Exotic smooth structures and symplectic forms on closed manifolds

Symplectic Structures on the cotangent bundles of open 4-manifolds

We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stein homotopic to the standard Stein structure on $T^*\mathbb{R}^4 = \mathbb{R}^8$. We use this to show that any exotic $\mathbb{R}^4$ embeds in the standard symplectic $\mathbb{R}^8$ as a Lagrangian submanifold. As a corollary, we show that $\mathbb{R}^8$ has uncountably many smoot…

Symplectic surfaces in a fixed homology class

The purpose of this paper is to investigate the following problem: For a fixed 2-dimensional homology class K in a simply connected symplectic 4-manifold, up to smooth isotopy, how many connected smoothly embedded symplectic submanifolds represent K? We show that when K can be represented by a symplectic torus, there are many instances when K can be representated by infinitely many non-isotopic symplectic tori.

Gromov Invariants and Symplectic Maps

Given a symplectomorphism f of a symplectic manifold X, one can form the `symplectic mapping cylinder' $X_f = (X \times R \times S^1)/Z$ where the Z action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$. In this paper we compute the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$ with other symplectic manifolds. This is done by expressing the Gromov invariants in terms of the Lefschetz zeta function of f and, in special cases, in terms of the Alexander polynomials of knots. The result is a large set of interesting non-Kahler symplectic manifolds with computational ways o…

Symplectic rational blowdowns

We prove that the rational blowdown, a surgery on smooth 4-manifolds introduced by Fintushel and Stern, can be performed in the symplectic category. As a consequence, interesting families of smooth 4-manifolds, including the exotic $K3$ surfaces of Gompf and Mrowka, admit symplectic structures.

Diffeomorphisms and almost complex structures on tori

We prove that there exist diffeomorphisms of tori, supported in a disc, which are not isotopic to symplectomorphisms with respect to any symplectic structure. This yields a partial negative answer to a question of Benson and Gordon about the existence of symplectic structures on tori with exotic differential structure.

A regularization approach to non-smooth symplectic geometry

We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly singular symplectic forms, including the continuous non-differentiable case, we prove the existence of generalized Darboux coordinates in the sense of a local non-smooth pull-back to the canonical symplectic form on the cotangent bundle.

The Donaldson geometric flow for symplectic four-manifolds

This is an exposition of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The original work appeared in [1].

An interesting symplectic 4-manifold with small Euler characteristic

In this article we construct a minimal symplectic 4-manifold R that has small Euler characteristic (e(R)=8) and two essential Lagrangian tori with nice properties. These properties make R particularly suitable for constructing interesting examples of symplectic manifolds with small Euler characteristic. In particular, we construct an exotic symplectic CP^2# 5(-CP^2), the smallest known minimal symplectic 4-manifold with pi_1=Z, the smallest known minimal symplectic 4-manifolds with pi_1=Z/a + Z/b for all a,b in Z, and the smallest known minimal symplectic 4-manifold with pi_1=Z^3. We use the p…

A perturbation and generic smoothness of the Vafa-Witten moduli spaces on closed symplectic four-manifolds

We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa--Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the $PU(2)$-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa--Witten equations, and prove that the moduli space of solutions to the perturbed Vafa-Witten equations on a closed symplectic four-manifold for the structure group $SU(2)$ or $SO(3)$ is a smooth manifold of dimension zero for a generic choice of the perturbation para…

D-branes and synthetic/$C^{\infty}$-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifolds

We lay down an elementary yet fundamental lemma concerning a finite algebraicness property of a smooth map from an Azumaya/matrix manifold with a fundamental module to a smooth manifold. This gives us a starting point to build a synthetic (synonymously, $C^{\infty}$-algebraic) symplectic geometry and calibrated geometry that are both tailored to and guided by D-brane phenomena in string theory and along the line of our previous works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]).

Smooth maps of a foliated manifold in a symplectic manifold

The immersions of a smooth manifold $M$ in a symplectic manifold $(N,σ)$ inducing a given closed form $ω$ on $M$ satisfy the $C^0$-dense $h$-principle in the space of all continuous maps which pull back the deRham cohomology class of $σ$ onto that of $ω$. In this paper we prove a foliated version of this result due to Gromov.

