Preprints | Published | Thesis | In progress | Other |

**Abstract**: We provide an interpretation of the space of symplectic diffeomorphism of a 4-dimensional real torus $T^4$, as the vanishing locus of a hyperKähler moment map. This construction leads to a notion of modified hyperKähler moment map flow, whose regular fixed points coincide with the symplectic diffeomorphisms of $T^4$ . Given a Euclidean triangulation of $T^4$ , we introduce a notion of polyhedral maps and polyhedral symplectic maps of the torus. We show that all the construction carried out in the smooth setting have analogues in the polyhedral setting. In this context, the hyperKähler modified moment map flow is an ordinary differential equation, which is always convergent. Under a non-degeneracy assumption of the limit, this provides an effective construction of polyhedral symplectic maps of the torus. This paper lays out the ground work for some experimental symplectic geometry. In an upcoming work with François Jauberteau, we write a computer program that runs a numerical version of the flow and generates non trivial examples of symplectic polyhedral maps of $T^4$.

**Abstract**: We prove that every smoothly immersed 2-torus of $\mathbb{R}^4$ can be approximated, in the $C^0$-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the surface can be approximated in the $C^1$-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic 2-tori of $\mathbb{R}^{2n}$.

**Abstract**: We show that isotropic polyhedral surfaces of a symplectic vector space V can be understood as the vanishing locus of a moment map as- sociated to a particular Hamiltonian torus action on a finite dimensional symplectic space. This interpretation allows to introduce an associated modified moment map flow, which is an ODE whose limits are essentially the (parametrized) isotropic surfaces. This approach provides an effec- tive method for constructing Lagrangian polyhedral surfaces of $\mathbb{R}^4$ . This paper is provided with a computer program which shows the modified flow as a real-time movie, in the case of a 2-dimensional torus.

*Most of my preprints are available on
www.arXiv.org *

**Abstract**: We consider smooth isotropic immersions from the 2-dimensional torus into R2n, for n≥2. When n=2 the image of such map is an immersed Lagrangian torus of R4. We prove that such isotropic immersions can be approximated by arbitrarily C0-close piecewise linear isotropic maps. If n≥3 the piecewise linear isotropic maps can be chosen so that they are piecewise linear isotropic immersions as well. The proofs are obtained using analogies with an infinite dimensional moment map geometry due to Donaldson. As a byproduct of these considerations, we introduce a numerical flow in finite dimension, whose limit provide, from an experimental perspective, many examples of piecewise linear Lagrangian tori in R4. The DMMF program, which is freely available, is based on the Euler method and shows the evolution equation of discrete surfaces in real time, as a movie.

**Abstract**: Hamiltonian stationary Lagrangian submanifolds (HSLAG) are a natural generalization of special Lagrangian manifolds (SLAG). The latter only make sense on Calabi-Yau manifolds whereas the former are defined for any almost Kähler manifold. Special Lagrangians, and, more specificaly, fibrations by special Lagrangians play an important role in the context of the geometric mirror symmetry conjecture. However, these objects are rather scarce in nature. On the contrary, we show that HSLAG submanifolds, or fibrations, arise quite often. Many examples of HSLAG fibrations are provided by toric Kähler geometry. In this paper, we obtain a large class of examples by deforming the toric metrics into non toric almost Kähler metrics, together with HSLAG submanifolds.

**Abstract**: Abstract: We construct new explicit toric scalar-flat Kähler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal Kähler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact type.

**Abstract**: Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature Kähler metrics on the iterated blowup are equivalent, for certain polarizations close to the boundary of the Kähler cone.

**Abstract**: We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface admits an extremal metric, then the smoothings also admit extremal metrics in nearby Kähler classes. In addition, we construct small Lagrangian stationary spheres which represent Lagrangian vanishing cycles for surfaces close to the singular one.

**Abstract**: Let X be a compact toric extremal Kähler manifold. Using the work of Székelyhidi, 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 $CP^1\times CP^1$.

**Abstract**: Let $(\mathcal {X},\Omega)$ be a closed polarized complex manifold, $g$ be an extremal metric on $\mathcal X$ that represents the Kähler class $\Omega$, and $G$ be a compact connected subgroup of the isometry group $Isom(\mathcal{X},g)$. Assume that the Futaki invariant relative to $G$ is nondegenerate at $g$. Consider a smooth family $(\mathcal{M}\to B)$ of polarized complex deformations of $(\mathcal{X},\Omega)\simeq (\mathcal{M}_0,\Theta_0)$ provided with a holomorphic action of $G$. Then for every $t\in B$ sufficiently small, there exists an $h^{1,1}(\mathcal{X})$-dimensional family of extremal Kähler metrics on $\mathcal{M}_t$ whose Kähler classes are arbitrarily close to $\Theta_t$. We apply this deformation theory to analyze the Mukai-Umemura 3-fold and its complex deformations.

**Abstract**: We present new constructions of Kaehler metrics with constant scalar curvature on complex surfaces, in particular on certain del Pezzo surfaces. Some higher dimensional examples are provided as well.

**Abstract**: We give a new construction of Einstein and Kaehler-Einstein manifolds which are asymptotically complex hyperbolic, inspired by the work of Mazzeo-Pacard in the real hyperbolic case. The idea is to develop a gluing theorem for 1-handle surgery at infinity, which generalizes the Klein construction for the complex hyperbolic metric.

**Abstract**: A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus we can produce numerous examples of Kaehler surfaces of constant scalar curvature with circle or toric symmetry.

**Abstract**: We prove a generalization of Bennequin's inequality for Legendrian knots in a 3-dimensional contact manifold, under the assumption that it is the boundary of a 4-dimensional manifold M and the version of Seiberg-Witten invariants introduced by Kronheimer and Mrowka is non-vanishing. 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.

**Abstract**: A new construction is presented of scalar-flat Kaehler 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 $\mathbb{CP}^2$ blown up at 10 suitably chosen points, admits a scalar-flat Kaehler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual 4-manifolds.

**Abstract**: We prove that every Einstein metric on the unit ball $B^4$ of $\mathbb{C}^2$, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more generally, if $M^4$ is a manifold with a CR-boundary at infinity, an adapted spinc-structure which has a non zero Kronheimer--Mrowka invariant and an asymptotically complex hyperbolic Einstein metric, we produce a solution of Seiberg--Witten equations with an strong exponential decay property.

**Abstract**: Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar curvature on M is equivalent to the parabolic polystability of E; moreover these metrics all come from finite volume quotients of $\mathbb{H}^2 \times \mathbb{CP}^1$. In order to prove the theorem, we must produce a solution of Seiberg-Witten equations for a singular metric g. We use orbifold compactifications $\overline M$ on which we approximate g by a sequence of smooth metrics; the desired solution for g is obtained as the limit of a sequence of Seiberg-Witten solutions for these smooth metrics.

http://yann.rollin.free.fr/these_rollin.pdf.

Y. Rollin,

Last update: Fri, 28 May 2021 09:01:38 +0200