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

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 $\mathbb{C}<sup>*$ 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 $\mathbb{C}<sup>*$-invariant, embedded complex submanifold. We deduce the existence of a Bialynicki-Birula decomposition for a $\mathbb{C}<sup>*$-invariant, closed, complex analytic subspace of complex manifold with a $\mathbb{C}<sup>*$ 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.