Yann Rollin

Non-minimal scalar-flat Kähler surfaces and parabolic stability

A new construction is presented of scalar-flat Kähler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that

Legendrian knots and monopoles

\mathbb{CP}^2

Construction of Kahler surfaces with constant scalar curvature

blown up at 10 suitably chosen points, admits a scalar-flat Kähler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual 4-manifolds.

Deformations of Extremal Toric Manifolds

We prove a generalization of Bennequin’s inequality for Legendrian knots in a 3‐ dimensional contact manifold.Y;/ , under the assumption that Y is the boundary of a 4‐dimensional manifold M and the version of Seiberg‐Witten invariants introduced by Kronheimer and Mrowka in [10] is nonvanishing. The proof requires an excision result for Seiberg‐Witten moduli spaces; then the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying inside M . 57R17, 57M25, 57M27, 57R57

Deformation of extremal metrics, complex manifolds and the relative Futaki invariant

The aim of this note is to present a new construction of Kähler metrics of constant scalar curvature (CSC) on complex surfaces. In order to introduce our results, let us introduce the terms “positive CSC” to mean “constant positive scalar curvature”, “zero CSC” for “(constant) zero scalar curvature” and “negative CSC” for “constant negative scalar curvature”. Our construction gives rise to many families of examples, but in this introduction we shall focus on Xk := k-fold blow-up of CP1 × CP1. We note that if k ≥ 1 then Xk can also be viewed as a k + 1-fold blow-up of CP2. Of course, the above description of Xk does not fix its complex structure: this will depend on the location of the centres of the blow-ups. Our first result gives positive CSC Kähler metrics in a family of Kähler classes on Xk , for k = 6, 7, 8, and for certain choices of complex structure. We note that if k ≤ 7 then Xk is Fano and the work of Tian [13] and others gives positive Kähler–Einstein metrics on Xk . Our result is new in that it produces CSC metrics on X8 as well as CSC metrics on X6 and X7 in Kähler classes that are “arbitrarily far” from c1(X): Theorem A. For k = 6, 7, 8, there exists a k-point blow-up X of CP1 × CP1 with no non-trivial holomorphic vector field and the following properties. Let F = {x} ×CP1 be a generic rational curve of CP1 × CP1. For every constant c > 0 and ε > 0, there exists a Kähler metric ω of strictly positive constant scalar curvature on X such that ∣∣∣∣ [ω] · F √[ω]2 − c ∣∣∣∣ ≤ ε. (1.1)

Wormholes in ACH Einstein manifolds

Let X be a compact toric extremal Kähler manifold. Using the work of Székelyhidi (Am. J. Math. 132(4):1077–1090, 2010), we provide a combinatorial criterion on the fan describing X to ensure the existence of complex deformations of X that carry extremal metrics. As an example, we find new CSC metrics on 4-points blow-ups of

ALE scalar-flat Kähler metrics on non-compact weighted projective spaces

\mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{1}

Constant Scalar Curvature Kahler Surfaces and Parabolic Polystability

.

Smoothing singular extremal K\"ahler surfaces and minimal Lagrangians

Let