Pure Mathematics (Differential Geometry and Topology)

Lead Research Organisation: University of Cambridge
Department Name: Pure Maths and Mathematical Statistics

Abstract

I work in symplectic topology, an area of differential geometry.
A symplectic structure on a manifold is a non-degenerate closed 2-form; these exist on all oriented surfaces, on cotangent bundles, and on smooth algebraic varieties.

The geometry of symplectic forms is interestingly different to volume-preserving geometry and blends topological features and rigidity features; the latter are explored by holomorphic curve invariants ("Floer theory"), which are based on analysis. My main research questions concern Lagrangian submanifolds, and symmetries of symplectic manifolds. At the moment I am focusing on a class of symplectomorphisms (i.e diffeomorphisms that preserve the symplectic structure) called Dehn twists.
Seidel has extensively studied Dehn twists in Lagrangian spheres. I have been studying Dehn twists in real and complex projective spaces.

My first result shows that in Stein manifolds (like affine algebraic varieties) one cannot have a product of such twists isotopic to the identity, which extends a result of Seidel who proved the corresponding theorem for twists in spheres. The theorem uses a monodromy description for projective space twists and an argument counting sections of Lefschetz fibrations over the 2-sphere.

I am now studying
(i) possible extensions of the above result to more general situations, e.g. closed manifolds rather than affine varieties, and
(ii) a particular model for a real projective plane twist via Lefschetz fibrations, conjectured by Ailsa Keating.
These should give more insight into the possible relations amongst twists in symplectic mapping class groups.

Publications

10 25 50

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509620/1 01/10/2016 30/09/2022
1804059 Studentship EP/N509620/1 01/10/2016 31/12/2020 Brunella Charlotte Torricelli
 
Description One side of my work specialises in trying to discern the behaviour of projective twists within the symplectic mapping class group for specific types of symplectic manifolds.

1) For (spherical) Dehn twists, I reproved the following result by Barth-Geiges-Zehmisch.
In a Liouville manifold M, there is no composition of Dehn twists around Lagrangian spheres L_1,.,L_m that is isotopic to the identity. The proof involves Picard-Lefschetz theory. We build a Lefschetz fibration from the data (M,(L_1,L_m)) and use the existence of an invariant (due to Seidel) encoding the count of pseudo-holomorphic sections of the fibration.

2)I previously claimed to have proven the generalisation for Lagrangian projective twists, a class of symplectomorphisms defined from Lagrangian projective spaces. This generalization, which is now in a conjectural state, followed an analogue argument, but involving Morse-Bott-Lefschetz fibrations. Unfortunately, the latter turned out to be more problematic than expected because of compactness issues.

Focusing on the same questions, I restricted to a narrower set of symplectic manifolds, constructed by "plumbing" of cotangent bundles of projective spaces. The construction comes from taking the quotient from an S^1-equivariant plumbing of spheres, via the Hopf map.

3)I proved that on such spaces, there is no composition of projective twists around the Lagrangian cores K_1, K_2 that can be isotoped to the identity. This implies that the projective twists t_{K_1}, t_{K_2} along the cores generate a free group in the symplectic mapping class group of the plumbing.
The proof relies on the theory of "Lagrangian correspondences", developped by Mau-Wehrheim-Woodward, a result by Ailsa Keating about free groups generated by Dehn twists (applied to the plumbing of spheres), and a symplectic Gysin sequence proved by Timothy Perutz. I use a Lagrangian correspondence that I call the "Hopf correspondence" which, given a projective Lagrangian submanifold of a symplectic manifold K\subset X, assigns a Lagrangian sphere embedded in an auxiliary symplectic manifold L\subset M.

(CC) For complex projective spaces, the correspondence can be established in the presence of a cohomology class \alpha \in H^2(X;Z) whose restriction to each Lagrangian K_i is a generator of H^*(K_i;Z).
(CR) For real projective spaces, the correspondence can be established in the presence of a cohomology class \alpha \in H^1(X;Z_2) whose restriction to each Lagrangian K_i is a generator of H^*(K_i;Z_2).


I have proved a theorem that approaches the conjecture 2), using the insight from 3):

4)For a Liouville manifold X containing (real or complex) projective Lagrangian submanifolds K_i and a class \alpha \in H^*(X;Z) satisfying condition (CC) or (CR), there is no composition of projective twists around the K_i isotopic to the identity.

The intuition behind this statement rests on a generalisation of some of the mechanisms appearing 3) and on a mirror symmetry prediction, following a conjecture by Huybrecht-Thomas (Proposition 2.7 of Huybrecht-Thomas << P-objects and autoequivalences of derived categories >>).

5) I have obtained "relative" results for both 1) and 4):
Under the same assumptions, and assuming the existence of a distinguished Lagrangian disc T, the map obtained as product of twists cannot preserve T (as Lagrangian/and or/ object of a wrapped Fukaya category).

6)I combine the "Hopf correspondence" with classical homotopy theory and the literature about the "nearby Lagrangian conjecture" to prove that there are examples of projective twists in Symp(T^*CP^n) whose isotopy class depends on a choice of "framing", i.e a choice of diffeomorphism L---> CP^n, where L is the Lagrangian zero section of T^*CP^n. These twists are called "exotic projective twists".

This phenomenon was observed by Evans-Dimitroglu-Rizell for Dehn twists, and my outcomes show that there are dimensions (possibly infinite many) n for which Symp(T^*CP^n) admits exotic projective twists. The same ideas lead to non-embedding theorems of certain exotic Lagrangian projective spaces into T^*CP^n (n=4,7).

7)I am completing the description of two local models to visualise the projective twists around RP^2 and CP^2. This means that I provide models for those symplectomorphisms in the symplectic mapping class groups of the cotangent bundles T^*RP^2 and T^*CP^2 respectively.

This has been done by Seidel for spherical twists, and by Perutz for fibred twists; but the latter case remains difficult to employ in practical applications, due to the setting requiring Morse-Bott-Lefschetz fibrations. These are a degenerate version of Lefschetz fibration, and can present many compactness-related issues (see 2)).

To achieve 7) I am using examples of Lefschetz fibrations with total space either of the cotangent bundles, that can be constructed from pencils of hyperplane sections in CP^2 (for the RP^2 twist) and CP^2xCP^2 (for the CP^2 twist). These fibrations agree with Lefschetz fibrations that were originally constructed by Joe Johns.
The projective twists are buit as symplectomorphisms of the total space of these fibrations, and using a Floer theoretical approach I show that they are (symplectically) of infinite order.
Exploitation Route My findings will contribute to the advance of research in the field of symplectic topology in pure mathematics. This branch of mathematics lies at the crossroads between differential geometry, algebraic topology and algebraic geometry, the results will have an impact beyond the "symplectic community". For example "mirror symmetry" is one of the most powerful links between the symplectic geometry (A) of a manifold with its algebro-geometric structures (B): we denote this relation symbolically by A<--->B.
Mirror symmetry is still conjectural, but it has been confirmed in many cases, providing tools for answering questions of A knowing B or vice versa, answering questions of B knowing A.
The latter case is exactly one instance in which symplectic topology happens to be useful to shed light on other areas of mathematics.
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