In Chapter 1 we recall some basic notions of complex and K ̈ahler geometry and we introduce
some technical results regarding cscK metrics that we will use intensively in successive chapters.
We also explain in detail what kind of result we want to prove and the strategy of the proof. We
warmly suggest to read section 1.6.2 where we give a detailed overview of the proof of Theorem
1.7.
In Chapter 2 we investigate the properties of particular linear differential operators on cscK
manifolds. More precisely we study their invertibility properties between weighted H ̈older spaces.
In Chapter 3 we begin the proof of our main result. With tools we introduced in chapter 2
we construct families, depending on some parameters, of cscK metrics on particular manifolds
with boundary.
In Chapter 4 we finish the proof we started in the preceding chapter. To conclude the proof
we perform the connected sum construction along the boundaries of the manifolds we chose in
chapter 3 and we glue the families of cscK metrics we constructed on them. To glue the families
of metrics we use the technique known as Cauchy data matching. We also discuss the proof of
Theorem 4.2. In Chapter 5 we look for examples of cscK orbifolds satisfying assumptions of Theorem 1.7.
We focus our attention on toric 3-folds and it turns out that there is no toric three-dimensional
orbifold satisfying our requests.
In Chapter 6 we discuss the extension of Theorem 1.7 to 2-dimensional orbifolds and the
relative technical issues. We discuss, moreover, some conjectures and ideas for future work.
