Notes on My PapersJ-holomorphic curves in a nef classWe study (reducible) curve configurations in this paper. It is shown that the sum of J-genus of each component is bounded above by the J-genus of the total class if we assume the class is J-nef, an analogue of nefness in algebraic geometry. For a spherical class and tamed J, it shows that each irreducible component is a smooth rational curve. This result seems new even when J is integrable. The nefness condition is needed because for instance the following example in CP2 blown up at two points: 3H-2E_2=(3H-E_1-E_2)+(E_1-E_2). The classes in parentheses are J-effective. The J-geuns of these 3 classes are 0, 1 and 0 respectively. J-symplectic cones of rational gour manifoldsThis paper studies Nakai-Moishezon type question and Donaldson's "tamed to compatible" for almost complex structure on rational four manifolds. Here, Nakai-Moishezon type question means the duality between the curve cone and almost Kahler cone when b+=1. The Kahler version is proved by Buchdahl, Lamari and Demailly-Paun (for general dimensions and pairing with all subvarieties). In this paper, we answer both questions for S2 bundles over S2 (Donaldson's question was known to be true for CP2 by results of Gromov and Taubes). We also give answers for other tamed J's, e.g. those ones with only -1-spheres and -K in their curve cones. On cohomological decomposability of almost-Kahler structuresIn this paper, we discuss the cohomological decomposability of J-invariant and J-anti-invariant cohomology groups in dimension higher than 4. In dimension 4, the decomposition holds for any almost complex structure. In this paper, we give examples showing it is even not correct for almost-Kahler structures. We also explore the upper bound for the dimension of J-anti-invariant cohomology group. It seems n(n-1) is the right bound when the dimension is 2n. From Taubes currents to almost Kahler formsTaubes current is introduced in Taubes' "tamed to compatible" paper as an intermediate stage to construct almost Kahler forms through subvariety-current-form strategy. This paper explores the second step: how to construct almost Kahler forms once we have Taubes currents. Taubes established this regularization result when b^+=1 and J is tamed. We prove in this paper that this could be done for any 4-dimensional almost complex manifold without the tameness assumption a priori. Morveover, the cohomology class could be kept. Similar result is established for higher dimensions under the assumption J is almost Kahler. A note on exact forms on almost complex manifoldsThe main goal of this note is to demonstrate if there is a J-compatible symplectic form (i.e. an almost Kahler form) then there is a great flexibility in constructing J-tamed forms. More precisely, in this circumstance, we have tamed symplectic forms with arbitrarily given J-anti-invariant parts. As byproducts, we reformulate Donaldson's tamed versus compatible question in terms of various spaces of exact forms. Another notable fact is about the representatives of J-(anti-)invariant homology groups H^J_{\pm} defined by currents. As its cohomology counterpart H_J^-, each class in H^J_- has a unique representative. While H^J_+ has infinitely many representatives. Most of the conclusions are no longer true in higher dimensions. On the J-anti-invariant cohomology of almost complex 4-manifoldsThis is the second paper of a series written with Tedi Draghici and Tian-Jun Li. we analyze the properties (especially the ranks) of J-(anti-)invariant cohomology groups. In dimension four, we calculate the ranks of these groups for an almost complex structure which is metric related to a complex structure. After the calculations, we propose two conjectures: the J-anti-invariant groups are generically of rank 0; if the rank is larger than 3, then the almost complex structure has to be integrable. We prove the first conjecure when b^+=1. Geometry of tamed almost complex structures in dimension 4This is mainly a survey paper on the geometry of tamed almost complex structures in dimension 4, motivated by Donaldson's tamed versus compatible questions and Taubes' fundamental constructions of pseudo-holomorphic submanifolds and almost Kahler forms. Besides three published papers, we also survey on the results proved in several preprints: J-symplectic cones of rational four manifolds (with Tian-Jun Li), From Taubes currents to almost K\"ahler forms, Configurations of negative curves for almost complex structures, On cohomological decomposability of almost-K\"ahler structures (with Angella and Tomassini). Additivity and Relative Kodaira DimensionsThis paper is dedicated to Yau on the occassion of his 60th birthday. Symplectic forms and cohomology decomposition of almost complex 4-manifoldsThis paper is a continuation of the previous paper "Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds". This is the first paper of a series written with Prof. Tedi Draghici and Tian-Jun Li. We mainly focus on dimension 4 in this paper. Especially, we prove that the cohomology groups $H_J^+$ and $H_J^-$ defined in "Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds" constitute a decomposition of real second cohomology group for 4-dimensional almost complex manifolds. This can be viewed as a generalization of the (real) Dolbeault docomposition for complex surfaces to general almost complex manifolds. A reformulation of Donaldson's original question in terms of pointwise estimates is also mentioned. The Kodaira Dimension of Lefschetz FibrationsThis paper is a joint work with my academic brother Josef. G. Dorfmeister . His father is a very famous mathematician. This paper studies (and defines) several different Kodaira dimensions and the relation between them. Especially, we proved the equivalence of holomorphic Kodaira dimension and symplectic Kodaira dimension when the manifold admits both complex and symplectic structures. We also calculate Kodaira dimensions for many Lefschetz fibrations and in turn define so called "Lefschetz fibration Kodaira dimension". The Lefschetz fibration Kodaira dimension lies in a framework of relative Kodaira dimension. It is easy to see that the discussion in the last section of "Additivity and Relative Kodaira Dimensions" goes well when we talk about Lefschetz fibration Kodaira dimension instead of the symplectic Kodaira dimension there. Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifoldsThis paper is written with my advisor Tian-Jun Li. It has two parts. The first part (now it is the second part) is the analysis of the relation of J-tamed cone and J-Kahler cone when J is a complex structure. Especially, this answers a question of Donaldson when J is a complex structure. This part can be viewed as Kodaira embedding Theorem and Nakai-Moishezon Theorem in the J-symplectic setting. The survey of this part, related background and questions is available as my oral paper. Some results on special stable vector bundles of rank 3 on algebraic curvesXiao-Jiang Tan is a Professor in Peking University. He is one of my undergraduate advisors. I still remember how Prof. Tan brings me into the gate of modern mathematics from the concept of Riemann surfaces.
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