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Weimin Chen

Publications and source records attributed to Weimin Chen.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2008

Symmetries and exotic smooth structures on a $K3$ surface

Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order $\geq 7$ is proved and nonexistence of smooth actions by several $K3$ groups is established (included among which is the binary tetrahedral group $T_{24}$ which has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of a prime order $\geq 5$ is explicitly determined, provided that the action is homologically nontrivial.

math.GT↗
arXiv2007

Symplectic symmetries of 4-manifolds

A study of symplectic actions of a finite group $G$ on smooth 4-manifolds is initiated. The central new idea is the use of $G$-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries. The main result in this paper is a complete description of the fixed-point set structure (and the action around it) of a symplectic cyclic action of prime order on a minimal symplectic 4-manifold with $c_1^2=0$. Comparison of this result with the case of locally linear topological actions is made. As an application of these considerations, the triviality of many such actions on a large class of 4-manifolds is established. In particular, we show the triviality of homologically trivial symplectic symmetries of a $K3$ surface (in analogy with holomorphic automorphisms). Various examples and comments illustrating our considerations are also included.

math.GT↗
arXiv2006

On a notion of maps between orbifolds, II. homotopy and CW-complex

This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem which asserts that a weak homotopy equivalence is a homotopy equivalence is extended to the orbifold category.

math.AT↗
arXiv2006

On a notion of maps between orbifolds, I. function spaces

This is the first of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we define the maps in the more general context of orbispaces, and establish several basic results concerning the topological structure of the space of such maps. In particular, we show that the space of such maps of C^r-class between smooth orbifolds has a natural Banach orbifold structure if the domain of the map is compact, generalizing the corresponding result in the manifold case. Motivations and applications of the theory come from string theory and the theory of pseudoholomorphic curves in symplectic orbifolds.

math.GT↗
arXiv2006

Smooth s-cobordisms of elliptic 3-manifolds

The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smoothly a product. We explain how the conjecture fits naturally into the program of Taubes of constructing symplectic structures on an oriented smooth 4-manifold from generic self-dual harmonic forms. The paper also contains an auxiliary result of independent interest, which generalizes Taubes' theorem SW --> Gr to the case of symplectic 4-orbifolds.

math.GT↗
arXiv2005

Pseudoholomorphic curves in four-orbifolds and some applications

The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on symplectic 4-manifolds concerning pseudoholomorphic curves and Seiberg-Witten theory. They form the technical backbone of the proof that a symplectic s-cobordism of elliptic 3-manifolds (with a canonical contact structure on the boundary) is smoothly a product. One interesting feature of the theory is that existence of pseudoholomorphic curves gives certain restrictions on the singular points of the 4-orbifold contained by the pseudoholomorphic curves. In the last section, we discuss applications (or potential ones) of the theory in some other problems such as symplectic finite group actions on 4-manifolds, symplectic circle actions on 6-manifolds, and algebraic surfaces with quotient singularities.

math.SG↗
arXiv2004

On a theorem of Peters on automorphisms of Kahler surfaces

For any Kahler surface which admits no nonzero holomorphic vectorfields, we consider the group of holomorphic automorphisms which induce identity on the second rational cohomology. Assuming the canonical linear system is without base points and fixed components, C.A.M. Peters showed that this group is trivial except when the Kahler surface is of general type and either $c_1^2=2c_2$ or $c_1^2=3c_2$ holds. Moreover, this group is a 2-group in the former case, and is a 3-group in the latter. The purpose of this note is to give further information about this group. In particular, we show that $c_1^2$ is divisible by the order of the group. Our argument is based on the results of C.H. Taubes on symplectic 4-manifolds, which are applied here in an equivariant setting.

math.AG↗
arXiv2004

Orbifold adjunction formula and symplectic cobordisms between lens spaces

Each lens space has a canonical contact structure which lifts to the distribution of complex lines on the three-sphere. In this paper, we show that a symplectic homology cobordism between two lens spaces, which is given with the canonical contact structure on the boundary, must be diffeomorphic to the product of a lens space with the unit interval. As one of the main ingredients in the proof, we also derive in this paper the adjunction and intersection formulae for pseudoholomorphic curves in an almost complex 4-orbifold, extending the relevant work of Gromov and McDuff in the manifold setting.

