Abstract. We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties (X, L); in particular for K-and Chow stability. For each type of stability this leads to a concept of slope µ for varieties and their subschemes; if (X, L) is semistable then µ(Z) ≤ µ(X) for all Z ⊂ X. We give examples such as curves, canonical models and CalabiYaus. We prove various foundational technical results towards understanding the converse, leading to partial results; in particular this gives a geometric (rather than combinatorial) proof of the stability of smooth curves.

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope µ for a projective manifold and for each of its subschemes, and show that if X is cscK then µ(Z) ≤ µ(X) for all subschemes Z.This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as del Pezzo surfaces and projective bundles. If P(E) → B is a projective bundle which admits a cscK metric in a rational Kähler class with sufficiently small fibres, then E is a slope semistable bundle (and B is a slope semistable polarised manifold). The same is true for all rational Kähler classes if the base B is a curve.We also show that the slope inequality holds automatically for smooth curves, canonically polarised and Calabi Yau manifolds, and manifolds with c1(X) < 0 and L close to the canonical polarisation. 1

Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle L over a compact manifold. Using this we associate weak geodesics to suitable filtrations of the algebra of sections of L. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how this recovers the weak geodesic ray of Phong-Sturm.Theorem 1.2. Suppose that F k,λ is left continuous and decreasing in λ and bounded (see (7.2)). Then there is a well-defined limitwhere the star denotes taking the upper semicontinuous regularization after the limit. Furthermore this limit is maximal except possibly for one critical value of λ, and its Legendre transform is a weak geodesic ray. CONVEX MOTIVATION 5Bedford-Taylor which says that such envelopes are maximal (Theorem 4.10). This is then extended to the case of a test curve of singularities, and in Section 6 we discuss the Legendre transform and prove Theorem 1.1. Following these analytic results, we move on to the algebraic picture. In Section 7 we associate a test curve to a suitable filtration of the coordinate ring of (X, L), and prove Theorem 1.2. We then recall how such filtrations arise from test configurations, and in Section 9 show how this agrees with the construction of Phong and Sturm.

Abstract. We formulate a notion of K-stability for Kähler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies K-semistability (resp. uniformly K-stable). In particular this shows that the existence of a constant scalar curvature Kähler metric implies K-semistability, and K-stability if one assumes the automorphism group is discrete. We also show how Stoppa's argument holds in the Kähler case, giving a simpler proof of this K-stability statement.

We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (nonreductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds.We then prove an orbifold version of Donaldson's theorem: the existence of an orbifold Kähler metric of constant scalar curvature implies K-semistability.By extending the notion of slope stability to orbifolds we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold Kähler metrics. We describe the manifold applications of this orbifold result, and show how many previously known results (Troyanov, Ghigi-Kollár, Rollin-Singer, the AdS/CFT Sasaki-Einstein obstructions of Gauntlett-Martelli-Sparks-Yau) fit into this framework.