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Weimin Chen

Publications and source records attributed to Weimin Chen.

At least 19 recordsLinked to original sources
arXiv2026

A note on the self-intersection of rational unicuspidal curve with one Puiseux pair

For any $(p,q)$, we proved the existence of a rational unicuspidal curve with one Puiseux pair $(p,q)$ in an algebraic surface, with a self-intersection which realizes the upper bound $m_{p,q}$ conjectured by the first-named author in \cite{C}, thus strengthening the symplectic version of the result in \cite{C}. These ``optimal" curves were obtained by examining two infinite families of rational bicuspidal curves, one in $CP^2$ and one in $CP^1\times CP^1$. In order to facilitate the computations, we derived certain recursive identities, and as a byproduct, we obtained a new formula for the bound $m_{p,q}$, which is more amenable to computations and gives us better insight concerning the nature of the bound. As a byproduct of this investigation, a formula for the last entry of the multiplicity sequence of the singularity was also found.

math.AG↗
arXiv2025

Symplectic configurations: a homological and computer-aided approach

Motivated by and extending the technical results in our earlier work on symplectic Calabi-Yau $4$-manifolds, a general and systematic approach for studying certain unions of symplectic embedded surfaces in a rational $4$-manifold $X=CP^2\# N\overline{CP^2}$ is formulated, which may find applications in a broader range of problems. A distinct feature of this method is that it is computer-aided. We address several fundamental theoretical questions concerning the computational aspect. On the other hand, we also establish a symplectic analog of Cremona transformations from algebraic geometry, which is another fundamental feature and a main technical tool of this method. For an illustration, we give a new proof that a certain line arrangement in $CP^2$, called Fano planes, cannot exist in the symplectic category. The nonexistence of Fano planes in the algebraic category follows from a theorem of Hirzebruch, while in the topological category, including the symplectic category, it was first proved by Ruberman and Starkston. Our proof for the symplectic category is independent to both, and is by combining the Cremona transformation technique with Gromov's theory of pseudoholomorphic curves.

math.SG↗
arXiv2025

Small symplectic $4$-manifolds via contact gluing and some applications

We introduce a streamlined procedure for constructing small symplectic $4$-manifolds via contact gluing, based on a technique invented by David Gay around 2000. We give several applications of this procedure, which include results concerning embeddings of singular Lagrangian $RP^2$s, or embeddings of lens spaces as a hypersurface of contact type, in small rational surfaces such as $CP^2\#\overline{CP^2}$ and $S^2\times S^2$, as well as results on the uniqueness or classification of $Q$-homology ball symplectic fillings. Further work on the classification of singular Lagrangian $RP^2$s is suggested. Moreover, our investigation on the $S^1$-invariant contact structures suggests an interesting and fairly strong upper bound for the self-intersection of a rational unicuspidal curve with one Puiseux pair $(p,q)$ in any algebraic surface (the bound depends only on the values $p,q$), and for the symplectic version, we prove the existence of an ``optimal" symplectic rational unicuspidal curve in a rational $4$-manifold which realizes the upper bound for any given Puiseux pair $(p,q)$. Our results also suggest a revisit of the ``symplectic divisorial capping" problem first considered by Li and Mak. Further applications of the techniques developed in this paper hinge upon better understandings on the tightness and fillability criterions of $S^1$-invariant contact structures as well as their (small) symplectic fillings.

math.GT↗
arXiv2024

Are We There Yet? Unraveling the State-of-the-Art Smart Contract Fuzzers

Given the growing importance of smart contracts in various applications, ensuring their security and reliability is critical. Fuzzing, an effective vulnerability detection technique, has recently been widely applied to smart contracts. Despite numerous studies, a systematic investigation of smart contract fuzzing techniques remains lacking. In this paper, we fill this gap by: 1) providing a comprehensive review of current research in contract fuzzing, and 2) conducting an in-depth empirical study to evaluate state-of-the-art contract fuzzers' usability. To guarantee a fair evaluation, we employ a carefully-labeled benchmark and introduce a set of pragmatic performance metrics, evaluating fuzzers from five complementary perspectives. Based on our findings, we provide direction for the future research and development of contract fuzzers.

cs.SE↗
arXiv2020

Finite group actions on symplectic Calabi-Yau $4$-manifolds with $b_1>0$

This is the first of a series of papers devoted to the topology of symplectic Calabi-Yau $4$-manifolds endowed with certain symplectic finite group actions. We completely determine the fixed-point set structure of a finite cyclic action on a symplectic Calabi-Yau $4$-manifold with $b_1>0$. As an outcome of this fixed-point set analysis, the $4$-manifold is shown to be a $T^2$-bundle over $T^2$ in some circumstances, e.g., in the case where the group action is an involution which fixes a $2$-dimensional surface in the $4$-manifold. Our project on symplectic Calabi-Yau $4$-manifolds is based on an analysis of the existence and classification of disjoint embeddings of certain configurations of symplectic surfaces in a rational $4$-manifold. This paper lays the ground work for such an analysis at the homological level. Some other result which is of independent interest, concerning the maximal number of disjointly embedded symplectic $(-2)$-spheres in a rational $4$-manifold, is also obtained.

