In this section we give a simple proof of the fact that the purity of the weight filtration of a complex projective variety
up to a certain degree, implies formality of the variety up to the same degree. A direct application is the formality of
the Malcev completion of the fundamental group of projective varieties with normal isolated singularities.

**Definition 3.1:** *Let $r\ge 0$ be an integer.
A morphism of cdgas $f:A\to B$ is called **
**r*-quasi-isomorphism if the induced morphism in cohomology ${H}^{i}(f):{H}^{i}(A)\to {H}^{i}(B)$ is an isomorphism
for all $i\le r$ and a monomorphism for $i=r+1$.

**Definition 3.2:** *A cdga $(A,d)$ over $\mathbf{k}$ is called **
**r*-formal if there is a string of *r*-quasi-isomorphisms from $(A,d)$
to its cohomology $({H}^{*}(A;\mathbf{k}),0)$ considered as a cdga with trivial differential.
We will say that a topological space *X* is *
**r*-formal
if the rational cdga ${\mathcal{\mathcal{A}}}_{\mathrm{pl}}(X)$ is *r*-formal.

The case $r=1$ is of special interest, since 1-formality implies that the rational Malcev completion of
${\pi}_{1}(X)$ can be computed directly from
the cohomology group ${H}^{1}(X;\mathbb{Q})$, together with the cup product

${H}^{1}(X;\mathbb{Q})\otimes {H}^{1}(X;\mathbb{Q})\to {H}^{2}(X;\mathbb{Q}).$

In this case we say that *
${\pi}_{1}(X)$ is formal*.
For $r=\mathrm{\infty}$ we recover the usual notion of formality, which in the case of simply connected (or more generally, nilpotent) spaces, implies
that the higher rational homotopy groups ${\pi}_{i}(X)\otimes \mathbb{Q}$, with $i>1$ can be computed directly from the cohomology ring ${H}^{*}(X;\mathbb{Q})$.
Note that if *X* is formal, then ${\pi}_{1}(X)$ is also formal.

**Theorem 3.3:** *
**Let **X* be a complex projective variety and let $r\mathrm{\ge}\mathrm{0}$ be an integer.
If the weight filtration on ${H}^{k}\mathit{}\mathrm{(}X\mathrm{;}\mathrm{Q}\mathrm{)}$ is pure of weight *k*, for all $\mathrm{0}\mathrm{\le}k\mathrm{\le}r$, then *X* is *r*-formal.

*We prove formality over $\u2102$ and apply independence of formality on the base field for cdgas with
cohomology of finite type (see [27], see also [17]).
Since the disjoint union of **r*-formal spaces is *r*-formal, we may assume that *X* is connected, so it has a mixed Hodge diagram $\mathcal{\mathcal{A}}(X)$ with ${H}^{0}(\mathcal{\mathcal{A}}{(X)}_{\mathbb{Q}})\cong \mathbb{Q}$.
By Theorem 2.4 it suffices to define a string of *r*-quasi-isomorphisms
of differential bigraded algebras from $({E}_{1}^{*,*}(X),{d}_{1})$ to $({E}_{2}^{*,*}(X),0)$.
Let *M* be the bigraded vector space given by ${M}^{0,q}=Ker({d}_{1}^{0,q})$ for all $q\ge 0$ and ${M}^{p,q}=0$ for all $p>0$ and all $q\ge 0$,
where ${d}_{1}^{p,q}:{E}_{1}^{p,q}(X)\to {E}_{1}^{p+1,q}(X)$ denotes the differential of ${E}_{1}^{*,*}(X)$.
Then ${M}^{*,*}$ is a bigraded sub-complex of $({E}_{1}^{*,*}(X),{d}_{1})$ with trivial differential. Denote by $\phi :(M,0)\to ({E}_{1}^{*,*}(X),{d}_{1})$ the inclusion.
Since $Ker({d}_{1}^{0,*})\times Ker({d}_{1}^{0,*})\subset Ker({d}_{1}^{0,*})$,
the multiplicative structure induced by φ on *M* is closed in *M*.
Hence φ is an inclusion of differential bigraded algebras.
On the other hand, we have
${E}_{2}^{0,q}(X)\cong {M}^{0,q}=Ker({d}_{1}^{0,q}).$
This gives an inclusion of bigraded algebras
$\psi :M\to {E}_{2}^{*,*}(X)$.
Assume that the weight filtration on ${H}^{k}(X;\mathbb{Q})$ is pure of weight *k*, for all $k\le r$.
We next show that both φ and ψ are *r*-quasi-isomorphisms. Indeed,
for every $p>0$ and every $q\ge 0$ such that $p+q\le r$ we have
${E}_{2}^{p,q}(X)=0$, while for every $q\ge 0$, we have ${E}_{2}^{0,q}(X)\cong {M}^{0,q}\cong {H}^{q}(X;\mathbb{Q})$.
Therefore the induced maps ${H}^{k}(\phi )$ and ${H}^{k}(\psi )$ are isomorphisms for all $k\le r$
and the maps ${H}^{k+1}(\phi )$ and ${H}^{k+1}(\psi )$ are monomorphisms.
∎

We highlight the two extreme cases $r=1$ and $r=\mathrm{\infty}$ in the following corollary.

