Is canonical class a topological invariant? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:58:45Z http://mathoverflow.net/feeds/question/70429 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70429/is-canonical-class-a-topological-invariant Is canonical class a topological invariant? jerrysciencemath 2011-07-15T12:11:33Z 2012-02-29T10:20:33Z <p>For a $n$-dim smooth projective complex algebraic variety $X$, we can form the complex line bundle $\Omega^n$ of holomorphic $n$-form on $X$. Let $K_X$ be the divisor class of $\Omega^n$, then $K_X$ is called the canonical class of $X$.</p> <p><strong>Question</strong>: Is homology class of $K_X$ in $H_{2n-2}(X)$ a topological invariant? If it's true, please tell me the idea of proof or some references. If not, please give me the counterexamples.</p> http://mathoverflow.net/questions/70429/is-canonical-class-a-topological-invariant/70437#70437 Answer by Francesco Polizzi for Is canonical class a topological invariant? Francesco Polizzi 2011-07-15T13:53:54Z 2012-02-29T10:20:33Z <p>For the question about <em>homeomorphisms</em> the answer is <em>no</em>, even if $X$ and $X'$ are algebraic surfaces. </p> <p>In fact, in his paper [Orientation reversing homeomorphisms in surface geography, Math. Ann. 292 (1992)], D. Kotschick proves the following result:</p> <blockquote> <p><strong>Theorem.</strong> There exist infinitely many pairs of simply connected algebraic surfaces of general type which are orientation-reversing homeomorphic (with respect to their complex orientations), but not diffeomorphic.</p> </blockquote> <p>He also makes a conjecture about <em>orientation-reversing diffeomorphic</em> algebraic surfaces. As I said in my comments before, by using Seiberg-Witten theory one proves that, given <em>any</em> diffeomorphism $\phi \colon X \to X'$ between two smooth $4$-manifolds, one has either $\phi(K_X)=K_{X'}$ or $\phi(K_X)=-K_{X'}$. </p> <p>Kotschick's conjecture is therefore the following:</p> <blockquote> <p><strong>Conjecture.</strong> If two algebraic surface with finite fundamental group are orientation-reversing diffeomorphic, then they are homeomorphic to a geometrically ruled rational surface. In particular, they are simply connected.</p> </blockquote> <p>I do not know the current state of this conjecture. </p> <p><strong>Added On February 29, 2012</strong>. D. Kotschick kindly informed me that he actually proved this conjecture in his paper <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&amp;aid=18825" rel="nofollow">Orientations and geometrizations of compact complex surfaces</a>, Bulletin of the London Mathematical Society <strong>29</strong> (1997), 145-149. </p> http://mathoverflow.net/questions/70429/is-canonical-class-a-topological-invariant/70438#70438 Answer by Dmitri for Is canonical class a topological invariant? Dmitri 2011-07-15T14:07:44Z 2011-07-15T14:07:44Z <p>It is well known that in dimension $3$ and higher there exist complex structures on diffeomerphic manifolds with totally different Chern classes (and Chern numbers).</p> <p>For the case of complex manifolds you can check </p> <p><a href="http://mathoverflow.net/questions/26586/can-one-bound-the-todd-class-of-a-3-dimensional-variety-polynomially-in-c-3/26598#26598" rel="nofollow">http://mathoverflow.net/questions/26586/can-one-bound-the-todd-class-of-a-3-dimensional-variety-polynomially-in-c-3/26598#26598</a></p> <p>For the case of complex projective manifolds the reference given in the same answer: </p> <p><a href="http://arxiv.org/PS_cache/arxiv/pdf/0903/0903.1587v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/0903/0903.1587v1.pdf</a></p> http://mathoverflow.net/questions/70429/is-canonical-class-a-topological-invariant/70445#70445 Answer by Tim Perutz for Is canonical class a topological invariant? Tim Perutz 2011-07-15T15:44:15Z 2011-07-16T01:20:47Z <p>This answer is about the case of complex surfaces $X$ and their diffeomorphisms (all my diffeos are assumed to be orientation-preserving!). </p> <p><b>(1) Examples of self-diffeomorphisms that reverse the sign of the canonical class.