By purely topological reasons we know that every simply connected, 4-dimensional CW-complex
is formal. We also know there exist non-formal 4-dimensional CW-complexes.
As mentioned in the introduction, thanks to deep results of
Simpson and Kapovich–Kollár we know that there exist non-formal complex projective surfaces.
In this section we prove that every complex projective surface with normal singularities is formal.
We generalize this result in two directions.
First, we prove formality for projective varieties of dimension *n* with only
isolated singularities whose link is $(n-2)$-connected. Second, we prove formality for those
projective varieties with normal isolated singularities admitting a resolution of singularities
with smooth exceptional divisor.

*Every normal complex projective surface is a formal topological space.*

#### Proof.

Let *X* be a normal complex projective surface, which we may assume to be connected.
We use the formulas for ${E}_{1}^{*,*}(X)$ given in Proposition 2.13.
Since normal singularities have codimension $\ge 2$, we have $dim(\mathrm{\Sigma})=0$.
Since $dim({D}^{(p)})=1-p$, we have ${E}_{1}^{p,q}(X)=0$ for all $q>4-2p$.
By Lemma 3.7 together with semi-purity, the weight filtration
on ${H}^{k}(X;\mathbb{Q})$ is pure of weight *k* for all $k\ne 2$.
We have

For all $p,q\ge 0$ take a section ${E}_{2}^{p,q}(X)\to Ker({d}_{1}^{p,q})\subset {E}_{1}^{p,q}(X)$ of the projection
$Ker({d}_{1}^{p,q})\twoheadrightarrow {E}_{2}^{p,q}(X)$.
This defines a morphism $\rho :({E}_{2}^{*,*}(X),0)\to ({E}_{1}^{*,*}(X),{d}_{1})$
of bigraded complexes which is a quasi-isomorphism. We next show that ρ is multiplicative.
By bidegree reasons, the only non-trivial products in ${E}_{2}^{*,*}(X)$ are the products

$Ker({d}_{1}^{0,q})\times Ker({d}_{1}^{0,{q}^{\prime}})\to Ker({d}_{1}^{0,q+{q}^{\prime}})$

induced by the cup product of ${H}^{*}(\stackrel{~}{X};\mathbb{Q})$.
Since ρ is the identity on $Ker({d}_{1}^{0,q})$, it preserves these products.
It also preserves the unit $1\in Ker({d}_{1}^{0,0})$.
It only remains to see that
the diagram

commutes.
By Proposition 2.13, the term ${E}_{1}^{0,1}(X)$ is given by the pull-back

and
${E}_{1}^{1,k}(X)\cong {H}^{k}({D}^{(1)};\mathbb{Q})\otimes \mathrm{\Lambda}(t)dt$ for $k\in \{1,2\}$.
Moreover, the differential ${d}_{1}^{0,1}:{E}_{1}^{0,1}(X)\to {E}_{1}^{1,1}(X)$ is given by $(x,a(t))\mapsto {a}^{\prime}(t)dt$ and the product
${E}_{1}^{0,1}(X)\times {E}_{1}^{1,1}(X)\to {E}_{1}^{1,2}(X)$
is given by $(x,a(t))\cdot b(t)dt=a(t)b(t)dt$.

Let $(x,a(t))\in {E}_{1}^{0,1}(X)$. Since $a(0)=0$, it follows that
$(x,a(t))\in Ker({d}_{1}^{0,1})$ if and only if $a(t)=0$.
Therefore we have $Ker({d}_{1}^{0,1})\cdot {E}_{1}^{1,1}(X)=0$, and the above diagram commutes.
This proves that the map

$\rho :({E}_{2}^{*,*}(X),0)\to ({E}_{1}^{*,*}(X),{d}_{1})$

is multiplicative.
Since *X* is connected, it has a mixed Hodge diagram $\mathcal{\mathcal{A}}(X)$ with ${H}^{0}(\mathcal{\mathcal{A}}{(X)}_{\mathbb{Q}})\cong \mathbb{Q}$. Hence
by Theorem 2.4 we have a string of quasi-isomorphisms of complex cdgas

