Return to Answer

As noticed by Sasha in his comment, the answer is no.

The following proof also shows that this result cannot be generalized for higher values of the degree.

For any smooth complex hypersurface of degree $d$, say $X_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have $$H^k(X_d)= H^k(\mathbb{P}^n) \quad \mathrm{for} \; k \neq n.$$

In particular, all the odd Betti numbers are zero, except possibly the middle Betti number when $n$ is odd. On the other hand, the Euler-Poincare characteristic of $X_d$ is equal to $$\chi(X_d)= \langle c_n(T_{X_d}), [X_d] \rangle =\frac{1}{d}[(1-d)^{n+2}-1]+n+2,$$ so for $n$ odd and $d=2$ the middle cohomology group must be zero too. Notice that for $n$ odd and $d >2$ one always has a non-zero middle Betti number. For instence, if $X \subset \mathbb{P}^4$ is a smooth cubic hypersurface, then $b_3(X)=10$.

A good reference for these results is Dimca's book "Singularities and topology of hypersurfaces", Chapter 5, which also considers the case of hypersurfaces in $\mathbb{P}^{n+1}$ with isolated singularities.

As noticed by Sasha in his comment, the answer is no.

The following proof also shows that this result cannot be generalized for higher values of the degree.

For any smooth complex hypersurface of degree $d$, say $X_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have $$H^k(X_d)= H^k(\mathbb{P}^n) \quad \mathrm{for} \; k \neq n.$$

In particular, all the odd Betti numbers are zero, except possibly the middle Betti number when $n$ is odd. On the other hand, the Euler-Poincare characteristic of $X_d$ is equal to $$\chi(X_d)= \langle c_n(T_{X_d}), [X_d] \rangle =\frac{1}{d}[(1-d)^{n+2}-1]+n+2,$$ so for $n$ odd and $d=2$ the middle cohomology group must be zero too. Notice that for $n$ odd and $d >2$ one always has a non-zero middle Betti number. For instence, if $X \subset \mathbb{P}^4$ is a smooth cubic hypersurface, then $b_3(X)=10$.

A good reference for these results is Dimca's book Singularities and topology of hypersurfaces, Chapter 5, which also considers the case of hypersurfaces in $\mathbb{P}^{n+1}$ with isolated singularities.

As noticed by Sasha in his comment, the answer is no.

The following proof also shows that this result cannot be generalized for higher values of the degree.

For any smooth complex hypersurface of degree $d$, say $X_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have

$H^k(X_d)= H^k(\mathbb{P}^n)$ for $k \neq n$.

In particular, all the odd Betti numbers are zero, except possibly the middle Betti number when $n$ is odd. On the other hand, the Euler-Poincare characteristic of $X_d$ is equal to

$\chi(X_d)= \langle c_n(T_{X_d}), [X_d] \rangle =\frac{1}{d}[(1-d)^{n+2}-1]+n+2$,

so for $n$ odd and $d=2$ the middle cohomology group must be zero too. Notice that for $n$ odd and $d >2$ one always has a non-zero middle Betti number. For instence, if $X \subset \mathbb{P}^4$ is a smooth cubic hypersurface, then $b_3(X)=10$.

A good reference for these results is Dimca's book "Singularities and topology of hypersurfaces", Chapter 5, which also considers the case of hypersurfaces in $\mathbb{P}^{n+1}$ with isolated singularities.

As noticed by Sasha in his comment, the answer is no.

The following proof also shows that this result cannot be generalized for higher values of the degree.

For any smooth complex hypersurface of degree $d$, say $X_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have $$H^k(X_d)= H^k(\mathbb{P}^n) \quad \mathrm{for} \; k \neq n.$$

In particular, all the odd Betti numbers are zero, except possibly the middle Betti number when $n$ is odd. On the other hand, the Euler-Poincare characteristic of $X_d$ is equal to $$\chi(X_d)= \langle c_n(T_{X_d}), [X_d] \rangle =\frac{1}{d}[(1-d)^{n+2}-1]+n+2,$$ so for $n$ odd and $d=2$ the middle cohomology group must be zero too. Notice that for $n$ odd and $d >2$ one always has a non-zero middle Betti number. For instence, if $X \subset \mathbb{P}^4$ is a smooth cubic hypersurface, then $b_3(X)=10$.

A good reference for these results is Dimca's book "Singularities and topology of hypersurfaces", Chapter 5, which also considers the case of hypersurfaces in $\mathbb{P}^{n+1}$ with isolated singularities.

As noticed by Sasha in his comment, the answer is no.

The following proof also shows that this result cannot be generalized for higher values of the degree.

For any smooth complex hypersurface of degree $d$, say $X_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have

$H^k(X_d)= H^k(\mathbb{P}^n)$ for $k \neq n$.

In particular, all the odd Betti numbers are zero, except possibly the middle Betti number when $n$ is odd. On the other hand, the Euler-Poincare characteristic of $X_d$ is equal to

$\chi(X_d)= \langle c_n(T_{X_d}), [X_d] \rangle =\frac{1}{d}[(1-d)^{n+2}-1]+n+2$,

so for $n$ odd and $d=2$ the middle cohomology group must be zero too. Notice that for $n$ odd and $d >2$ one always has a non-zero middle Betti number. For instence, if $X \subset \mathbb{P}^4$ is a smooth cubic hypersurface, then $b_3(X)=4$.

A good reference for these results is Dimca's book "Singularities and topology of hypersurfaces", Chapter 5, which also considers the case of hypersurfaces in $\mathbb{P}^{n+1}$ with isolated singularities.

As noticed by Sasha in his comment, the answer is no.

The following proof also shows that this result cannot be generalized for higher values of the degree.

For any smooth complex hypersurface of degree $d$, say $X_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have

$H^k(X_d)= H^k(\mathbb{P}^n)$ for $k \neq n$.

In particular, all the odd Betti numbers are zero, except possibly the middle Betti number when $n$ is odd. On the other hand, the Euler-Poincare characteristic of $X_d$ is equal to

$\chi(X_d)= \langle c_n(T_{X_d}), [X_d] \rangle =\frac{1}{d}[(1-d)^{n+2}-1]+n+2$,

so for $n$ odd and $d=2$ the middle cohomology group must be zero too. Notice that for $n$ odd and $d >2$ one always has a non-zero middle Betti number. For instence, if $X \subset \mathbb{P}^4$ is a smooth cubic hypersurface, then $b_3(X)=10$.

A good reference for these results is Dimca's book "Singularities and topology of hypersurfaces", Chapter 5, which also considers the case of hypersurfaces in $\mathbb{P}^{n+1}$ with isolated singularities.