</b> </p> <p>Take $X=\mathbb{C}P^1\times \mathbb{C}P^1$. Let $\tau$ be reflection in the equator of $S^2=\mathbb{C}P^1$. Then $\tau \times \tau$ preserves orientation and acts as $-I$ on $H^2(X)$. It therefore sends $K_X$ to $-K_X$.</p> <p>One can also realise the automorphism $-I$ of $H^2(X)$ by a diffeomorphism when $X$ is the blow-up of the projective plane at $k$ points, $k = 2,3,\dots,9$. This follows from a result of C.T.C. Wall from</p> <p><i>Diffeomorphisms of 4-manifolds</i>, J. London Math. Soc. 39 (1964) 131–140, MR0163323</p> <p>Wall says that if $N$ is a simply connected, closed oriented 4-manifold with $b_2(N)&lt;9$, and $X$ is the connected sum of $N$ with $S^2 \times S^2$, then all automorphisms of the intersection form of $X$ are realised by diffeos. To apply this, recall that the 1-point blow-up of $\mathbb{C}P^1\times \mathbb{C}P^1$ is the 2-point blow up of the projective plane. (Wall's strategy, by the way, is to factor the automorphism into reflections along hyperplanes, and to realise those.)</p> <p><b>(2) Results from Seiberg-Witten theory.</b> </p> <p>These results tie complex geometry amazingly closely to differential topology. They say that the unsigned pair $\pm K_X$ is invariant under diffeomorphisms (Witten <a href="http://arxiv.org/abs/hep-th/9411102" rel="nofollow">http://arxiv.org/abs/hep-th/9411102</a> and others); so too is the Kodaira dimension; so too are the plurigenera (Friedman-Morgan <a href="http://arxiv.org/abs/alg-geom/9502026" rel="nofollow">http://arxiv.org/abs/alg-geom/9502026</a>). </p> <p>In Kodaira dimension $&lt;2$, one can take this further and prove that oriented-diffeomorphic surfaces are actually deformation-equivalent (to be safe, let me specify the simply connected case). But that's <i>not</i> the explanation in general: there are pairs of simply connected general-type surfaces that are diffeomorphic (by diffeos preserving the canonical class), which are not deformation-equivalent (Catanese-Wajnryb <a href="http://arxiv.org/abs/math/0405299" rel="nofollow">http://arxiv.org/abs/math/0405299</a>).</p> <p><b>(3) How it happens.</b></p> <p>The Seiberg-Witten invariant (for an oriented 4-manifold with $b^+(X)>1$) is a map $$SW: Spin^c(X)\to\mathbb{Z}$$ defined on the $H^2(X)$-torsor of $Spin^c$-structures. The overall sign is equivalent to a "homology orientation". It's natural under diffeomorphisms. It's also invariant under "conjugation" $\mathfrak{s}\mapsto \bar{\mathfrak{s}}$ of $Spin^c$-structures.</p> <p>For algebraic surfaces, there's a canonical spin-c structure $\mathfrak{s}$, so $Spin^c(X)$ is identified with $H^2(X)$. Witten (http://arxiv.org/abs/hep-th/9411102) observed that the elliptic equations that define $SW$ simplify drastically in the algebraic case; in evaluating $SW$ on a cohomology class represented by a complex line bundle $L\to X$, you're led to consider a moduli space of pairs consisting of a holomorphic structure on the line bundle and a holomorphic section of it, with an obstruction bundle on the moduli space. Conjugation-invariance becomes Serre duality. </p> <p>For general type surfaces, $\pm SW(\mathfrak{s}) = \pm SW(\bar{\mathfrak{s}}) = \pm 1$; all other spin-c structures have vanishing invariant. Since $c_1(\mathfrak{s})=-c_1(\bar{\mathfrak{s}})=-K$, one deduces diffeomorphism-invariance of $\pm K$. For lower Kodaira dimension, a more complicated analysis is needed.</p>