Complex hypersurface in complex projective space

Question

Apparently two smooth complex hypersurfaces in the complex projective space that have the same degree are diffeomorphic. Does anyone know where the proof of this can be found ? Is there a counterexample for symplectic manifolds?

Answer 1

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 Topology of Nonsingular Complex Hypersurfaces by Kulkarni and Wood.

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.

Answer 2

A more general version of the question made by the OP is the following

Question. 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 $H_2$ ?

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 "The topology of smooth divisors and the arithmetic of abelian varieties" for a thorough discussion.

Positive answer when $H^1(X, \mathcal O_X)=0$. 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 $$ 0 \to \mathbb Z \to \mathcal O_X \to \mathcal O_X^* \to 0 $$ implies the Chern class morphism $c: H^1(X,\mathcal O_X^*) \to H^2(X, \mathbb Z)$ is injective.

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.

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 $\pi$.

Counter-example. 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 $\Gamma$.

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 $B_1 \times C_2$.

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.

Another question. The example above suggests the following conjecture made by Totaro in the very same paper.

Conjecture. 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 $H_2$.

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 $p>0$.

Another evidence toward this conjecture is presented here where an analogous statement for divisors in compact Kahler manifolds is proved after replacing "deformation equivalent" by "diffeomorphic".

Answer 3

There is a proof of this result in Gompf and Stipsicz's book "4-Manifolds and Kirby Calculus": See Claim 1.3.11.