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Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold $X$ with an embedding $S^2\times S^2\setminus \Delta\to X$ on to a dense open subset.

Of course $X=S^2\times S^2$ is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to $S^2\times S^2$, with $h^2(X)=2$?

And more generally, in four-dimensional, replace $S^2$ by a compact Riemann surface and the same question.

Even more generally, let $S^k\Sigma$ be the $k$-th symmetric product of a compact Riemann surface, which is a $2k$-dimensional manifold. Consider $S^k\Sigma\times S^l\Sigma\setminus \Delta$, where $\Delta= \{ (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset \}$. What kind of compatification of $S^k\Sigma\times S^l\Sigma\setminus \Delta$ can we have? (How to describe the minimum condition in this case?)

p.s. Many blow-up constructions give candidate compactifications, but they will increase the second betti number $h^2$. So I restrict this question to "minimal" compactifications.

Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold $X$ with an embedding $S^2\times S^2\setminus \Delta\to X$ on to a dense open subset.

Of course $X=S^2\times S^2$ is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to $S^2\times S^2$, with $h^2(X)=rank(H^2(X))=2$?

And more generally, in four-dimensional, replace $S^2$ by a compact Riemann surface and the same question.

Even more generally, let $S^k\Sigma$ be the $k$-th symmetric product of a compact Riemann surface, which is a $2k$-dimensional manifold. Consider $S^k\Sigma\times S^l\Sigma\setminus \Delta$, where $\Delta= \{ (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset \}$. What kind of compatification of $S^k\Sigma\times S^l\Sigma\setminus \Delta$ can we have? (How to describe the minimum condition in this case?)

p.s. Many blow-up constructions give candidate compactifications, but they will increase the second betti number $h^2$. So I restrict this question to "minimal" compactifications.

Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold $X$ with an embedding $S^2\times S^2\setminus \Delta\to X$ on to a dense open subset.

Of course $X=S^2\times S^2$ is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to $S^2\times S^2$?

And more generally, in four-dimensional, replace $S^2$ by a compact Riemann surface and the same question.

Even more generally, let $S^k\Sigma$ be the $k$-th symmetric product of a compact Riemann surface, which is a $2k$-dimensional manifold. Consider $S^k\Sigma\times S^l\Sigma\setminus \Delta$, where $\Delta= \{ (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset \}$. What kind of compatification of $S^k\Sigma\times S^l\Sigma\setminus \Delta$ can we have?

Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold $X$ with an embedding $S^2\times S^2\setminus \Delta\to X$ on to a dense open subset.

Of course $X=S^2\times S^2$ is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to $S^2\times S^2$, with $h^2(X)=2$?

And more generally, in four-dimensional, replace $S^2$ by a compact Riemann surface and the same question.

Even more generally, let $S^k\Sigma$ be the $k$-th symmetric product of a compact Riemann surface, which is a $2k$-dimensional manifold. Consider $S^k\Sigma\times S^l\Sigma\setminus \Delta$, where $\Delta= \{ (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset \}$. What kind of compatification of $S^k\Sigma\times S^l\Sigma\setminus \Delta$ can we have? (How to describe the minimum condition in this case?)

p.s. Many blow-up constructions give candidate compactifications, but they will increase the second betti number $h^2$. So I restrict this question to "minimal" compactifications.

Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open 4-fold. By a compactification of it, I mean a closed 4-fold $X$ with an embedding $S^2\times S^2\setminus \Delta\to X$ on to a dense open subset.

Of course $X=S^2\times S^2$ is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to $S^2\times S^2$?

And more generally, in four-dimensional, replace $S^2$ by a compact Riemann surface and the same question.

Even more generally, let $S^k\Sigma$ be the $k$-th symmetric product of a compact Riemann surface, which is a $2k$-dimensional manifold. Consider $S^k\Sigma\times S^l\Sigma\setminus \Delta$, where $\Delta= \{ (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset \}$. What kind of compatification of $S^k\Sigma\times S^l\Sigma\setminus \Delta$ can we have?

Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold $X$ with an embedding $S^2\times S^2\setminus \Delta\to X$ on to a dense open subset.

Of course $X=S^2\times S^2$ is an obvious candidate. My question is that do we have other compactifications which is not (diffeo)homeomorphic to $S^2\times S^2$?

And more generally, in four-dimensional, replace $S^2$ by a compact Riemann surface and the same question.

Even more generally, let $S^k\Sigma$ be the $k$-th symmetric product of a compact Riemann surface, which is a $2k$-dimensional manifold. Consider $S^k\Sigma\times S^l\Sigma\setminus \Delta$, where $\Delta= \{ (x, y)\in S^k\Sigma\times S^l\Sigma| x\cap y\neq \emptyset \}$. What kind of compatification of $S^k\Sigma\times S^l\Sigma\setminus \Delta$ can we have?