Complex Hypersurface in Complex Projective Space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:41:12Z http://mathoverflow.net/feeds/question/58875 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58875/complex-hypersurface-in-complex-projective-space Complex Hypersurface in Complex Projective Space unknown (yahoo) 2011-03-18T21:33:38Z 2011-03-21T12:30:52Z <p>Apparently 2 smooth complex hypersurface in complex projective space that have the same degree are diffeomorphic. Does anyone know where the proof of this can be found ? Is there a counter example for symplectic manifolds?</p> http://mathoverflow.net/questions/58875/complex-hypersurface-in-complex-projective-space/58944#58944 Answer by Dylan Wilson for Complex Hypersurface in Complex Projective Space Dylan Wilson 2011-03-19T23:56:41Z 2011-03-19T23:56:41Z <p>Here is a proof of the much stronger result that, given any two nonsingular degree d hypersurfaces in $CP^{n+1}$, there exists a diffeomorphism $CP^{n+1} \rightarrow CP^{n+1}$ isotopic to the identity that restricts to a diffeomorphism of the two hypersurfaces. This is taken from the paper <em>Topology of Nonsingular Complex Hypersurfaces</em> by Kulkarni and Wood.</p> <blockquote> <p>Let $X$ (resp. $Y$) be defined by the polynomial $p(z)$ (resp. $q(z)$). Then the polynomial $f(t,z) = t_0p(z) + t_1q(z)$ of homogeneous bidegree $(1,d)$ defines a hypersurface $F$ in $CP^1 \times CP^{n+1}$. The set $S \subset CP^1 \times CP^{n+1}$ of points $[t,z]$ at which $F \cap [t] \times CP^{n+1}$ is singular is a closed algebraic set. So the projection $\pi(S)$ of $S$ onto $CP^1$ is also a closed algebraic set and since evidently $\pi(S) \ne CP^1$, $\pi(S)$ is zero dimensional hence a finite set of points. Let $I$ be a smooth arc in $CP^1$ from $[1,0]$ to $[0,1]$ in the complement of $\pi(S)$. Then $\pi^{-1}(I)= I \times CP^{n+1}$ contains the smooth submanifold $M = \pi^{-1}(1) \cap F$ of real codimension $2$ such that $\pi: M \rightarrow I$ is a product bundle: $\phi: I \times X \rightarrow M$. Thus $M$ may be regarded as the graph of an isotopy of $X$ in $CP^{n+1}$. Let $\partial/\partial t$ be the vector field on $I \times X$ tangential to the first factor. Then $\phi_{*}(\partial/\partial t)$, which is tangent to $M$, extends to a vector field $V$ on $I \times CP^{n+1}$. The integral flow of this vector field gives the desired ambient isotopy.</p> </blockquote> http://mathoverflow.net/questions/58875/complex-hypersurface-in-complex-projective-space/59027#59027 Answer by jvp for Complex Hypersurface in Complex Projective Space jvp 2011-03-21T01:48:26Z 2011-03-21T12:30:52Z <p>A more general version of the question made by the OP is the following</p> <blockquote> <p><strong>Question.</strong> Let $H_1$ and $H_2$ be two smooth connect hypersurfaces in a projective manifold $X$. Suppose $H_1$ is homologous to $H_2$, or equivalently that the Chern classes of $\mathcal O_X(H_1)$ and $\mathcal O_X(H_2)$ coincide. Is it true that $H_1$ is diffeomorphic to $H_2$ ?</p> </blockquote> <p>A similar question was proposed by Fulton: can we determine the Betti numbers of a smooth divisor as a function of $X$ and of its Chern class ? See Totaro's "<a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.mmj/1030132736" rel="nofollow">The topology of smooth divisors and the arithmetic of abelian varieties</a>" for a thorough discussion.</p> <p><strong>Positive answer when $H^1(X, \mathcal O_X)=0$.