most recent 30 from http://mathoverflow.net 2013-05-24T22:37:06Z http://mathoverflow.net/feeds/question/42744 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42744/diffeomorphic-kahler-manifolds-with-different-hodge-numbers Sándor Kovács 2010-10-19T06:44:02Z 2010-10-19T07:37:10Z

<p><a href="http://mathoverflow.net/questions/42709/question-about-hodge-number/42735#42735" rel="nofollow">This</a> question made me wonder about the following:</p> <p>Are there orientedly diffeomorphic Kähler manifolds with different Hodge numbers?</p> <p>It seems that this would require that those manifolds are not deformation equivalent. However, there are examples by Catanese and Manetti that that happens already for smooth projective surfaces. </p>

http://mathoverflow.net/questions/42744/diffeomorphic-kahler-manifolds-with-different-hodge-numbers/42746#42746 Greg Kuperberg 2010-10-19T07:12:04Z 2010-10-19T07:37:10Z

<p>This question was <a href="http://www.mathkb.com/Uwe/Forum.aspx/research/766/Hodge-numbers" rel="nofollow">debated in another forum</a> a few years ago. The result was <a href="http://www.iecn.u-nancy.fr/~gaillard/DIVERS/hodgenumbers.pdf" rel="nofollow">a note by Frédéric Campana</a> in which he describes a counterexample as a corollary of another construction. In 1986 Gang Xiao found two simply connected complex surfaces $S$ and $S'$ (that is, complex dimension 2), with different Hodge numbers, that are <em>homeomorphic</em> by Freedman's classification. The homeomorphism has to be orientation-reversing, but $S \times S$ and $S' \times S'$ are orientedly diffeomorphic and of course still have different Hodge numbers. Freedman's difficult classification is not essential to the argument, because in 8 real dimensions you can use standard surgery theory to establish the diffeomorphism.</p> <p>Campana also explains that Borel and Hirzebruch found the first counterexample in 1959, in 5 complex dimensions.</p>