$({\mathcal{\mathcal{A}}}_{\mathrm{pl}}(X),d)\otimes \u2102\stackrel{\sim}{\u27f7}({E}_{1}^{*,*}(X),{d}_{1})\otimes \u2102\stackrel{\sim}{\u27f5}({E}_{2}^{*,*}(X),0)\otimes \u2102\cong ({H}^{*}(X;\u2102),0).$

To conclude that *X* is formal it suffices to apply descent of formality of cdgas from $\u2102$ to $\mathbb{Q}$.
∎

The following is an example of a normal projective surface with isolated singularities and non-trivial weight filtration on ${H}^{2}$
(cf. [28, §7], see also [3]).

Let *C* be a curve of degree $d\ge 3$ with $n>0$ nodes in $\u2102{\mathbb{P}}^{2}$. The genus of *C* is given by $g=(d-1)(d-2)/2-n$.
Choose a smooth projective curve ${C}^{\prime}$ of
degree ${d}^{\prime}=d+1$ intersecting *C* transversally at smooth points of *C*, so that $|C\cap {C}^{\prime}|=d{d}^{\prime}$
and consider the blow-up $\stackrel{~}{X}=B{l}_{C\cap {C}^{\prime}}\u2102{\mathbb{P}}^{2}$ of $\u2102{\mathbb{P}}^{2}$ at the $d{d}^{\prime}$ points of $C\cap {C}^{\prime}$.
Then the proper transform $\stackrel{~}{C}$ of *C* has negative self-intersection $|\stackrel{~}{C}\cap \stackrel{~}{C}|=d(d-{d}^{\prime})=-d$
and we may consider the blow-down *X* of $\stackrel{~}{C}$ to a point.
Explicitly, assume that the curve *C* is given by $f(x,y,z)=0$, and that ${C}^{\prime}$
is given by $g(x,y,z)=0$. Then *X* is the projective variety
defined by the equation $wf(x,y,z)+g(x,y,z)=0,$
which has a normal isolated singularity at $(0,0,0,1)$.
Here $(x,y,z,w)$ are the homogeneous coordinates in $\u2102{\mathbb{P}}^{3}$.
The normalization of $\stackrel{~}{C}$ is a smooth projective curve of genus *g* and $\stackrel{~}{X}$ is homeomorphic to the connected sum of
$d{d}^{\prime}+1$ projective planes.
Deligne’s weight spectral sequence can be written as

Hence ${H}^{2}(X;\mathbb{Q})$ has a non-trivial weight filtration:

${\mathrm{Gr}}_{0}^{W}{H}^{2}(X;\mathbb{Q})\cong {\mathbb{Q}}^{n},{\mathrm{Gr}}_{1}^{W}{H}^{2}(X;\mathbb{Q})\cong {\mathbb{Q}}^{2g}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mathrm{Gr}}_{2}^{W}{H}^{2}(X;\mathbb{Q})\cong {\mathbb{Q}}^{d{d}^{\prime}}.$

Since *X* is simply connected (see for example [10, Corollary V.2.4]), we may compute the rational homotopy groups of *X* with their weight filtration from a
bigraded minimal model $\rho :M\stackrel{\sim}{\to}{E}_{2}^{*,*}(X)$ of the bigraded algebra ${E}_{2}^{*,*}(X)$.
The weight filtration on ${\pi}_{i}:={\pi}_{i}(X)\otimes \mathbb{Q}$
satisfies ${\mathrm{Gr}}_{q}^{W}{\pi}_{p+q}\cong Hom(Q{(M)}^{p,q},\mathbb{Q})$,
where the term $Q{(M)}^{p,q}$ denotes the indecomposables of *M* of bidegree $(p,q)$.

