arXiv subjects
Publications and source records attributed to Tian-Jun Li.
Lefschetz fibration is the symplectic analogue of stable holomorphic fibration in complex geometry. A 4-dimensional stable holomorphic fibration satisfies the famous Parshin-Arakelov inequality. In this note we present an analogous inequality for a 4-dimensional Lefschetz fibration.
In this paper we set up the family Seiberg-Witten theory. It can be applied to the counting of nodal pseudo-holomorphic curves in a symplectic 4-manifold (especially a Kahler surface). A new feature in this theory is that the chamber structure plays a more prominent role. We derive some wall crossing formulas measuring how the family Seiberg-Witten invariants change from one chamber to another.
In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least -1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least -1 in any 4-manifold admitting a symplectic structure.
Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b^+=1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms. Similar results hold when M is not minimal.
Two lens spaces are given to show that Ohtsuki's $τ$ for rational homology spheres does not determine Kirby-Melvin's $\{τ_r^{'}, r odd\geq3\}$
In this note, we first derive explicit formulas for Kirby-Melvin's three-manifold invariants $τ_r^{'}$ for all Lens spaces from our results for another set of invariants $ξ_r$ defined by the first author. Then, as a corollary, we obtain formulas for Othsuki's $τ$ for all Lens spaces.