Research

My research interest lies in low dimensional topology and symplectic topology. In particular I am interested in symplectic normal crossing divisors and its applications to symplectic topology, for instantce symplectic fillings, contact structures, Hamiltonian torus actions, etc. Recently, I have also been thinking about symplectic cohomology and birational geometry of affine varieties.

Papers and preprints: (arXiv, GoogleScholar)

[6] Almost complex geometry of symplectic log Calabi-Yau pairs with applications to almost toric fibrations (with Shengzhen Ning) Submitted. Preprint.

Given a symplectic log Calabi-Yau pair, we study the almost complex structures that are adapted to it. For these almost complex structures, we study the behavior of the curve cone, prove a Nakai-Moishezon criterion and describe the almost K\"{a}hler cone for the generic ones. As an application, we prove a decomposition theorem for Seiberg-Witten non-trivial classes. This leads to an alternative proof of the existence of almost toric fibrations on $c_1$-positive rational surfaces shown in [5].

[5] Almost toric presentations of symplectic log Calabi-Yau pairs (with Tian-Jun Li, Shengzhen Ning) Submitted. arXiv:2303.09964

We show that any symplectic log Calabi-Yau divisor can be realized as the boundary divisor of an almost toric fibration. This realization is canonical once we choose an extra data called the framing on the space of LCY. This is achieved by considering the symplectic analogue of the toric model used in the algebraic geometrical settings. 

[4] Finite group actions on symplectic rational surfaces (with Weiwei Wu, Shuo Zhang) In preparation.

[3] Enumerative aspect of symplectic log Calabi-Yau divisors and almost toric fibrations (with Tian-Jun Li, Shengzhen Ning) To appear Israel Journal of Mathematics. arXiv:2203.08544

we are interested in the isotopy classes of symplectic log Calabi-Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and prove the stability, finiteness and rigidity results. Motivated by the problem of counting toric actions, we obtain a general counting formula of symplectic log Calabi-Yau divisors in a restrictive region of c1-nef cone. A detailed count in the case of 2- and 3-point blow-ups of complex projective space for all symplectic forms is also given. In our framework the complexity of the combinatorics of analyzing Delzant polygons is reduced to the arrangement of homology classes. Then we study its relation with almost toric fibrations. We raise the problem of realizing all symplectic log Calabi-Yau divisors by some almost toric fibrations and verify it together with another conjecture of Symington in a special region.

[2] Circular spherical divisors and their contact topology (with Tian-Jun Li, Cheuk Yu Mak) To appear Communications in Analysis and Geometry. arXiv:2002.10504

This paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors D that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such D. When D is anticanonical and convex, we give explicit betti number bounds for Stein fillings of its boundary contact torus bundle.

[1] Local geometry of symplectic divisors with applications to contact torus bundles (with Tian-Jun Li) To appear Proceedings of the International Congress of Chinese Mathematicians (Beijing 2019). arXiv:2101.05981 (arxiv version contains slightly more stuff than the published version)

In this note we study the contact geometry of symplectic divisors. We show the contact structure induced on the boundary of a divisor neighborhood is invariant under toric and interior blow-ups and blow-downs. We also construct an open book decomposition on the boundary of a concave divisor neighborhood and apply it to the study of universally tight contact structures of contact torus bundles.

[0] Symplectic divisors in dimension four. PhD Thesis. ProQuest

This is my PhD thesis, which contains mostly preliminary results from papers [1], [2], [3] above. But it also contains a bit more stuff on achiral Lefschetz fibrations.

Work in progress:.

Expository writings:

Research Talks:

Selected Expository Talks: