Related papers: Corks, covers, and complex curves

For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$ (for $d=7$ under some assumption). We also count plane rational quartic curves…

We prove that there is an infinite sequence of pairs of plane cuspidal curves $C_{m,1}$ and $C_{m,2}$, such that the pairs $(\Bbb CP^2, C_{m,1})$ and $(\Bbb CP^2, C_{m,2})$ are diffeomorphic, but $C_{m,1}$ and $C_{m,2}$ have non-equivalent…

In this paper, we investigate the possibility of constructing isomonodromic deformations of logarithmic connections on curves by using ramified covers. We give new examples and prove a classification result.

We study the geometry of the simplest type of compact arithmetic quotients of the hyperbolic 2-ball $\mathbb{B}^2$, which has a moduli interpretation for certain types of abelian varieties of dimension 6 with $\mathcal{O}_F$-endomorphism,…

We study those real $\mathcal{C}^\infty$ foliations in complex surfaces whose leaves are holomorphic curves. The main motivation is to try and understand these foliations in neighborhoods of curves: can we expect the space of foliations in…

Given a closed complex hypersurface $Z\subset \mathbb{C}^{N+1}$ $(N\in\mathbb{N})$ and a compact subset $K\subset Z$, we prove the existence of a pseudoconvex Runge domain $D$ in $Z$ such that $K\subset D$ and there is a complete proper…

We prove that the envelope of meromorphy of any imbedded symplectic sphere in $CP^2$ coincides with the whole $CP^2$. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as…

We provide the first information on diffeotopy groups of exotic smoothings of R^4: For each of uncountably many smoothings, there are uncountably many isotopy classes of self-diffeomorphisms. We realize these by various explicit group…

We prove that there exist infinitely many embedded tori with a common geometric dual in $T^4\#(S^2\times S^2)$ that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations. These…

These results complete our paper in Hiroshima Mathematical Journal, vol. 42, pp. 37-75. Let C be a covering of the plane by disjoint complete folding curves which satisfies the local isomorphism property. We show that C is locally…

The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular…

One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair…

The author recently proved the existence of an infinite order cork: a compact, contractible submanifold $C$ of a 4-manifold and an infinite order diffeomorphism $f$ of $\partial C$ such that cutting out $C$ and regluing it by distinct…

We consider the subspace of the homology of a covering space spanned by lifts of simple closed curves. Our main result is the existence of unbranched covers of surfaces where this is a proper subspace. More generally, for a fixed finite…

We discuss corks, and introduce new objects which we call plugs. Though plugs are fundamentally different objects, they also detect exotic smooth structures in 4-manifolds like corks. We discuss relation between corks, plugs and rational…

We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More…

We study smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli, i.e., embeddings pairwise homeomorphic to the standard embeddings of R^2 and R^2-int D^2 in R^4. We encounter two uncountable…

Rational curves on Hilbert schemes of points on $K3$ surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up…

Rationally convex topological embeddings of compact surfaces (closed or with boundary) into $\mathbb{C}^2$ are constructed.

This is primarily an exposition, combining work of several authors (Curtis, Hsiang, Freedman, Stong, Matveyev, and Bizaca), of the proof that a smooth 5-dimensional h-cobordism between simply connected 4-manifolds is a product off of a…