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Hard Lefschetz implies Poincaré duality with real coefficients, in particular $H^0\cong H^{2n}$ and for a non-compact connected manifold that never happens ($H^{2n}=0$).

For example: $M:=\mathbb{C}P^2\setminus\mathbb{C}P^1$ is contractible, but hard Lefschetz theorem (as stated) should have given an isomorphism between $H^0(M)=\mathbb{R}$ and $H^4(M)=0$.

UPD. Poincaré duality modification for non-compact manifolds (pointed out by BS.) suggests a Lefschetz-type map $\mu: H_c^{n-k}\to H^{n+k}$ defined by multiplication by $\omega^k$: $H_c^{n-k}\to H_c^{n+k}$ and subsequent composition with the canonical map from $H_c$ to $H$. $\mu$, hovewer, needs not be an isomorphism:

for $\mathbb{C}^2\setminus \{0\}$ Lefschetz map is identically zero, but $H^{3}=H_c^1$ are one-dimensional.

So, all in all, hard Lefschetz theorem is wrong for non-compact Kahler manifolds.

Hard Lefschetz implies Poincaré duality with real coefficients, in particular $H^0\cong H^{2n}$ and for a non-compact connected manifold that never happens ($H^{2n}=0$).

For example: $M:=\mathbb{C}P^2\setminus\mathbb{C}P^1$ is contractible, but hard Lefschetz theorem (as stated) should have given an isomorphism between $H^0(M)=\mathbb{R}$ and $H^4(M)=0$.

UPD. Poincaré duality modification for non-compact manifolds (pointed out by BS.) suggests a Lefschetz-type map $\mu: H_c^{n-k}\to H^{n+k}$ defined by multiplication by $\omega^k$: $H_c^{n-k}\to H_c^{n+k}$ and subsequent composition with the canonical map from $H_c$ to $H$. $\mu$, hovewer, needs not be an isomorphism:

for $\mathbb{C}^2\setminus \{0\}$ Lefschetz map is identically zero, but $H^{3}=H_c^1$ are one-dimensional.

So, all in all, hard Lefschetz theorem fails for non-compact Kahler manifolds.

Hard Lefschetz implies Poincaré duality with real coefficients, in particular $H^0\cong H^{2n}$ and for a non-compact connected manifold that never happens ($H^{2n}=0$).

For example: $M:=\mathbb{C}P^2\setminus\mathbb{C}P^1$ is contractible, but hard Lefschetz theorem (as stated) should have given an isomorphism between $H^0(M)=\mathbb{R}$ and $H^4(M)=0$.

UPD. As BS. pointed out, Poincaré duality modification for non-compact manifolds suggests a Lefschetz-type map $\mu: H_c^{n-k}\to H^{n+k}$ defined by multiplication by $\omega^k$: $H_c^{n-k}\to H_c^{n+k}$ and subsequent composition with the canonical map from $H_c$ to $H$. $\mu$, hovewer, needs not be an isomorphism:

for $\mathbb{C}^2\setminus \{0\}$ Lefschetz map is identically zero, but $H^{3}=H_c^1$ are one-dimensional.

So, all in all, hard Lefschetz theorem is wrong for non-compact Kahler manifolds.

Hard Lefschetz implies Poincaré duality with real coefficients, in particular $H^0\cong H^{2n}$ and for a non-compact connected manifold that never happens ($H^{2n}=0$).

For example: $M:=\mathbb{C}P^2\setminus\mathbb{C}P^1$ is contractible, but hard Lefschetz theorem (as stated) should have given an isomorphism between $H^0(M)=\mathbb{R}$ and $H^4(M)=0$.

UPD. Poincaré duality modification for non-compact manifolds (pointed out by BS.) suggests a Lefschetz-type map $\mu: H_c^{n-k}\to H^{n+k}$ defined by multiplication by $\omega^k$: $H_c^{n-k}\to H_c^{n+k}$ and subsequent composition with the canonical map from $H_c$ to $H$. $\mu$, hovewer, needs not be an isomorphism:

for $\mathbb{C}^2\setminus \{0\}$ Lefschetz map is identically zero, but $H^{3}=H_c^1$ are one-dimensional.

So, all in all, hard Lefschetz theorem is wrong for non-compact Kahler manifolds.

Hard Lefschetz implies Poincaré duality with real coefficients, in particular $H^0\cong H^{2n}$ and for a non-compact connected manifold that never happens ($H^{2n}=0$).

For example: $M:=\mathbb{C}P^2\setminus\mathbb{C}P^1$ is contractible, but hard Lefschetz theorem (as stated) should have given an isomorphism between $H^0(M)=\mathbb{R}$ and $H^4(M)=0$.

UPD. As BS. pointed out, Poincaré duality modification for non-compact manifolds suggests a Lefschetz-type map by multiplication by $\omega^k$:  $H_c^{n-k}\to H_c^{n+k}$ and then composing with the canonical map from $H_c$ to $H$: $\mu: H_c^{n-k}\to H^{n+k}$. $\mu$, hovewer, needs not be an isomorphism:

for $\mathbb{C}^2\setminus \{0\}$ Lefschetz map is identically zero, but $H^{3}=H_c^1$ are one-dimensional.

So, all in all, hard Lefschetz theorem is wrong for non-compact Kahler manifolds.

Hard Lefschetz implies Poincaré duality with real coefficients, in particular $H^0\cong H^{2n}$ and for a non-compact connected manifold that never happens ($H^{2n}=0$).

For example: $M:=\mathbb{C}P^2\setminus\mathbb{C}P^1$ is contractible, but hard Lefschetz theorem (as stated) should have given an isomorphism between $H^0(M)=\mathbb{R}$ and $H^4(M)=0$.

UPD. As BS. pointed out, Poincaré duality modification for non-compact manifolds suggests a Lefschetz-type map $\mu: H_c^{n-k}\to H^{n+k}$ defined by multiplication by $\omega^k$:  $H_c^{n-k}\to H_c^{n+k}$ and subsequent composition with the canonical map from $H_c$ to $H$. $\mu$, hovewer, needs not be an isomorphism:

for $\mathbb{C}^2\setminus \{0\}$ Lefschetz map is identically zero, but $H^{3}=H_c^1$ are one-dimensional.

So, all in all, hard Lefschetz theorem is wrong for non-compact Kahler manifolds.