Related papers: Special Lagrangian cones in C^3 and primitive harm…
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current…
E.B. Vinberg developed a theory of homogeneous convex cones $C \subset V= \mathbb{R}^n$, which has many applications. He gave a construction of such cones in terms of non-associative rank $n$ matrix T-algebras $\cal{T}$, that consist of…
We find new obstructions on the topology of monotone Lagrangian submanifolds of $C^{n}$ under some hypothesis on the homology of their universal cover. In particular we show that nontrivial connected sums of manifolds of odd dimensions do…
We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal…
We classify and expose all the gradient Ricci solitons on complete surfaces, open or closed, with curvature bounded below, and possibly with a discrete set of cone-like singular points that arise naturally. We give a precise qualitative…
We study exotic Lagrangian tori in dimension four. In certain Stein domains $B_{dpq}$ (which naturally appear in almost toric fibrations) we find $d+1$ families of monotone Lagrangian tori which are mutually distinct, up to…
A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal…
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…
We study special linear systems of surfaces of $\mathbb{P}^3$ interpolating nine points in general position having a quadric as fixed component. By performing degenerations in the blown-up space, we interpret the quadric obstruction in…
In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals.…
It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the…
We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems…
We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play…
Every Maslov-zero Lagrangian torus in a K3 surface has non-trivial homology class. This note aims to extend this result to Lagrangian tori with Maslov indices congruent to zero modulo 4. Conversely, we show that every homologically…
We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification…
In a toric symplectic manifold, regular fibres of the moment map are Lagrangian tori which are called toric fibres. We discuss the question which two toric fibres are equivalent up to a Hamiltonian diffeomorphism of the ambient space. On…
This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map…
This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and…
The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One…
We consider here square tilings of the plane. By extending the formalism introduced in [3] we build a correspondence between plane maps endowed with an harmonic vector and square tilings satisfying a condition of regularity. In the case of…