn{ae}ss and Steinness indices on compact pseudoconvex CR manifolds of hypersurface type in terms of the DAngelo 1-form. When the CR manifold bounds a domain in a complex manifold, under certain additional non-degeneracy condition, those indices are shown to coincide with the original Diederich--Forn{ae}ss and Steinness indices of the domain, and CR invariance of the original indices follows.
Let $varphi$ be a quasi-psh function on a complex manifold $X$ and let $Ssubset X$ be a complex submanifold. Then the multiplier ideal sheaves $mathcal{I}(varphi|_S)subsetmathcal{I}(varphi)|_{S}$ 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.
In these lectures, we give a pedagogical introduction to the superconformal index. This is the writeup of the lectures given at the Winter School YRISW 2020 and is to appear in a special issue of JPhysA. The lectures are at a basic level and are geared towards a beginning graduate student interested in working with the superconformal index.
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