The cohomology ring of $\stackrel{~}{X}$ is given by
${H}^{*}(\stackrel{~}{X};\mathbb{Q})\cong \mathbb{Q}[a,{b}_{1},\mathrm{\dots},{b}_{d{d}^{\prime}}]$ with
${a}^{2}=T$, ${b}_{i}^{2}=-T$ and ${b}_{i}\cdot {b}_{j}=0$ for all $i\ne j$. Here
*T* denotes the top class of $\stackrel{~}{X}$, *a* is the hyperplane class and ${b}_{i}$ correspond to the exceptional divisors.
Let ${\gamma}_{i}:=a-d\cdot {b}_{i}$. Then
${E}_{2}^{0,2}(X)\cong Ker({d}_{1}^{0,2})\cong \mathbb{Q}[{\gamma}_{1},\mathrm{\dots},{\gamma}_{d{d}^{\prime}}]$ with
${\gamma}_{i}^{2}=T(1-{d}^{2})$ and
${\gamma}_{i}\cdot {\gamma}_{j}={d}^{2}T$ for $i\ne j$.
Hence we may write

${E}_{2}^{*,*}(X)\cong \mathbb{Q}[{\alpha}_{1},\mathrm{\dots},{\alpha}_{n},{\beta}_{1},\mathrm{\dots},{\beta}_{2g},{\gamma}_{1},\mathrm{\dots},{\gamma}_{d{d}^{\prime}}],$

where the generators have bidegree
$|{\alpha}_{i}|=(2,0)$, $|{\beta}_{i}|=(1,1)$ and $|{\gamma}_{i}|=(0,2)$.
By bidegree reasons, the only non-trivial products are given by
${\gamma}_{i}^{2}=T(1-{d}^{2})$ and
${\gamma}_{i}\cdot {\gamma}_{j}={d}^{2}T$, for all $i\ne j$.
We compute the first steps of a minimal model for ${E}_{2}^{*,*}(X)$. Let ${M}_{2}$ be the free bigraded algebra

${M}_{2}=\mathrm{\Lambda}({\overline{\alpha}}_{1},\mathrm{\dots},{\overline{\alpha}}_{n},{\overline{\beta}}_{1},\mathrm{\dots},{\overline{\beta}}_{2g},{\overline{\gamma}}_{1},\mathrm{\dots},{\overline{\gamma}}_{d{d}^{\prime}})$

with trivial differential generated by elements of bidegree
$|{\overline{\alpha}}_{i}|=(2,0)$, $|{\overline{\beta}}_{i}|=(1,1)$ and $|{\overline{\gamma}}_{i}|=(0,2)$.
Then the map ${\rho}_{2}:{M}_{2}\to {E}_{2}^{*,*}(X)$ given by $\overline{x}\mapsto x$ is a
2-quasi-isomorphism of bigraded algebras. Hence we have

${\mathrm{Gr}}_{0}^{W}{\pi}_{2}\cong {\mathbb{Q}}^{n},{\mathrm{Gr}}_{1}^{W}{\pi}_{2}\cong {\mathbb{Q}}^{2g}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mathrm{Gr}}_{2}^{W}{\pi}_{2}\cong {\mathbb{Q}}^{d{d}^{\prime}}.$

Let ${M}_{3}={M}_{2}{\otimes}_{d}\mathrm{\Lambda}({V}_{3,0},{V}_{2,1},{V}_{1,2},{V}_{0,3},{V}_{-1,4})$ where
${V}_{i,j}$ are the graded vector spaces of pure bidegree $(i,j)$, and $d:{V}_{i,j}\to {M}_{2}^{i+1,j}$ are the differentials
given by

