Jarek Kȩdra | Searcharxiv

arXiv2019

Remarks on monotone Lagrangians in $\mathbf{C}^n$

We derive some restrictions on the topology of a monotone Lagrangian submanifold $L\subset\mathbf{C}^n$ by making observations about the topology of the moduli space of Maslov 2 holomorphic discs with boundary on $L$ and then using Damian's theorem which gives conditions under which the evaluation map from this moduli space to $L$ has nonzero degree. In particular we prove that an orientable 3-manifold admits a monotone Lagrangian embedding in $\mathbf{C}^3$ only if it is a product, which is a variation on a theorem of Fukaya. Finally we prove an h-principle for monotone Lagrangian immersions.

math.SG↗

arXiv2003

Symplectically aspherical manifolds

The main subjects of the paper is studying the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups $π_1(M)$ of symplectically aspherical manifolds $M$ with $π_2(M)=0$ and $π_2(M)\neq 0$. Relations between these classes are discussed. We show that several important classes of groups can be realized in both classes, while some of groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general research of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical, etc.

math.SG↗