Non-bimeromorphic compactifications - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:39:27Z http://mathoverflow.net/feeds/question/86000 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86000/non-bimeromorphic-compactifications Non-bimeromorphic compactifications diverietti 2012-01-18T14:55:26Z 2012-01-18T17:49:16Z <p>Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify this subset with $X$) and such that $\overline X\setminus X$ is a proper analytic subset (with no conditions on its codimension). </p> <p>In particular, given two (different) compactifications of $X$, they always contain biholomorphic dense open subsets. </p> <p>My question is: given two compactifications of $X$, are they necessarily bimeromorphic?</p> <p>More precisely, does the biholomorphism between the two dense open subsets always extend to a global bimeromorphic map?</p> <p>I guess the answer is no, but after a moment of reflection I cannot find any counterexample. Perhaps, it would suffice to look at compactifications of $\mathbb C^2$... </p> http://mathoverflow.net/questions/86000/non-bimeromorphic-compactifications/86004#86004 Answer by Francesco Polizzi for Non-bimeromorphic compactifications Francesco Polizzi 2012-01-18T15:37:13Z 2012-01-18T15:37:13Z <p>As you guessed, the answer is <strong>no</strong>. </p> <p>A counterexample in dimension $2$ can be found in Vo Van Tan's paper <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0195&amp;DMDID=dmdlog32" rel="nofollow">On the compactification of strongly pseudoconvex surfaces III</a>: the Stein surface $\mathbb{C}^* \times \mathbb{C}^*$ admits two algebraic compactifications $M_1$ and $M_2$ which are <em>not bimeromorphic</em>. </p> <p>In fact, $M_1=\mathbb{P}^1 \times \mathbb{P}^1$ is a rational surface whereas $M_2$ is a $\mathbb{P}^1$-bundle over an elliptic curve.</p> http://mathoverflow.net/questions/86000/non-bimeromorphic-compactifications/86009#86009 Answer by jvp for Non-bimeromorphic compactifications jvp 2012-01-18T16:07:47Z 2012-01-18T16:23:44Z <p>Another example, different but diffeomorphic to Polizzi's, is given by<br> $E \times \mathbb C$, where $E$ is a fixed elliptic curve say $\mathbb C^* / \lbrace z \mapsto 2z \rbrace$. It can be compactified as the projective surface $E \times \mathbb P^1$ or as the (non-Kähler) Hopf surface $\mathbb C^2-{0} / \lbrace (z,w) \mapsto (2z,2w) \rbrace$. </p> <p>Notice that the two compactifications have fields of meromorphic functions of different transcendence degree over $\mathbb C$, i.e., they have different algebraic dimensions. </p>