${V}_{3,0}=\mathbb{Q}\u3008{x}_{ij}\u3009,d{x}_{ij}={\overline{\alpha}}_{i}{\overline{\alpha}}_{j},1\le i\le j\le n,$${V}_{2,1}=\mathbb{Q}\u3008{y}_{ij}\u3009,d{y}_{ij}={\overline{\alpha}}_{i}{\overline{\beta}}_{j},1\le i\le n,1\le j\le 2g,$${V}_{1,2}=\mathbb{Q}\u3008{z}_{ij},{w}_{kl}\u3009,d{z}_{ij}={\overline{\alpha}}_{i}{\overline{\gamma}}_{j},d{w}_{kl}={\overline{\beta}}_{k}{\overline{\beta}}_{l},1\le i\le n,1\le j\le d{d}^{\prime},1\le k\le l\le 2g,$${V}_{0,3}=\mathbb{Q}\u3008{\tau}_{ij}\u3009,d{\tau}_{ij}={\overline{\beta}}_{i}{\overline{\gamma}}_{j},1\le i\le n,1\le j\le 2g,$${V}_{-1,4}=\mathbb{Q}\u3008{\xi}_{ij}\u3009,d{\xi}_{ij}={\overline{\gamma}}_{i}{\overline{\gamma}}_{j},1\le i\le d{d}^{\prime},(i,j)\ne (1,1).$

Then the extension ${\rho}_{3}:{M}_{3}\to {E}_{2}^{*,*}(X)$ of ${\rho}_{2}$ given by ${V}_{i,j}\mapsto 0$ is a 3-quasi-isomorphism.
The formula
${\mathrm{Gr}}_{p}^{W}{\pi}_{3}\cong Hom({V}_{3-p,p},\mathbb{Q})$ gives

${\mathrm{Gr}}_{0}^{W}{\pi}_{3}\cong {\mathbb{Q}}^{\frac{n(n+1)}{2}},{\mathrm{Gr}}_{1}^{W}{\pi}_{3}\cong {\mathbb{Q}}^{2g\cdot n},{\mathrm{Gr}}_{2}^{W}{\pi}_{3}\cong {\mathbb{Q}}^{d{d}^{\prime}\cdot n+g(2g+1)},{\mathrm{Gr}}_{3}^{W}{\pi}_{3}\cong {\mathbb{Q}}^{d{d}^{\prime}\cdot 2g},{\mathrm{Gr}}_{4}^{W}{\pi}_{3}\cong {\mathbb{Q}}^{\frac{d{d}^{\prime}(d{d}^{\prime}+1)}{2}-1}.$

For example, we may take *C* to be the nodal cubic curve given by $f(x,y,z)={y}^{2}z-{x}^{2}z-{x}^{3}$
and ${C}^{\prime}$ a smooth plane quartic. Then $d{d}^{\prime}=12$, $g=0$ and $n=1$. This gives

${\mathrm{Gr}}_{0}^{W}{\pi}_{2}\cong \mathbb{Q},{\mathrm{Gr}}_{1}^{W}{\pi}_{2}=0\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mathrm{Gr}}_{2}^{W}{\pi}_{2}\cong {\mathbb{Q}}^{12};$${\mathrm{Gr}}_{0}^{W}{\pi}_{3}\cong \mathbb{Q},{\mathrm{Gr}}_{1}^{W}{\pi}_{3}=0,{\mathrm{Gr}}_{2}^{W}{\pi}_{3}\cong {\mathbb{Q}}^{12},{\mathrm{Gr}}_{3}^{W}{\pi}_{3}=0\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mathrm{Gr}}_{4}^{W}{\pi}_{3}\cong {\mathbb{Q}}^{77}.$

The following is a generalization of Theorem 4.1 to projective varieties of arbitrary dimension.

*Let **X* be a complex projective variety of dimension *n* with normal isolated singularities.
Denote by Σ the singular locus of *X*, and for each $\sigma \mathrm{\in}\mathrm{\Sigma}$ let
${L}_{\sigma}$ denote the link of σ in *X*. If ${\stackrel{\mathrm{~}}{H}}^{k}\mathit{}\mathrm{(}{L}_{\sigma}\mathrm{;}\mathrm{Q}\mathrm{)}\mathrm{=}\mathrm{0}$ for all $k\mathrm{\le}n\mathrm{-}\mathrm{2}$
for every $\sigma \mathrm{\in}\mathrm{\Sigma}$, then *X* is a formal topological space.