**Corollary 3.4:** *
**Let **X* be a complex projective variety.

*(1)*

*
**If the weight filtration on *
${H}^{1}(X;\mathbb{Q})$
* is pure of weight *
1
*, then *
${\pi}_{1}(X)$
* is formal.*

*(2)*

*
**If the weight filtration on *
${H}^{k}(X;\mathbb{Q})$
* is pure of weight *
*k*
*, for all *
$k\ge 0$
* then *
*X*
* is formal.
*

*
***Example 3.5:** *Let **X* be a complex projective variety of dimension *n*.
Assume that *X* is a $\mathbb{Q}$-homology manifold (for all $x\in X$, ${H}_{\{x\}}^{k}(X;\mathbb{Q})=0$ for $k\ne 2n$ and
${H}_{\{x\}}^{2n}(X;\mathbb{Q})\cong \mathbb{Q}$). Then the weight filtration on ${H}^{k}(X;\mathbb{Q})$ is pure of weight *k*, for all $k\ge 0$
(see [8, Theorem 8.2.4]). Hence *X* is formal.
Examples of such varieties are given by weighted projective spaces or more generally *V*-manifolds (see [10, Appendix B]),
surfaces with ${A}_{1}$-singularities, the Cayley cubic or the Kummer surface.

In fact, purity of the weight filtration is strongly related to Poincaré duality:
if *X* is a complex projective variety whose rational cohomology satisfies Poincaré duality, then
the weight filtration on ${H}^{k}(X;\mathbb{Q})$ is pure of weight *k*, for all $k\ge 0$.
Indeed, the Poincaré duality maps
${H}^{k}(X;\mathbb{Q})\cong ({H}^{2n-k}{(X;\mathbb{Q})}^{*})(-n)$ are compatible with mixed Hodge structures.
The weights on the left-hand (resp. right-hand) side are $\le k$ (resp. $\ge k$),
hence equal to *k*.
Therefore such varieties are formal (cf. [16, Theorem 5]).
Another well-known result relating purity and Poincaré duality is the purity of the weight filtration
on the (middle perversity) intersection cohomology $I{H}^{*}(X;\mathbb{Q})$ of a projective variety *X*.
Furthermore, Weber [29] showed that for a complex projective variety
*X*, the image of the map ${H}^{k}(X;\mathbb{Q})\to I{H}^{k}(X;\mathbb{Q})$ is isomorphic to the pure term ${\mathrm{Gr}}_{k}^{W}{H}^{k}(X;\mathbb{Q})$.

The purity of the weight filtration in cohomology does not imply Poincaré duality, as
shown by the following example.

**Example 3.6:** *Let $C\subset \u2102{\mathbb{P}}^{N}$ be a smooth curve of genus **g* and consider the projective cone $X={P}_{c}C$ over *C*.
The Betti numbers of *X* are ${b}_{0}=1$, ${b}_{1}=0$, ${b}_{2}=1$, ${b}_{3}=2g$ and ${b}_{4}=1$.
The weight filtration on ${H}^{k}(X;\mathbb{Q})$ is pure of weight *k*, for all $k\ge 0$, and hence *X* is formal, but
${H}^{*}(X;\mathbb{Q})$ does not satisfy Poincaré duality.

In [1], it is shown that the weight filtration on ${H}^{1}(X;\mathbb{Q})$ of a normal complex projective variety *X* is pure of weight 1.
In the case of isolated singularities, the proof is a standard argument in mixed Hodge theory.

**Lemma 3.7:** *
**Let **X* be a normal complex projective variety with isolated singularities. Then the weight filtration on ${H}^{\mathrm{1}}\mathit{}\mathrm{(}X\mathrm{;}\mathrm{Q}\mathrm{)}$ is pure weight 1.

*Let Σ denote the singular locus of **X* and $f:\stackrel{~}{X}\to X$ a resolution such that $D:={f}^{-1}(\mathrm{\Sigma})$ is a
simple normal crossings divisor.
Since *X* is normal, by Zariski’s main Theorem
we have ${H}^{0}(D;\mathbb{Q})\cong {H}^{0}(\mathrm{\Sigma};\mathbb{Q})$. Since $dim\mathrm{\Sigma}=0$, we have ${H}^{k}(\mathrm{\Sigma})=0$ for all $k>0$. This gives a Mayer–Vietoris long exact sequence
$0\stackrel{}{\to}{H}^{1}(X;\mathbb{Q})\stackrel{{f}^{*}}{\to}{H}^{1}(\stackrel{~}{X};\mathbb{Q})\stackrel{{j}^{*}}{\to}{H}^{1}(D;\mathbb{Q})\stackrel{}{\to}{H}^{2}(X;\mathbb{Q})\stackrel{{f}^{*}}{\to}\mathrm{\cdots}$
which is strictly compatible with the weight filtration (see for example [24, Corollary-Definition 5.37]).
Since $\stackrel{~}{X}$ is smooth and projective, its weight filtration is pure.
Hence the weight filtration on ${H}^{1}(X;\mathbb{Q})$ is pure weight 1.
∎

**Corollary 3.8:** *
**The fundamental group ${\pi}_{\mathrm{1}}\mathit{}\mathrm{(}X\mathrm{)}$ of every
normal complex projective variety **X* with isolated singularities is formal.

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