arXiv subjects

Jonathan David Evans

Publications and source records attributed to Jonathan David Evans.

25 records · Page 2Linked to original sources

arXiv2017

Markov numbers and Lagrangian cell complexes in the complex projective plane

We study Lagrangian embeddings of a class of two-dimensional cell complexes $L_{p,q}$ into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type $\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into $\mathbf{CP}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\ldots,N$, cannot be made disjoint unless $N\leq 3$ and the $p_i$ form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a $\mathbf{Q}$-Gorenstein smoothing whose general fibre is $\mathbf{CP}^2$.

math.SG↗

arXiv2015

Pseudoholomorphic tori in the Kodaira-Thurston manifold

The Kodaira-Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov-Witten invariants which count pseudoholomorphic tori in the Kodaira-Thurston manifold. For a fixed symplectic form the Gromov-Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov-Witten computation for a non-Kähler manifold.

math.SG↗

arXiv2015

Exotic spheres and the topology of symplectomorphism groups

We show that, for certain families $ϕ_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $ϕ^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We show that these phenomena persist for Dehn twists along the standard matching spheres of the $A_m$-Milnor fibre. The nontriviality is detected by considering the action of symplectomorphisms on the space of parametrised Lagrangian submanifolds. We find related examples of symplectic mapping classes for $T^*(S^n\times S^1)$ and of an exotic symplectic structure on $T^*(S^n\times S^1)$ standard at infinity.

math.SG↗

arXiv2014

Floer cohomology of the Chiang Lagrangian

We study holomorphic discs with boundary on a Lagrangian submanifold $L$ in a Kaehler manifold admitting a Hamiltonian action of a group $K$ which has $L$ as an orbit. We prove various transversality and classification results for such discs which we then apply to the case of a particular Lagrangian in $\mathbf{CP}^3$ first noticed by Chiang. We prove that this Lagrangian has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5, in which case it generates the split-closed derived Fukaya category as a triangulated category.

math.SG↗

arXiv2014

Symplectic mapping class groups of some Stein and rational surfaces

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundle of $\mathbf{RP}^{2}$ and of $\mathbf{C}^*\times\mathbf{C}$. We also make progress in the case of the $A_n$-Milnor fibres: here we can show that the (compactly supported) Hamiltonian group is contractible and that the symplectic mapping class group embeds in the braid group on $n$-strands.

math.SG↗

arXiv2012

The infimum of the Nijenhuis energy

We prove that on any symplectic manifold whose symplectic form represents a rational cohomology class there exists a sequence of compatible almost complex structures whose Nijenhuis energy (the $L^2$-norm of the Nijenhuis tensor) tends to zero. The sequence is obtained by stretching the neck around a Donaldson hypersurface.

math.SG↗

arXiv2010

Lagrangian spheres in Del Pezzo surfaces

Lagrangian spheres in the symplectic Del Pezzo surfaces arising as blow-ups of the complex projective plane in 4 or fewer points are classified up to Lagrangian isotopy. Unlike the case of the 5-point blow-up, there is no Lagrangian knotting.

math.SG↗