#### Proof.

The link ${L}_{\sigma}$ of $\sigma \in \mathrm{\Sigma}$ in *X* is a smooth connected real manifold of dimension $2n-1$.
Let $L={\bigsqcup}_{\sigma \in \mathrm{\Sigma}}{L}_{\sigma}$. Then ${H}^{0}(L;\mathbb{Q})\cong {H}^{0}(\mathrm{\Sigma};\mathbb{Q})$.
Assume that ${\stackrel{~}{H}}^{k}({L}_{\sigma};\mathbb{Q})=0$ for all $k\le n-2$. By Poincaré duality the only non-trivial rational cohomology groups of *L* are
in degrees $0,n-1,n$ and $2n-1$. Let ${X}_{\mathrm{reg}}=X-\mathrm{\Sigma}$. From the Mayer–Vietoris exact sequence

$\mathrm{\cdots}\to {H}^{k-1}(L;\mathbb{Q})\to {H}^{k}(X;\mathbb{Q})\to {H}^{k}({X}_{reg};\mathbb{Q})\oplus {H}^{k}(\mathrm{\Sigma};\mathbb{Q})\to {H}^{k}(L;\mathbb{Q})\to \mathrm{\cdots}$

it follows that the map ${H}^{k}(X;\mathbb{Q})\to {H}^{k}({X}_{reg};\mathbb{Q})$ is an isomorphism whenever $k<n-1$ or $n+1<k<2n-1$,
and injective for $k=n-1$.
Since ${H}^{k}(X;\mathbb{Q})$ has weights in $\{0,1,\mathrm{\dots},k\}$ and ${H}^{k}({X}_{\mathrm{reg}};\mathbb{Q})$ has weights in $\{k,k+1,\mathrm{\dots},2k\}$,
and the morphism ${H}^{k}(X;\mathbb{Q})\to {H}^{k}({X}_{\mathrm{reg}};\mathbb{Q})$ is strictly compatible with the weight filtrations,
it follows that for $k\ne n,n+1$, the weight filtration on ${H}^{k}(X;\mathbb{Q})$ is pure of weight *k*.
Furthermore, by semi-purity we have that ${H}^{n+1}(X;\mathbb{Q})$ is pure of weight $n+1$.
Therefore the only non-trivial weights of ${H}^{*}(X;\mathbb{Q})$ are in degree $k=n$.
The weight spectral sequence for *X* has the form

where the bullets denote the non-trivial elements.
Consider the quasi-isomorphism of complexes

$\rho :({E}_{2}^{*,*}(X),0)\to ({E}_{1}^{*,*}(X),{d}_{1})$

defined by taking sections of the projections $Ker({d}_{1}^{p,q})\twoheadrightarrow {E}_{2}^{p,q}(X)$.
We next show that ρ is multiplicative.
Note that by bidegree reasons, the only non-trivial products of ${E}_{2}^{*,*}(X)$ are between elements of the first column. Since ρ is the identity on
${E}_{2}^{0,*}(X)\cong Ker({d}_{1}^{0,q})$,
it preserves these products. It also preserves the unit $1\in {E}_{2}^{0,0}(X)$.
Since ${E}_{1}^{p,q}(X)=0$ for all $q>2(n-p)$, we have ${E}_{1}^{p,n-p}(X)\cdot {E}_{1}^{{p}^{\prime},n-{p}^{\prime}}(X)=0$ for all $p,{p}^{\prime}>0$.
Therefore it only remains to show that for $p,q>0$, the following diagram commutes:

By Proposition 2.13 and since ${H}^{q}(\mathrm{\Sigma})=0$ for $q>0$, the term ${E}_{1}^{0,q}(X)$ is given by

while for $p>0$ we have ${E}_{1}^{p,n-p}(X)\cong {E}_{1}^{p,n-p}(D)\otimes \mathrm{\Lambda}(t)\oplus {E}_{1}^{p-1,n-p}(D)\otimes \mathrm{\Lambda}(t)dt$.
The proof now follows as in the proof of Theorem 4.1.
∎

#### (Complete intersections)

Let *X* be a complete intersection of dimension $n>1$.
Assume that the singular locus $\mathrm{\Sigma}=\{{\sigma}_{1},\mathrm{\dots},{\sigma}_{N}\}$ is a finite number of points.
The link
of ${\sigma}_{i}$ in *X* is $(n-2)$-connected
(this result is due to Milnor [20] in the case of hypersurfaces
and to Hamm [18] for general complete intersections).
Therefore by Theorem 4.3, *X* is formal.
Note that in particular, every complex hypersurface with isolated singularities is formal.

