Tian Jun Li

Birational cobordism invariance of uniruled symplectic manifolds

A symplectic manifold

Quaternionic bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero

(M,\omega)

Symplectic Parshin-Arakelov inequality

is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternbergs notion of cobordism as the symplectic analogue of the birational equivalence.

Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

The Kodaira dimension of a non-minimal manifold is defined to be that of any of its minimal models. It is shown in [12] that, if ω is a Kahler form on a complex surface (M,J), then κ(M,ω) agrees with the usual holomorphic Kodaira dimension of (M,J). It is also shown in [12] that minimal symplectic 4−manifolds with κ = 0 are exactly those with torsion canonical class, thus can be viewed as symplectic Calabi-Yau surfaces. Known examples of symplectic 4−manifolds with torsion canonical class are either Kahler surfaces with (holomorphic) Kodaira dimension zero or T 2−bundles over T 2 ([10], [12]). They all have small Betti numbers and Euler numbers: b+ ≤ 3, b ≤ 19 and b1 ≤ 4; and the Euler number is between 0 and 24. It is speculated in [12] that these are the only ones. In this paper we prove that it is true up to rational homology.

Spherical Lagrangians via ball packings and symplectic cutting

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.

GENERALIZED GAUSSIAN SUMS CHERN-SIMONS-WITTEN-JONES INVARIANTS OF LEN-SPACES

University of Minnesota Ph.D. dissertation. July 2012. Major: Mathematics. Advisor: Tian-Jun Li. 1 computer file (PDF); iii, 61 pages.

Smooth minimal genera for small negative classes in CP2#nCP2 with n⩽9

In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian,

Calabi-Yau caps, uniruled caps and symplectic fillings: CALABI-YAU CAPS, UNIRULED CAPS AND SYMPLECTIC FILLINGS

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