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Let me elaborate on my comment, adding some details and references.

The space $\overline{\mathcal{M}}_{g, \, p}$ is an almost Kähler $V$-manifold. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.

Then the assertion follows by applying the following result, that is Theorem 2.43, p. 57 of Peters-Steenbrink's book Mixed Hodge Structures.

Theorem. Let $X$ be an almost Kähler $V$-manifold. Then $H^k(X, \, \mathbf{Q})$ admits a pure Hodge structure of weight $k$.

Let me elaborate on my comment, adding some details and references.

The space $\overline{\mathcal{M}}_{g, \, p}$ is an almost Kähler $V$-manifold. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.

Then the assertion follows by applying the following result, that is Theorem 2.43, p. 57 of Peters-Steenbrink's book Mixed Hodge Structures.

Theorem. Let $X$ be an almost Kähler $V$-manifold. Then $H^k(X, \, \mathbb{Q})$ admits a pure Hodge structure of weight $k$.

I elaborate on my comment, adding some details and references.

The space $\overline{\mathcal{M}}_{g, \, p}$ is an almost Kähler $V$-manifold. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.

Then the assertion follows by applying the following result, that is Theorem 2.43, p. 57 of Peters-Steenbrink's book Mixed Hodge Structures.

Theorem. Let $X$ be an almost Kähler $V$-manifold. Then $H^k(X, \, \mathbf{Q})$ admits a pure Hodge structure of weight $k$.

Let me elaborate on my comment, adding some details and references.

The space $\overline{\mathcal{M}}_{g, \, p}$ is an almost Kähler $V$-manifold. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.

Then the assertion follows by applying the following result, that is Theorem 2.43, p. 57 of Peters-Steenbrink's book Mixed Hodge Structures.

Theorem. Let $X$ be an almost Kähler $V$-manifold. Then $H^k(X, \, \mathbf{Q})$ admits a pure Hodge structure of weight $k$.

The space $\overline{\mathcal{M}}_{g, p}$ is an almost-Kähler $V$-manifold. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.

Then the assertion follows by applying the following result, that is Theorem 2.43, p. 47 of Peters-Steenbrink's book Mixed Hodge Structures.

Theorem. Let $X$ be an almost-Kähler $V$-manifold. Then $H^k(X, \, \mathbf{Q})$ admits a pure Hodge structure of weight $k$.

I elaborate on my comment, adding some details and references.

The space $\overline{\mathcal{M}}_{g, \, p}$ is an almost Kähler $V$-manifold. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.

Then the assertion follows by applying the following result, that is Theorem 2.43, p. 57 of Peters-Steenbrink's book Mixed Hodge Structures.

Theorem. Let $X$ be an almost Kähler $V$-manifold. Then $H^k(X, \, \mathbf{Q})$ admits a pure Hodge structure of weight $k$.