Related papers: Tamed to compatible when b^(2+) = 1 and b^1 = 2

Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat…

We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic $2$-form which tames the underlying almost complex structure.

Let $(X,\omega)$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures $\mathcal{J}_{\omega}$ using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional…

We show that any compact half-conformally flat manifold of negative type, with bounded $L^2$ energy, sufficiently small scalar curvature, and a non-collapsing assumption, has all betti numbers bounded. We show that this result is optimal…

We introduce the secondary Stiefel-Whitney class $\tilde w_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class $\tilde…

The present work is concerned with the study of four-dimensional irreducible holomorphic symplectic manifolds with second Betti number $23$. We describe their birational geometry and their relations to EPW sextics.

Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b^+=1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is…

We prove results exploring the relationship between the fundamental group and the second Betti number of minimal symplectic fillings of lens spaces. These results unify and generalize several disparate facts appearing in the literature. The…

In this note, we show that for a closed almost-K\"{a}hler manifold $(X,J)$ with the almost complex structure $J$ satisfies $\dim\ker P_{J}=b_{2}-1$ the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern…

A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set,…

This is the sequel to our first paper concerning the balanced embedding of a non-compact complex manifold into an infinite-dimensional projective space. We prove the uniqueness of such an embedding. The proof relies on fine estimates of the…

The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial n-complex into $R^{2n}$ for $n\neq 2$, and it was recently shown to be incomplete for $n=2$. We use algebraic-topological…

We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are…

We prove that a positive definite smooth four-manifold with $b_2^+ \geq 2$ and having either no 1-handles or no 3-handles cannot admit a symplectic structure.

We point out a gap in the proof of the Davis--Januszkiewicz theorem on cohomology of small covers of simple polytopes, and give a correction to this proof. We use this theorem to compute explicitly the Betti numbers for a wide class of…

A compact oriented 4-manifold is defined to be of ``superconformal simple type'' if certain polynomials in the basic classes (constructed using the Seiberg-Witten invariants) vanish identically. We show that all known 4-manifolds of…

A new graph, called the symplectic inner product graph $Spi\big(2\nu,q\big)$, over a finite field $\mathbb{F}_q$ is introduced. We show that $Spi\big(2\nu,q\big)$ is connected with diameter $4$ if and only if $\nu\geq2$ and the automorphism…

We extend a result of M. Katz on conformal systoles to all four-manifolds with b^+=1 which have odd intersection form. The same result holds for all four-manifolds with b^+=1 with even intersection form and which are symplectic or satisfy…

We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable…

Symplectic 4-manifolds $(X,\omega)$ with $b_+{=}1$ are roughly classified by the canonical class $K$ and the symplectic form $\omega$ depending upon the sign of $K^2$ and $K\cdot \omega$. Examples are known for each category except for the…