Related papers: Lagrangian Surplusection Phenomena

This note presents basic restrictions on the topology "general" Lagrangian surfaces of hyper-K\"ahler $4$-folds and a remark on the interaction of a Lagrangian subvariety with a Lagrangian fibration of the associated hyper-K\"ahler variety.

Let X be a projective hyperk\"ahler manifold containing a Lagrangian subtorus L. We study intersections of deformations of L, defining a canonical almost holomorphic map called L-reduction, which is not birational if and only if X admits an…

We find lower bounds on the number of intersection points between two relatively exact Hamiltonian isotopic Lagrangians. The bounds are given in terms of the cuplength of the Lagrangian in various multiplicative generalised cohomology…

Intersecting hypersurfaces in classical Lovelock gravity are studied exploiting the description of the Lovelock Lagrangian as a sum of dimensionally continued Euler densities. We wish to present an interesting geometrical approach to the…

We consider open symplectic manifolds which admit dilations (in the sense previously introduced by Solomon and the author). We obtain restrictions on collections of Lagrangian submanifolds which are pairwise disjoint (or pairwise…

We review a certain problem on covering triangles in the plane. Equivalently, it can be viewed as a family of 'isobilliard' inequalities in convex shapes, and as a special case of Viterbo's conjecture in symplectic geometry. We give an…

Let X be an irreducible symplectic manifold and Def(X) the Kuranishi space. Assume that X admits a Lagrangian fibration. We prove that X can be deformed preserving a Lagrangian fibration. More precisely, there exists a smooth hypersurface H…

The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the…

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit…

Let X be a compact hyperk\"ahler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not…

We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can…

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian…

In this paper we use Floer theory to study topological restrictions on Lagrangian embeddings in closed symplectic manifolds. One of the phenomena arising from our results is ``homological rigidity'' of Lagrangian submanifolds. Namely, in…

We prove a degenerate homological Arnol'd conjecture on Lagrangian intersections beyond the case studied by A. Floer and H. Hofer via a new version of Lagrangian Ljusternik--Schnirelman theory. We introduce the notion of (Lagrangian)…

Given a compact Lagrangian $L$ in a semipositive convex-at-infinity symplectic manifold $W$, we establish a cup-length estimate for the action values of $L$ associated to a Hamiltonian isotopy whose spectral norm is smaller than some…

A theory of gravity in higher dimensions is considered. The usual Einstein-Hilbert action is supplemented with Lovelock terms, of higher order in the curvature tensor. These terms are important for the low energy action of string theories.…

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of…

Lagrangian $k$-surgery modifies an immersed Lagrangian submanifold by topological $k$-surgery while removing a self-intersection. Associated to a $k$-surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism…

Given two weighted k-uniform hypergraphs G, H of order n, how much (or little) can we make them overlap by placing them on the same vertex set? If we place them at random, how concentrated is the distribution of the intersection? The aim of…

In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y are a mirror pair of such manifolds, then X should possess a special Lagrangian torus…