math.GT↗
arXiv2020

On a class of symplectic $4$-orbifolds with vanishing canonical class

A study of certain symplectic $4$-orbifolds with vanishing canonical class is initiated. We show that for any such symplectic $4$-orbifold $X$, there is a canonically constructed symplectic $4$-orbifold $Y$, together with a cyclic orbifold covering $Y\rightarrow X$, such that $Y$ has at most isolated Du Val singularities and a trivial orbifold canonical line bundle. The minimal resolution of $Y$, to be denoted by $\tilde{Y}$, is a symplectic Calabi-Yau $4$-manifold endowed with a natural symplectic finite cyclic action, extending the deck transformations of the orbifold covering $Y\rightarrow X$. Furthermore, we show that when $b_1(X)>0$, $\tilde{Y}$ is a $T^2$-bundle over $T^2$ with symplectic fibers, and when $b_1(X)=0$, $\tilde{Y}$ is either an integral homology $K3$ surface or a rational homology $T^4$; in the latter case, the singular set of $X$ is completely classified. To further investigate the topology of $X$, we introduce a general successive symplectic blowing-down procedure, which may be of independent interest. Under suitable assumptions, the procedure allows us to successively blow down a given symplectic rational $4$-manifold to $CP^2$, during which process we can canonically transform a given configuration of symplectic surfaces to a "symplectic arrangement" of pseudoholomorphic curves in $CP^2$. The procedure is reversible; by a sequence of successive blowing-ups in the reversing order, one can recover the original configuration of symplectic surfaces up to a smooth isotopy.

math.GT↗
arXiv2020

Resolving symplectic orbifolds with applications to finite group actions

We associate to each symplectic $4$-orbifold $X$ a canonical smooth symplectic resolution $π: \tilde{X}\rightarrow X$, which can be done equivariantly if $X$ comes with a symplectic $G$-action by a finite group. Moreover, we show that the resolutions of the symplectic $4$-orbifolds $X/G$ and $\tilde{X}/G$ are in the same symplectic birational equivalence class; in fact, the resolution of $\tilde{X}/G$ can be reduced to that of $X/G$ by successively blowing down symplectic $(-1)$-spheres. To any finite symplectic $G$-action on a $4$-manifold $M$, we associate a pair $(M_G,D)$, where $π: M_G\rightarrow M/G$ is the canonical resolution of the quotient orbifold and $D$ is the pre-image of the singular set of $M/G$ under $π$. We propose to study the group action on $M$ by analyzing the smooth or symplectic topology of $M_G$ as well as the embedding of $D$ in $M_G$. In this paper, an investigation on the symplectic Kodaira dimension $κ^s$ of $M_G$ is initiated. In particular, we conjecture that $κ^s(M_G)\leq κ^s(M)$. The inequality is verified for several classes of symplectic $G$-actions, including any actions on a rational surface or a symplectic $4$-manifold with $κ^s=0$.

math.SG↗
arXiv2017

Symplectic rational $G$-surfaces and equivariant symplectic cones

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.

math.SG↗
arXiv2015

$G$-minimality and invariant negative spheres in $G$-Hirzebruch surfaces

In this paper a study of $G$-minimality, i.e., minimality of four-manifolds equipped with an action of a finite group $G$, is initiated. We focus on cyclic actions on $CP^2\# \overline{CP^2}$, and our work shows that even in this simple setting, the comparison of $G$-minimality in the various categories, i.e., locally linear, smooth, and symplectic, is already delicate and interesting. For example, we show that if a symplectic $Z_n$-action on $CP^2\# \overline{CP^2}$ has an invariant locally linear topological $(-1)$-sphere, then it must admit an invariant symplectic $(-1)$-sphere, provided that $n=2$ or $n$ is odd. For the case where $n>2$ and even, the same conclusion holds under a stronger assumption, i.e., the invariant $(-1)$-sphere is smoothly embedded. Along the way of these proofs we develop certain techniques for producing embedded invariant $J$-holomorphic two-spheres of self-intersection $-r$ under a weaker assumption of an invariant smooth $(-r)$-sphere for $r$ relatively small compared with the group order $n$. We then apply the techniques to give a classification of $G$-Hirzebruch surfaces (i.e., Hirzebruch surfaces equipped with a homologically trivial, holomorphic $G=Z_n$-action) up to orientation-preserving equivariant diffeomorphisms. The main issue of the classification is to distinguish non-diffeomorphic $G$-Hirzebruch surfaces which have the same fixed-point set structure. An interesting discovery is that these non-diffeomorphic $G$-Hirzebruch surfaces have distinct equivariant Gromov-Taubes invariant, giving the first examples of such kind. Going back to the original question of $G$-minimality, we show that for $G=Z_n$, a minimal rational $G$-surface is minimal as a symplectic $G$-manifold if and only if it is minimal as a smooth $G$-manifold.

