Blowing up Kähler manifolds with constant scalar curvature II
Claudio Arezzo, Frank Pacard
TL;DR
The paper extends existence results for constant scalar curvature Kähler ($Kcsc$) metrics on blow-ups $\mathrm{Bl}_{p_1,...,p_m} M$ of a compact $Kcsc$ manifold $M$, focusing on base manifolds with holomorphic vector fields and giving sufficient conditions on the blow-up points to preserve $Kcsc$ using a gluing construction with a zero-scalar-curvature model on $N=\mathrm{Bl}_0 \mathbb{C}^n$. The approach analyzes deformations of the scalar curvature under $\omega = \omega_M + i\partial\bar{\partial}\varphi$, yielding the linear operator ${\mathbb L}_M\varphi = -\tfrac{1}{2}\Delta_M^2\varphi - \operatorname{Ric}_M \cdot \nabla_M^2\varphi$ with kernel spanned by constants and $\xi_1,...,\xi_d$; solvability and gluing conditions are controlled by the matrix $M(p_1,...,p_m)$ and nondegeneracy requirements such as $C_1 = d$ and $C_2 \neq 0$. The paper develops weighted Hölder spaces, deficiency spaces, and bi-harmonic extensions, and uses a fixed-point argument to construct CSC Kähler forms on the glued manifolds $M_\varepsilon$ after gluing in scaled copies of $N$, under the nondegeneracy condition ${\mathfrak M}(p_1,...,p_m) \neq 0$ and related positivity constraints. Explicit examples include optimal results for $M = \mathbb{P}^n$ and for $\mathbb{P}^n \times M$, with the kernel interpreted in terms of holomorphic vector fields and their lifting to the blow-up, and symmetry considerations yield invariant constructions enabling CSC metrics on blow-ups of projective space.
Abstract
In this paper we continue our study about the existence of Kaehler metrics of constant scalar curvature (Kcsc) on blow ups at points of compact manifolds with Kcsc metrics started in math.DG/0411522. In this second part we deal with the case of base manifolds with holomorphic vector fields and we give sufficient conditions for the position of points to be blown up to preserve the Kcsc property. These results are obtained by a gluing procedure. We give explicit examples of our construction, in particular getting optimal results in the cases of P^n and P^n x M, where M is a Kcsc compact manifold with discrete automorphism group.