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We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\o …

There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions (e.g. complex manifo …

For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a co …

Does any integer surgery on a right or left trefoil knot give the $S^1$-bundle over $T^2$ with Euler number $1$?

We know that a Brieskorn homology 3-spheres $\Sigma(p,q,r)$ admit a free $S^1$-action, which makes it a Seifert fibered spaces with three singular fibers: $M(b;r_1,r_2,r_3)$. How should one get from $ …

It is a theorem of Iwase that every 4-manifold can be obtained from a connected sum of a number of $\pm CP^2$ and $S^1\times B^3$ by surgery along tori: Iwase, Dehn surgery along a torus T2-knot II, …

If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values) may change? under …

I have a somewhat general and vague question in mind. Is there anything in literature related to Affine varieties as examples of Stein manifolds? I know that there is a topological approach to Stein m …

I have a basic question which I am not able to figure out. If we do a Gluck twist on a nullhomologous 2-sphere in a 4-manifold, it is said that it does not change its intersection form. But as far as …