*Let **X* be a projective variety with only isolated singularities. Assume that there exists a resolution of singularities $f\mathrm{:}\stackrel{\mathrm{~}}{X}\mathrm{\to}X$ such that
the exceptional divisor $D\mathrm{=}{f}^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Sigma}\mathrm{)}$ is smooth. Then *X* is a formal topological space.

#### Proof.

We may assume that *X* is connected.
By Proposition 2.13 the multiplicative weight spectral sequence is given by

where

The differential ${d}_{1}:{E}_{1}^{0,*}(X)\to {E}_{1}^{1,*}(X)$ is given by $(x,a(t))\mapsto {a}^{\prime}(t)dt$.
The non-trivial products of ${E}_{1}^{*,*}(X)$ are the maps
${E}_{1}^{0,q}(X)\times {E}_{1}^{0,{q}^{\prime}}(X)\to {E}_{1}^{0,q+{q}^{\prime}}(X)$
given by $(x,a(t))\cdot (y,b(t))=(x\cdot y,a(t)\cdot b(t))$
and the maps
${E}_{1}^{0,q}(X)\times {E}_{1}^{1,{q}^{\prime}}(X)\to {E}_{1}^{1,q+{q}^{\prime}}(X)$
given by $(x,a(t))\cdot b(t)dt=a(t)\cdot b(t)dt$.
The unit is $({1}_{\stackrel{~}{X}},{1}_{D})\in {E}_{1}^{0,0}(X)$.
By computing the cohomology of ${E}_{1}^{*,*}(X)$ we find

with the non-trivial products being $Ker{({j}^{*})}^{q}\times Ker{({j}^{*})}^{{q}^{\prime}}\to Ker{({j}^{*})}^{q+{q}^{\prime}}$.
Now define a quasi-isomorphism $\rho :({E}_{2}^{*,*}(X),0)\to ({E}_{1}^{*,*}(X),{d}_{1})$ of bigraded complexes as follows:
Let $\rho :{E}_{2}^{0,q}(X)\to {E}_{1}^{0,q}(X)$ be defined by the inclusion, for all $q\ge 0$.
Define $\rho :{E}_{2}^{1,q}(X)\to {E}_{1}^{1,q}(X)$ by taking a section
$Coker{({j}^{*})}^{q}\to {H}^{q}(D)$ of the projection
${H}^{q}(D)\twoheadrightarrow Coker{({j}^{*})}^{q}$, for all $q>0$.

To see that ρ is a morphism of bigraded algebras it suffices to observe that it preserves
the unit and that
$\rho (Ker({j}^{*}))\cdot {E}_{1}^{1,*}(X)=0$.
Since *X* is connected, it has a mixed Hodge diagram $\mathcal{\mathcal{A}}(X)$ with ${H}^{0}(\mathcal{\mathcal{A}}{(X)}_{\mathbb{Q}})\cong \mathbb{Q}$.
Hence by Theorem 2.4 we have a string of quasi-isomorphisms of complex cdgas

$({\mathcal{\mathcal{A}}}_{\mathrm{pl}}(X),d)\otimes \u2102\stackrel{\sim}{\u27f7}({E}_{1}^{*,*}(X),{d}_{1})\otimes \u2102\stackrel{\sim}{\u27f5}({E}_{2}^{*,*}(X),0)\otimes \u2102\cong ({H}^{*}(X;\u2102),0).$

To conclude that *X* is formal it suffices to apply descent of formality of cdgas from $\u2102$ to $\mathbb{Q}$.
∎

Projective varieties with only isolated ordinary multiple points are formal. Projective cones over smooth projective varieties are formal.

