Uniform Stability of Twisted Constant Scalar Curvature Kähler Metrics

We introduce a norm on the space of test configurations, called the minimum norm. We conjecture that uniform K-stability is equivalent to the existence of a constant scalar curvature Kähler metric. This uniformity is analogous to coercivity of the Mabuchi functional. We show that a test configuration has zero minimum norm if and only if it has zero L2 -norm, if and only if it is almost trivial. We prove the existence of a twisted constant scalar curvature Kähler metric that implies uniform twisted K-stability with respect to the minimum norm. We give algebro-geometric proofs of uniform K-stability in the general type and Calabi-Yau cases, as well as Fano case under an alpha invariant condition. Our results hold for nearby line bundles, and in the twisted setting.The author was funded by a studentship associated to an EPSRC Career Acceleration Fellowship (EP/J002062/1).This is the author accepted manuscript. The final version is available from Oxford University Press via http://dx.doi.org/10.1093/imrn/rnv29

Stability conditions for polarised varieties

We introduce an analogue of Bridgeland's stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical K\"ahler metric, modelled on the constant scalar curvature K\"ahler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically Z-stable admits Z-critical K\"ahler metrics in the large volume regime. We also prove a local converse, and explain how these results can be viewed in terms of local wall crossing. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang-Mills equation.Comment: v2: linear analysis section corrected, minor changes otherwise. 65 page

Stability conditions in geometric invariant theory

We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic notion of a central charge on a scheme with a group action, and ultimately to a notion of a stability condition on a stack analogous to that on an abelian category. We use these ideas to introduce an axiomatic notion of a stability condition for polarised schemes, defined in such a way that K-stability is a special case. In the setting of axiomatic geometric invariant theory on a smooth projective variety, we produce an analytic counterpart to stability and explain the role of the Kempf-Ness theorem. This clarifies many of the structures involved in the study of deformed Hermitian Yang-Mills connections, Z-critical connections and Z-critical K\"ahler metrics.Comment: 26 pages, appendix by Andr\'es Ib\'a\~nez N\'u\~nez added, other minor correction

The universal structure of moment maps in complex geometry

We introduce a natural, geometric approach to constructing moment maps in finite and infinite-dimensional complex geometry. This is applied to two settings: K\"ahler manifolds and holomorphic vector bundles. Our new approach exploits the existence of universal families and the theory of equivariant differential forms. In the setting of K\"ahler manifolds we first give a new, geometric proof of Donaldson-Fujiki's moment map interpretation of the scalar curvature. Associated to arbitrary products of Chern classes of the manifold-namely to a central charge-we then introduce a geometric PDE determining a Z-critical K\"ahler metric, and show that these general equations also satisfy moment map properties. On holomorphic vector bundles, using a similar strategy we give a geometric proof of Atiyah-Bott's moment map interpretation of the Hermitian Yang-Mills condition. We go on to give a new, geometric proof that the PDE determining a Z-critical connection-associated to a choice of central charge on the category of coherent sheaves-can be viewed as a moment map; deformed Hermitian Yang-Mills connections are a special case, in which our work gives a geometric proof of a result of Collins-Yau. Our main assertion is that this is the canonical way of producing moment maps in complex geometry-associated to any geometric problem along with a choice of stability condition-and hence that this accomplishes one of the main steps towards producing PDE counterparts to stability conditions in large generality.Comment: 41 page

Alpha Invariants and K-Stability for General Polarizations of Fano Varieties

We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian’s alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polarised case, which is the algebraic counterpart of Tian’s analytic criterion implying the existence of a K¨ahler-Einstein metric. As an application, we give new K-stable polarisations of a general degree one del Pezzo surface. We also prove a corresponding result for log K-stability.Supported by a studentship associated to an EPSRC Career Acceleration Fellowship (EP/J002062/1)This is the author accepted manuscript. The final version is available from Oxford University Press via http://dx.doi.org/10.1093/imrn/rnu16