math.GT↗
arXiv2015

Fixed-point free circle actions on 4-manifolds

This paper is concerned with fixed-point free $S^1$-actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which support a fixed-point free smooth (resp. locally linear) $S^1$-action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free $S^1$-actions under some further conditions on the fundamental group. The connection between the classification of the $S^1$-manifolds and the fundamental group is given by a certain decomposition, called fiber-sum decomposition, of the $S^1$-manifolds. More concretely, each fiber-sum decomposition naturally gives rise to a Z-splitting of the fundamental group. There are two technical results in this paper which play a central role in our considerations. One states that the Z-splitting is a canonical JSJ decomposition of the fundamental group in the sense of Rips and Sela. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free $S^1$-action on the 4-manifold must be infinite, unless the 4-manifold is the mapping torus of a periodic diffeomorphism of some elliptic 3-manifold. The paper ends with two questions concerning the topological nature of the smooth classification and the Seiberg-Witten invariants of 4-manifolds admitting a smooth fixed-point free $S^1$-action.

math.GT↗
arXiv2014

Finite symmetries of $S^4$

This paper discusses topological and locally linear actions of finite groups on $S^4$. Local linearity of the orientation preserving actions on $S^4$ forces the group to be a subgroup of $SO(5)$. On the other hand, orientation reversing topological actions of "exotic" groups $G$ (i.e. $G\not\subset O(5)$) on $S^4$ are constructed, and local linearity and stable smoothability of the actions are studied.

math.GT↗
arXiv2013

Seiberg-Witten invariants of 3-orbifolds and non-Kähler surfaces

A formula is given which computes the Seiberg-Witten invariant of a 3-orbifold from the invariant of the underlying manifold. As an application, we derive a formula for the Seiberg-Witten invariant of a non-Kähler complex surface, which was originally due to O. Biquard \cite{Biq} and S.R. Williams \cite{W} independently.

math.GT↗
arXiv2013

Hurwitz-type bound, knot surgery, and smooth $\s^1$-four-manifolds

In this paper we prove several related results concerning smooth $\Z_p$ or $\s^1$ actions on 4-manifolds. We show that there exists an infinite sequence of smooth 4-manifolds $X_n$, $n\geq 2$, which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each $X_n$ supports no smooth $\s^1$-actions but admits a smooth $\Z_n$-action. In order to construct such manifolds, we devise a method for annihilating smooth $\s^1$-actions on 4-manifolds using Fintushel-Stern knot surgery, and apply it to the Kodaira-Thurston manifold in an equivariant setting. Finally, the method for annihilating smooth $\s^1$-actions relies on a new obstruction we derived in this paper for existence of smooth $\s^1$-actions on a 4-manifold: the fundamental group of a smooth $\s^1$-four-manifold with nonzero Seiberg-Witten invariant must have infinite center. We also include a discussion on various analogous or related results in the literature, including locally linear actions or smooth actions in dimensions other than four.

math.GT↗
arXiv2011

Seifert fibered four-manifolds with nonzero Seiberg-Witten invariant

The main result of this paper asserts that if a Seifert fibered 4-manifold has nonzero Seiberg-Witten invariant, the homotopy class of regular fibers has infinite order. This is a nontrivial obstruction to smooth circle actions; as applications, we show how to destroy smooth circle actions on a 4-manifold by knot surgery, without changing the integral homology, intersection form, and even the Seiberg-Witten invariant. Results concerning classification of Seifert fibered complex surfaces or symplectic 4-manifolds are included. We also show that every smooth circle action on the 4-torus is smoothly conjugate to a linear action.

math.GT↗
arXiv2011

Symmetric symplectic homotopy K3 surfaces

A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.

math.GT↗
arXiv2010

Group actions on 4-manifolds: some recent results and open questions

A survey of finite group actions on symplectic 4-manifolds is given with a special emphasis on results and questions concerning smooth or symplectic classification of group actions, group actions and exotic smooth structures, and homological rigidity and boundedness of group actions. We also take this opportunity to include several results and questions which did not appear elsewhere.

math.GT↗
arXiv2010

On the orders of periodic diffeomorphisms of 4-manifolds

This paper initiated an investigation on the following question: Suppose a smooth 4-manifold does not admit any smooth circle actions. Does there exist a constant $C>0$ such that the manifold support no smooth $\Z_p$-actions of prime order for $p>C$? We gave affirmative results to this question for the case of holomorphic and symplectic actions, with an interesting finding that the constant $C$ in the holomorphic case is topological in nature while in the symplectic case it involves also the smooth structure of the manifold.

math.GT↗