#### (Segre cubic)

The Segre cubic *S* is a simply connected projective threefold with ten singular points, and is described by
the set of points $({x}_{0}:{x}_{1}:{x}_{2}:{x}_{3}:{x}_{4}:{x}_{5})$ of $\u2102{\mathbb{P}}^{5}$:

$S:\{{x}_{0}+{x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}=0,{x}_{0}^{3}+{x}_{1}^{3}+{x}_{2}^{3}+{x}_{3}^{3}+{x}_{4}^{3}+{x}_{5}^{3}=0\}.$

A resolution $f:{\overline{\mathcal{\mathcal{M}}}}_{0,6}\to S$ of *S* is given by the moduli space ${\overline{\mathcal{\mathcal{M}}}}_{0,6}$ of stable rational curves with six marked points,
and ${f}^{-1}(\mathrm{\Sigma})={\bigsqcup}_{i=1}^{10}\u2102{\mathbb{P}}^{1}\times \u2102{\mathbb{P}}^{1}$, where $\mathrm{\Sigma}=\{{\sigma}_{1},\mathrm{\dots},{\sigma}_{10}\}$ is the singular locus of *S*.
We have

Hence *S* has a non-trivial weight filtration, with $0\ne {\mathrm{Gr}}_{2}^{W}{H}^{3}(S;\mathbb{Q})\cong {\mathbb{Q}}^{5}$.
By Theorem 4.5 and since *S* is simply connected,
we may compute the rational
homotopy groups ${\pi}_{*}(S)\otimes \mathbb{Q}$ with their weight filtration
from a minimal model of ${E}_{2}^{*,*}(S)$.
Since *S* is a hypersurface of $\u2102{\mathbb{P}}^{4}$, the map $S\to \u2102{\mathbb{P}}^{4}$ induces an isomorphism ${H}^{k}(\u2102{\mathbb{P}}^{4})\cong {H}^{k}(S)$ for $k\ne 3,4$
(see [10, Theorems V.2.6 and V.2.11]).
We deduce that

${E}_{2}^{*,*}(S)\cong \mathbb{Q}[a,{b}_{1},\mathrm{\cdots},{b}_{5},{c}_{0},\mathrm{\cdots},{c}_{5},e]$

with the only non-trivial products ${a}^{2}={c}_{0}$ and ${a}^{3}=e$. The bidegrees are given by
$|a|=(0,2)$, $|{b}_{i}|=(1,2)$, $|{c}_{i}|=(0,4)$ and $e=(0,6)$.
In low degrees we obtain

${\mathrm{Gr}}_{2}^{W}{\pi}_{2}\cong \mathbb{Q},$${\mathrm{Gr}}_{2}^{W}{\pi}_{3}\cong {\mathbb{Q}}^{5},$${\mathrm{Gr}}_{4}^{W}{\pi}_{4}\mathit{\hspace{1em}}\cong {\mathbb{Q}}^{5},$${\mathrm{Gr}}_{3}^{W}{\pi}_{5}\cong {\mathbb{Q}}^{10},$${\mathrm{Gr}}_{5}^{W}{\pi}_{5}\cong {\mathbb{Q}}^{5},$${\mathrm{Gr}}_{4}^{W}{\pi}_{6}\cong {\mathbb{Q}}^{25},$${\mathrm{Gr}}_{5}^{W}{\pi}_{6}\cong {\mathbb{Q}}^{25},$${\mathrm{Gr}}_{4}^{W}{\pi}_{7}\mathit{\hspace{1em}}\cong {\mathbb{Q}}^{40},$${\mathrm{Gr}}_{5}^{W}{\pi}_{7}\cong {\mathbb{Q}}^{50},$${\mathrm{Gr}}_{7}^{W}{\pi}_{7}\cong {\mathbb{Q}}^{26}.$

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