Bialynicki-Birula theory, Morse-Bott theory, and resolution of singularities for analytic spaces

TL;DR

This paper extends Bialynicki-Birula and Morse-Bott theories to complex analytic spaces with holomorphic ^* actions, including non-compact cases and singularities, providing new decompositions and geometric insights.

Contribution

It develops the theory of Bialynicki-Birula decompositions for complex analytic spaces with ^* actions, including non-compact and singular cases, and explores their functorial properties.

Findings

Extended Bialynicki-Birula decompositions to non-compact complex manifolds.

Established functorial properties under blowup along ^*-invariant submanifolds.

Derived geometric consequences for fixed point data and singularities.

Abstract

Our goal in this work is to develop aspects of Bialynicki-Birula and Morse-Bott theory that can be extended from the classical setting of smooth manifolds to that of complex analytic spaces with a holomorphic $\mathbb{C}^*$ action. We extend prior results on existence of Bialynicki-Birula decompositions for compact, complex K\"ahler manifolds to non-compact complex manifolds and develop functorial properties of the Bialynicki-Birula decomposition, in particular with respect to blowup along a $\mathbb{C}^*$-invariant, embedded complex submanifold. We deduce the existence of a Bialynicki-Birula decomposition for a $\mathbb{C}^*$-invariant, closed, complex analytic subspace of complex manifold with a $\mathbb{C}^*$ action; derive geometric consequences for the positivity of the Bialynicki-Birula nullity, co-index, and index at a fixed point; and we develop stronger versions of these results by applying resolution of singularities for analytic spaces.