# Notes on My Papers

## J-holomorphic curves in a nef class

We 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.

To prove the result, we treat the curve configuration as weighted graph. Then new techniques curve expansion and curve combination are introduced to rearrange the graphs. These help us to rebuild the multiply covered part of the curve configurations.

When the stratum of the reducible part has codimension one in the moduli space, it is extremely interesting to interpret the curve combination moves as combinatorial blow-downs. A classification of these configurations is thus obtained: they are either successive infinitely near blow-ups of a single smooth curve, or successive infinitely near blow-ups of a comb configuration along the spike curve.

## J-symplectic cones of rational gour manifolds

This 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.

Taubes gives positive answer to Donaldson's question for a generic tamed almost complex structure. We could summarize his strategy as subvariety-current-form. Namely, one first constructs Taubes currents, i.e. a closed J-invariant positive current acting on J-invariant forms as an almost Kahler form, from families of subvarieties, then smooths it to a form. The second part is the easier part, for which we have careful study in my previous paper "the J-anti-invariant cohomology of almost complex 4-manifolds". For the first part, usually we don't have sufficient subvarieties to construct Taubes currents. That's the reason why Taubes chooses the "generic" set.

However for rational four manifolds, using the special properties of rational curves and a generalized version of Taubes currents, we could prove both questions for the listed cases among many others. The techniques contain two parts: 1. Adapting Taubes' technique to any tamed almost complex structures on rational surfaces; 2. Geometry and Combinatorics for rational curves and its curve cone.

## On cohomological decomposability of almost-Kahler structures

In 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 forms

Taubes 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 manifolds

The 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-manifolds

This 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.

We also prove the semi-continuity of these groups under deformation of almost complex structures.

## Geometry of tamed almost complex structures in dimension 4

This 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).

In this paper, we also produce several new results. For example, we show that if \alpha is a closed J-anti-invariant 2-form and the the square of the norm is Morse-Bott, then the zero set of \alpha is J-holomorphic. We give an estimate of the rank of J-(anti-)invariant group by Seiberg-Witten theory as well.

## Additivity and Relative Kodaira Dimensions

This paper is dedicated to Yau on the occassion of his 60th birthday.

In this paper, we define the notion of "relative Kodaira dimension" for a 4-dimensional symplectic manifold M relative to an embedded symplectic divisor F. To define it, we first need the notion of a relative minimal model. We call (M,F) is relatively minimal, if F is a (possibly disconnected) surface intersecting nontrivially with each exceptional class and has no sphere components. Then a somewhat surprising result shows that the relative minimal model is actually UNIQUE. A preliminary classification of pairs (M,F) is also given. The detailed classification will be given later in a separate note (part of it is included in my thesis).

For the additivity part. We introduce the notion of relative Kodaira dimension for a surface. This notion is used to show that Kodaira dimensions are additive for a fibration. Two facts are notable. One is that the symplectic Kodaira dimension for dimension 4 is a covering invariant. The other is that we can write the 4-dimensional symplectic Kodaira dimension as the sum of the "formal Kodaira dimension" of K.K and K.\omega. In turn, we obtain the additivity for a Lefschetz pencil. The last interesting fact also indicates the relations with divisor contraction in algebraic geometry.

## Symplectic forms and cohomology decomposition of almost complex 4-manifolds

This 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.

Professor Tedi Draghici is now at Florida International University . His home country is beautiful Romania .

## The Kodaira Dimension of Lefschetz Fibrations

This 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 manifolds

This 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.

The second part (now it is the first and the main part) is the study of similar problems when J is not integrable. We extend the results for comparing J-tamed cone and J-compatible cone to the greatest generality under the assumption it is almost Kahler. Especially, for all the 4-dimensional almost Kahler manifolds, J-compatible cone and the group $H_J^-$ span J-tamed cone. To prove this result, we defined several new homology and cohomology groups and study the properties and duality there. Especially, we introduce the notion of pure and full almost complex structures. In the paper "Symplectic forms and cohomology decomposition of almost complex 4-manifolds" joint with Tedi Draghici and Tian-Jun Li, we continue exploring the properties of these groups and proved that any 4-D almost complex manifold is "smoothly" pure and full for example. A six-dimensional almost complex manifold which is not pure (and full) is given in a recent preprint written by Fino and Tomassini.

## Some results on special stable vector bundles of rank 3 on algebraic curves

Xiao-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.

The other collaborator is my friend Bohan Fang. He is now a graduate student in Northwestern University. His advisor is Eric Zaslow.

This paper gives first series of results on the existence and classification of rank 3 stable vector bundles. Specifically, we give necessary and sufficient conditions for the degrees when we have 3 or 4 independent holomorphic sections and give the classifications for the critical cases.