Coisotropic Displacement and Small Subsets of a Symplectic Manifold

We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing result for neighborhoods of products of unit spheres. 3. Existence of a "badly squeezable" set in $\mathbb{R}^{2n}$ of Hausdorff dimension at most $d$, for every $n\geq2$ and $d\geq n$. 4. Existence of a stably exotic symplectic form on $\mathbb{R}^{2n}$, for every $n\geq2$. 5. Non-triviality of a new capacity, which is based on the minimal symplectic area of a…

Poisson deformations of symplectic quotient singularities

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence). We prove further that if G i…

Exotic smooth structures on nonpositively curved symmetric spaces

We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the change of the smooth structure we use a tangential map from the locally symmetric space its dual compact type twin.

Scalar Curvature Functions of Almost-Kähler Metrics

For a closed smooth manifold $M$ admitting a symplectic structure, we define a smooth topological invariant $Z(M)$ using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce $Z(M, [[ω]])$ depending on symplectic deformation equivalence class $[[ω]]$. We first prove that there exists a 6-dimensional smooth manifold $M$ with more than one deformation equivalence classes with different signs of $Z(M, [[ω]] )$. Using $Z$ invariants, we set up a Kazdan-Warner type problem of classifying symplectic manifolds into three categories. We finally prove t…

辛流形的PL逼近

PL approximations of symplectic manifolds

本文定义了赋予分段常数Whitney辛形式的PL辛流形，证明光滑辛流形可被PL辛流形进行$C^0$-逼近，并证明其体积形式可被逼近PL流形的体积形式三角剖分。

聆听奇异光滑结构

Hearing Exotic Smooth Structures

本文研究了紧致李群等距作用下闭黎曼流形上黎曼拉普拉斯算子的基本谱，证明了特定的同胚但不微分同胚的同伦球与连通和可共享相同的基本谱及不变标量曲率函数环。

R^2被非有限辛群作用所得商空间的辛解消

Symplectic resolutions of the quotient of R^2 by a non-finite symplectic group

本研究将Du Val辛解消从代数几何拓展到实微分几何，构造了R^2在两类无限离散辛群作用下商空间的光滑辛解消，证明对应实Du Val奇异簇的极小解消均为辛解消。

The Normalized Ricci Flow on Four-Manifolds and Exotic Smooth Structures

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with uniformly bounded sectional curvature. In dimension 4, there exist many compact topological manifolds admitting distinct smooth structures, i.e., exotic smooth structures. Interestingly, in this article, the difference between existence and non-existence of non-singular solutions to …

Model and Set-Theoretic Aspects of Exotic Smoothness Structures on $\mathbb{R}^4$

Model-theoretic aspects of exotic smoothness were studied long ago uncovering unexpected relations to noncommutative spaces and quantum theory. Some of these relations were worked out in detail in later work. An important point in the argumentation was the forcing construction of Cohen but without a direct application to exotic smoothness. In this article we assign the set-theoretic forcing on trees to Casson handles and characterize small exotic smooth $R^4$ from this point of view. Moreover, we show how models in some Grothendieck toposes can help describing such differential structures in d…

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 generaliz…

An essential relation between Einstein metrics, volume entropy, and exotic smooth structures

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positive minimal volume entropy. 2) They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality, with a minimal volume entropy term. 3) They nevertheless admit infinitely many distinct smooth structures for which no compatible Einstein metric exists.

具有指定基本群的奇异不可定向四维流形

Exotic non-orientable four-manifolds with prescribed fundamental group

本文证明具有指数2子群的有限表示群可作为闭光滑不可定向四维流形的基本群，该流形通过Gluck扭结获得奇异光滑结构，其定向2-覆盖微分同胚，且两种结构在添加不同流形后表现出不等价与稳定化特性。

The global moduli theory of symplectic varieties

We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which does not use the existence of a hyperkähler metric or twistor deformations.

A smooth counterexample to the Hamiltonian Seifert conjecture in R^6

A smooth counterexample to the Hamiltonian Seifert conjecture for six-dimensional symplectic manifolds is found. In particular, we construct a smooth proper function on the symplectic 2n-dimensional vector space, 2n > 4, such that one of its non-singular level sets carries no periodic orbits of the Hamiltonian flow. The function can be taken to be C^0-close and isotopic to a positive-definite quadratic form so that the level set in question is isotopic to an ellipsoid. This is a refinement of previously known constructions giving such functions for 2n > 6. The proof is based on a new version o…