</strong> The argument sketched by Jack Huizenga in the comments shows that the answer is yes if we further assume that $H^1(X, \mathcal O_X)=0$. In this case, the line-bundles $\mathcal O_X(H_1)$ and $\mathcal O_X(H_2)$ coincide since the exponential sequence $$0 \to \mathcal O_X \to \mathcal O_X^* \to \mathbb Z \to 0$$ implies the Chern class morphism $c: H^1(X,\mathcal O_X^*) \to H^2(X, \mathbb Z)$ is injective. </p> <p>Thus $H_1$ and $H_2$ are both members of the same linear system, and we can consider the incidence variety $$Z = \lbrace (x,[ \sigma ] ) \in X \times \mathbb P H^0(X, \mathcal O_X(H_1)) ; \sigma(x)=0 \rbrace$$ which comes with a natural morphism $\pi : Z \to \mathbb PH^0(X, \mathcal O_X(H_1))$. The subset $U \subset \mathbb P H^0(X,\mathcal O_X(H_1))$ corresponding to sections with smooth zeros is clearly open in the Zariski topology, and non-empty since $H_1$ and $H_2$ are smooth. Consequently $U$ is also connected. </p> <p>Take a path $\gamma : [0 ,1] \to U$ connecting the sections defining $H_1$ and $H_2$. The real variety $Y = \pi^{-1} ( \gamma [0,1])$ now has as boundaries $H_1$ and $H_2$ and comes with a submersion $\pi:Y \to [0,1]$. If we consider the gradient of $\pi$ ( for any Riemmanian metric on $Y$ ) then its flow will define a diffeomorphism between any two fibers of $\pi$.</p> <p><strong>Counter-example.</strong> If $H^1(X,\mathcal O_X) \neq 0$ then $H_1$ and $H_2$ are not necessarily diffeomorphic. The following example appears in Totaro's paper. Let $C_1$ and $C_2$ be two smooth curves of genus $>1$. Let $B_1 \to C_1$ and $B_2 \to C_2$ be two non-trivial double coverings. The group $\Gamma = (\mathbb Z / 2 \mathbb Z)^2$ acts freely on $B_1 \times B_2$ with quotient $C_1 \times C_2$. Let $\Gamma$ act on $\mathbb P^1$ through the automorphism $x \mapsto -x$ and $x \mapsto 1/x$, and take $X$ as the quotient of $B_1 \times B_2 \times \mathbb P^1$ by $\Gamma$.</p> <p>Clearly $X$ is smooth ( since the action is free ) and comes with a fibration $\pi: X \to \mathbb P^1$. This fibration has exactly two non-reduced fibers and each has multiplicity two. One has support equal to $C_1 \times B_2$ while the support of the other is $B_1 \times C_2$.</p> <p>It can be verified that $H^2(X,\mathbb Z)$ is torsion free. Consequently the support of the two non-reduced fibers have the same Chern classes. For a general choice of $C_1$ and $C_2$ they are not diffeomorphic. </p> <p><strong>Another question.</strong> The example above suggests the following conjecture made by Totaro in the very same paper. </p> <blockquote> <p><strong>Conjecture.</strong> Let $H_1$ and $H_2$ be two smooth connect hypersurfaces in a projective manifold $X$. Suppose $H_1$ is homologous to $H_2$, or equivalently that the Chern classes of $\mathcal O_X(H_1)$ and $\mathcal O_X(H_2)$ coincide. Then there exists an étale cyclic covering of $H_1$ which is deformation equivalent to an étale cyclic covering of $H_2$. </p> </blockquote> <p>Evidence toward this conjecture is also presented by Totaro. If the Picard variety of $X$ is isogenous to a product of elliptic curves then there exists étale coverings of $H_1$ and $H_2$ which deformation equivalent via passage to some characteristic $p>0$.</p> <p>Another evidence toward this conjecture is presented <a href="http://ams.impa.br/mathscinet/search/publdoc.html?pg1=INDI&amp;s1=684880&amp;vfpref=html&amp;r=21&amp;mx-pid=2177196" rel="nofollow">here</a> where an analogous statement for divisors in compact Kahler manifolds is proved after replacing "deformation equivalent" by "diffeomorphic".</p>