arXiv research

Jet bundles as higher-order polarised $k$-contact manifolds

Develops k-contact geometry theory for field theories.

The paper studies Einstein-Hilbert functional and its relation to K-semistability.

We construct a moduli space of polarised manifolds which admit a constant scalar curvature Kähler metric. We show that this space admits a natural Kähler metric.

The study explores polarized deformations of SKT Calabi-Yau manifolds using Aeppli classes.

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 $μ(Z)\leμ(X)$ for all subschemes $Z$. This gives man…

Proves a conjecture for Calabi-Yau manifolds.

Analytic K-semistability connects curvature to metric existence.

The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.

Study connects mirror symmetry invariants to K-stability for toric manifolds.

It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant…

Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-Kähler) Hermitian metric $ω$, we introduce the concept of deformations of $X$ that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1…

Introduces stability conditions for polarized varieties, linking to K-stability.

The study finds resonance points in polarised curves with polynomial conserved quantities.

The study connects norms and filtrations on section rings of projective manifolds.

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we conjecture to be equivalent to the existence of an extremal metric in the polaris…

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 polari…

The paper studies HYM connections on stable vector bundles over Kähler manifolds.

For a stratified group $G$, we construct a class of polarised Lie groups, which we call modifications of $G$, that are locally contactomorphic to it. Vice versa, we show that if a polarised group is locally contactomorphic to a stratified group $G$, whose Lie algebra has finite Tanaka prolongation, then it must be a mo…

We show that a polarised manifold with a constant scalar curvature Kähler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson.

We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.

Introduces valuative stability for polarised varieties, equivalent to K-stability.

We define a new notion of "b-stability" for a polarised algebraic variety, adapted to the existence problem for Kahler-Einstein metrics on Fano manifolds.

Paper proves isoperimetric inequality for Minkowski spacetime.

Study shows Calabi-Yau metrics converge to a specific form under certain conditions.

We identify the difference between the CM polarisation and the Chow polarisation on the ``Hilbert scheme''. As a consequence, we give a numerical criterion for the CM stability as in Mumfords' G.I.T.. Also, we write down an explicit formula for the generalised futaki invariant interms of weights and multiplicities of t…

Uniform estimates for Calabi-Yau degenerations proved.

Equivalence proven between divisorial stability and quotient log divisorial stability.

Fix a K3 lattice $Λ$ of rank two and $L\inΛ$ a big and nef divisor that is positive enough. We prove that the generic $Λ$-polarised K3 surface has an integral nodal rational curve in the linear system $|L|$, in particular strengthening previous work of the first named author. The technique is by degeneration, and also …

Research explores flat subspaces in complex projective manifolds using Okounkov bodies.

Being inspired by Ross' construction of unstable products of certain smooth curves, we show that the product $C\times C$ of every smooth curve $C$ of genus at least 2 is not slope semistable with respect to certain polarisations. Besides, we produce examples of Kodaira-fibred surfaces of nonzero signature, which are no…

We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising…

We define a quantisation of the J-flow over a projective complex manifold. As corollaries, we obtain new proofs of uniqueness of critical points of the J-flow and that these critical points achieve the absolute minimum of an associated energy functional. We show that the existence of a critical point of the J-flow impl…

Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.

We investigate Chow stability of projective bundles P(E) where E is a strictly Gieseker stable bundle over a base manifold that has constant scalar curvature. We show that, for suitable polarisations L, the pair (P(E),L) is Chow stable and give examples for which it is not asymptotically Chow stable.

Study the limit of Calabi-Yau metrics with degenerate skeletons.

We prove that various GIT semistabilities of polarized varieties imply semi-log-canonicity.

Suppose that a polarised Kähler manifold $(X,L)$ admits an extremal metric $ω$. We prove that there exists a sequence of Kähler metrics $\{ ω_k \}_k$, converging to $ω$ as $k \to \infty$, each of which satisfies the equation $\bar{\partial} \text{grad}^{1,0}_{ω_k} ρ_k (ω_k)=0$; the $(1,0)$-part of the gradient of the B…

We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a very recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manif…

Proves constant scalar curvature Kähler metrics are very general.

Let (X,L) be a polarised manifold. We show that K-stability and asymptotic Chow stability of the blowup of X along a 0-dimensional cycle are closely related to Chow stability of the cycle itself, for polarizations making the exceptional divisors small. This can be used to give (almost) a converse to a result of Arezzo …

Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective…

The paper constructs contact-hyperbolic manifolds with large automorphism groups.

In this work we define a deformation theory for the Coupled Kähler-Yang-Mills equations in arXiv:1102.0991, generalizing work of Székelyhidi on constant scalar curvature Kähler metrics. We use the theory to find new solutions of the equations via deformation of the complex structure of a polarised manifold endowed with…

The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, ξ_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, ξ_N),$ provided $M$ contact embeds in $(N, ξ_N)$ with a trivial normal bundle and the contact s…

Study of higher-dimensional contact manifolds and their properties.

The study explores new metric structures on manifolds, linking them to Einstein metrics.