math.GT↗
arXiv2001

A Homotopy Theory of Orbispaces

In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was the prediction that a certain physicist's Euler number of the orbifold must be equal to the Euler number of any of its crepant resolutions. This was soon related to the so called McKay correspondence in mathematics (cf. [McK]). Later developments include stringy Hodge numbers (cf. [Z], [BD]), mirror symmetry of Calabi-Yau orbifolds (cf. [Ro]), and most recently the Gromov-Witten invariants of symplectic orbifolds (cf. [CR1-2]). One common feature of these studies is that certain contributions from singularities, which are called ``twisted sectors'' in physics, have to be properly incorporated. This is called the ``stringy aspect'' of an orbifold (cf. [R]). This paper makes an effort to understand the stringy aspect of orbifolds in the realm of ``traditional mathematics''.

math.AT↗
arXiv2001

Orbifold Gromov-Witten Theory

In this article, we introduce the notion of good map and use it to establish Gromov-Witten theory for orbifolds.

math.AG↗
arXiv2001

A New Cohomology Theory for Orbifold

Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality and has a natural ring structure. Some examples of orbifold cohomology ring are computed.

math.AG↗
arXiv2000

Symplectic Lefschetz fibrations on S^1 x M^3

In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold must fiber over a circle, and up to a self-diffeomorphism of the four-manifold, the symplectic structure is deformation equivalent to the canonical symplectic structure determined by the fibration of the three-manifold over the circle.

math.DG↗
arXiv2000

Various regimes of flux motion in Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ single crystals

Four regimes of vortex motion were identified in the magnetoresistance of Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ single crystals: (1) thermally activated flux flow (TAFF) in samples with surface defects caused by thermal annealing; (2) TAFF-like plastic motion of highly entangled vortex liquid at low temperatures, with $U_{pl} \sim (1-T/T_c)/H^{1/2}$; (3) pure free flux flow above the region of (2) in clean and optimally doped samples; or, in its place, (4) a combination of (2) and (3). This analysis gives an overall picture of flux motion in Bi cuprates.

cond-mat.supr-con↗
arXiv2000

Experimental fitting to the bipolaronic model of the normal-state resistance of Bi_2Sr_2CaCu_2O_8 single crystals

Normal-state resistance data from Bi_2Sr_2CaCu_2O_8 single crystals were fitted to the (bi)polaronic conduction model, $R=R_0 (T+σ_b T^2)/(1+bT)$, with satisfactory agreement over a wide temperature range. The fluctuating conduction region is found to be much narrower than that in the usual sense, as is the case for a charged Bose-gas. We estimate the effective (bi)polaron mass to be ~ 4m_e.

cond-mat.supr-con↗
arXiv2000

Pseudo-holomorphic curves and the Weinstein conjecture

We established existence of periodic Reeb orbits for a large class of tight contact structures on closed 3-manifolds, notably the Stein fillable structures, based on a fundamental theorem of Cliff Taubes on symplectic 4-manifolds.

dg-ga↗
arXiv2000

The Seiberg-Witten theory of homology 3-spheres

In this thesis we study the Seiberg-Witten theory of an oriented homology 3-sphere. The goal is to extract topological invariants - the Seiberg-Witten invariants - by counting the solutions to the Seiberg-Witten equations on the manifold. The first question we consider is whether the Seiberg-Witten invariants depend on the geometric or analytic data involved in their definition. In the first main result of this thesis, we completely determine the dependence of the Seiberg-Witten invariants on the data involved in their definition. In particular, we show that even for the simplest manifold, the 3-sphere $S^3$, the Seiberg-Witten invariants take infinitely many different values. The rest of this thesis is devoted to understanding the Seiberg-Witten invariants in a specific geometric setting - the surgery setting. In that context we prove a gluing formula, which identifies the Seiberg-Witten invariants as certain ``homological intersection numbers''.

dg-ga↗