Related papers: Stability of the symplectomorphism group of ration…

Let $(X,\omega)$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures $\mathcal{J}_{\omega}$ using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional…

A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact…

We use algebraic arc complexes to prove a homological stability result for symplectic groups with slope 2/3 for rings with finite unitary stable rank. Symplectic groups are here interpreted as the automorphism groups of formed spaces with…

Kollar and Ruan proved symplectic deformation invariance for uniruledness of Kaehler manifolds. Zhiyu Tian proved the same for rational connectedness in dimension < 4. Kollar conjectured this in all dimensions. We prove Kollar's conjecture,…

We prove that symplectic ball packing stability holds for every compact, connected symplectic $4$-manifold with smooth boundary. This follows from a stronger result: the full volume of any such manifold can be filled by a single symplectic…

We show that continuous bounded group cohomology stabilizes along the sequences of real or complex symplectic Lie groups, and deduce that bounded group cohomology stabilizes along sequences of lattices in them, such as…

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $\iota:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_{\iota}$. In this paper, we study…

We establish various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify…

This paper studies groups of symplectomorphisms of ruled surfaces for symplectic forms with varying cohomology class. This class is characterized by the ratio R of the size of the base to that of the fiber. By considering appropriate spaces…

We continue our previous work to prove that for any non-minimal ruled surface $(M,\omega)$, the stability under symplectic deformations of $\pi_0, \pi_1$ of $Symp(M,\omega)$ is guided by embedded $J$-holomorphic curves. Further, we prove…

We prove packing stability for any closed symplectic manifold with rational cohomology class. This will rely on a general symplectic embedding result for ellipsoids which assumes only that there is no volume obstruction and that the domain…

In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces…

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable…

We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points. We…

Let $M$ be either $S^2\times S^2$ or the one point blow-up $\cp# \bcp$ of $\cp$. In both cases $M$ carries a family of symplectic forms $\om_\la$, where $\la > -1$ determines the cohomology class $[\om_\la]$. This paper calculates the…

A key question for $4$-manifolds $M$ admitting symplectic structures is to determine which cohomology classes $\alpha\in H^2(M,\mathbb R)$ admit a symplectic representative. The collection of all such classes, the symplectic cone $\mathcal…

In this paper we prove a stability theorem for block diffeomorphisms of 2d-dimensional manifolds that are connected sums of S^d x S^d. Combining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet's lemma of…

We will study homological stability of the diffeomorphism groups of the manifolds $W_{g,1}:=D^{2n} \# (S^n \times S^n)^{\#g }$ using $E_k$-algebras. This will lead to new improvements in the stability results, especially when working with…

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic…

In this paper, we study the curve cone of an almost complex $4$-manifold which is tamed by a symplectic form. In particular, we prove the cone theorem as in Mori theory for all such manifolds using the Seiberg-Witten theory. For small…