arXiv subjects
Publications and source records attributed to Tian-Jun Li.
Let $M$ be a closed, oriented, smooth $4-$manifold with intersection form $Γ$, $A(Γ)$ the automorphism group of $Γ$ and $D(M)$ the subgroup induced by orientation-preserving diffeomorphisms of $M$. In this note we study the question when $D(M)$ is of infinite index in $A(Γ)$ for a symplectic 4-manifold.
We show that for any nonzero class $A$ in $H_2(X; \mathbb{Z}_2)$ in a rational 4-manifold $X$, $A$ is represented by a nonorientable embedded Lagrangian surface L (for some symplectic structure) if and only if $P(A)\equiv (L) (mod\ 4)$; where $P(A)$ denotes the mod 4 valued Pontrjagin square of $A$.
We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $\mathbb{C}P^2\# N\overline{\mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.
We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. In particular, contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. On the other hand, we also prove that, given any contact 3-manifold, there is a lower bound of $2χ+3σ$ for all its minimal symplectic fillings. This is motivated by the bound of Stipsicz for Stein fillings. Finally, we discuss various aspects of exact self cobordisms of fillable 3-manifolds.
We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinitely family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admitting minimal strong fillings of arbitrarily large $b_2$. Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/adjunction contact structures to present a unified picture to the existing finiteness results on the topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find new classes of contact 3-manifolds with the finiteness property and extend Wand's obstruction of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero. structures with complexity zero.
As a candidate of quantum gravity in ultrahigh energy, the $(3+1)$-dimensional Hořava-Lifshitz (HL) gravity with critical exponent $z\ne 1$, indicates anisotropy between time and space at short distance. In the paper, we investigate the most general $z=d$ Hořava-Lifshitz gravity in arbitrary spatial dimension $d$, with a generic dynamical Ricci flow parameter $λ$ and a detailed balance violation parameter $ε$. In arbitrary dimensional generalized HL$_{d+1}$ gravity with $z\ge d$ at long distance, we study the topological neutral black hole solutions with general $λ$ in $z=d$ HL$_{d+1}$, as well as the topological charged black holes with $λ=1$ in $z=d$ HL$_{d+1}$. The HL gravity in the Lagrangian formulation is adopted, while in the Hamiltonian formulation, it reduces to Dirac$-$De Witt's canonical gravity with $λ=1$. In particular, the topological charged black holes in $z=5$ HL$_6$, $z=4$ HL$_5$, $z=3,4$ HL$_4$ and $z=2$ HL$_3$ with $λ=1$ are solved. Their asymptotical behaviors near the infinite boundary and near the horizon are explored respectively. We also study the behavior of the topological black holes in the $(d+1)$-dimensional HL gravity with $U(1)$ gauge field in the zero temperature limit and finite temperature limit, respectively. Thermodynamics of the topological charged black holes with $λ=1$, including temperature, entropy, heat capacity, and free energy are evaluated.
This is a survey on the various notions of Kodaira dimension in low dimensional topology. The focus is on progress after the 2006 survey [78].
We study the symplectic analogue of log Calabi-Yau surfaces and show that the symplectic deformation classes of these surfaces are completely determined by the homological information.
We generalize Bangert's non-hyperbolicity result for uniformly tamed almost complex structures on standard symplectic $R^{2n}$ to asymtotically standard symplectic manifolds.
We study Nakai-Moishezon type question and Donaldson's "tamed to compatible" question for almost complex structures on rational four manifolds. By extending Taubes' subvarieties--current--form technique to $J-$nef genus $0$ classes, we give affirmative answers of these two questions for all tamed almost complex structures on $S^2$ bundles over $S^2$ as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.
Taubes established fundamental properties of $J-$holomorphic subvarieties in dimension 4 in \cite{T1}. In this paper, we further investigate properties of reducible $J-$holomorphic subvarieties. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is $J-$nef. For a spherical class, it has particularly strong consequences. It is shown that, for any tamed $J$, each irreducible component is a smooth rational curve. We also completely classify configurations of maximal dimension. To prove these results we treat subvarieties as weighted graphs and introduce several combinatorial moves.
We derive an adjunction inequality for any smooth, closed, connected, oriented 4-manifold $X$ with $b^+=1$. This inequality depends only on the cohomology algebra and generalizes the inequality of Strle in the case of $b_1=0$. We demonstrate that the inequality is especially powerful when $2\tilde χ+3σ\geq 0$, where $\tilde χ$ is the modified Euler number taking account of the cup product on $H^1$.
We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $ω$-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.
We establish various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres in symplectic $4-$manifolds with $κ=-\infty$. This involves the explicit construction of spheres in rational manifolds via a new construction technique called the tilted transport.
In this paper we prove that the Torelli part of the symplectomorphism groups of the $n$-point ($n\leq 4$) blow-ups of the projective plane is trivial. Consequently, we determine the symplectic mapping class group. It is generated by reflections on $K_ω- $spherical class with zero $ω$ area.
We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed stable symplectic hypersurface V in a symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders (in addition to insertions on X\V) for stable V. We obtain invariants of the deformation class of (X,V,w). Two large issues must be tackled to define such invariants: (1) Curves lying in the hypersurface V and (2) genericity results for almost complex structures constrained to make V pseudo-holomorphic (or almost complex). Moreover, these invariants are refined to take into account rim tori decompositions. In the latter part of the paper, we extend the definition to disconnected submanifolds and construct relative Gromov-Taubes invariants.
In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S^2 or RP^2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural extension of McDuff's connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.