Return to Question

Let $H_{\mathbf{Q}}$ and $H_{\mathbf{Q}}^'$ be two pure Hodge structures of weight $n$ and $n'$ respectively. How do you prove the following simple fact:

fact: If $n>n'$ and $f:H_{\mathbf{Q}}\rightarrow H_{\mathbf{Q}}^'$ is a morphism which respects the filtrations over $\mathbf{C}$, then $f=0$.

I don't quite see how to use the assumption $n>n'$...

added

My original question was related to the fact that asking the morphism to be compatible with the torus action seems to be a stronger condition that only asking for the filtration to be preserved. And I guess that in general this is all you can say. This reflexion was motivated by Deligne Scholie 5.1 in Hodge 1 which says the following:

Scholie 5.1 Soit $H$ et $H'$ des structures de Hodge de poids $n$ et $n'$ avec $n>n'$.Soit $f:H_{Q}\rightarrow H_Q'$ un morphisme tel que $f:H_C\rightarrow H_C'$ respecte $F$. Alors $f=0$.

Q: So how should I interpret Scholie 5.1?

Let $H_{\mathbf{Q}}$ and $H_{\mathbf{Q}}'$ be two pure Hodge structures of weight $n$ and $n'$ respectively. How do you prove the following simple fact:

fact: If $n>n'$ and $f:H_{\mathbf{Q}}\rightarrow H_{\mathbf{Q}}'$ is a morphism which respects the filtrations over $\mathbf{C}$, then $f=0$.

I don't quite see how to use the assumption $n>n'$...

added

My original question was related to the fact that asking the morphism to be compatible with the torus action seems to be a stronger condition that only asking for the filtration to be preserved. And I guess that in general this is all you can say. This reflexion was motivated by Deligne Scholie 5.1 in Hodge 1 which says the following:

Scholie 5.1 Soit $H$ et $H'$ des structures de Hodge de poids $n$ et $n'$ avec $n>n'$.Soit $f:H_{\mathbf{Q}}\rightarrow H_{\mathbf{Q}}'$ un morphisme tel que $f:H_C\rightarrow H_C'$ respecte $F$. Alors $f=0$.

Q: So how should I interpret Scholie 5.1?

Let $H_{\mathbf{Q}}$ and $H_{\mathbf{Q}}^'$ be two pure Hodge structures of weight $n$ and $n'$ respectively. How do you prove the following simple fact:

fact: If $n>n'$ and $f:H_{\mathbf{Q}}\rightarrow H_{\mathbf{Q}}^'$ is a morphism which respects the filtrations over $\mathbf{C}$, then $f=0$.

I don't quite see how to use the assumption $n>n'$...

added

My original question was related to the fact that asking the morphism to be compatible with the torus action semms to be a stronger condition that only asking for the filtration to be preserved. And I guess that in general this is all you can say. This reflexion was motivated by Deligne Scholie 5.1 in Hodge 1 which says the following:

Scholie 5.1 Soit $H$ et $H'$ des structures de Hodge de poids $n$ et $n'$ avec $n>n'$.Soit $f:H_{Q}\rightarrow H_Q'$ un morphisme tel que $f:H_C\rightarrow H_C'$ respecte $F$. Alors $f=0$.

Q: So how should I interpret Scholie 5.1?

Let $H_{\mathbf{Q}}$ and $H_{\mathbf{Q}}^'$ be two pure Hodge structures of weight $n$ and $n'$ respectively. How do you prove the following simple fact:

fact: If $n>n'$ and $f:H_{\mathbf{Q}}\rightarrow H_{\mathbf{Q}}^'$ is a morphism which respects the filtrations over $\mathbf{C}$, then $f=0$.

I don't quite see how to use the assumption $n>n'$...

added

My original question was related to the fact that asking the morphism to be compatible with the torus action seems to be a stronger condition that only asking for the filtration to be preserved. And I guess that in general this is all you can say. This reflexion was motivated by Deligne Scholie 5.1 in Hodge 1 which says the following:

Scholie 5.1 Soit $H$ et $H'$ des structures de Hodge de poids $n$ et $n'$ avec $n>n'$.Soit $f:H_{Q}\rightarrow H_Q'$ un morphisme tel que $f:H_C\rightarrow H_C'$ respecte $F$. Alors $f=0$.

Q: So how should I interpret Scholie 5.1?

Let $H_{\mathbf{Q}}$ and $H_{\mathbf{Q}}^'$ be two pure Hodge structures of weight $n$ and $n'$ respectively. How do you prove the following simple fact:

fact: If $n>n'$ and $f:H_{\mathbf{Q}}\rightarrow H_{\mathbf{Q}}^'$ is a morphism which respects the filtrations over $\mathbf{C}$, then $f=0$.

I don't quite see how to use the assumption $n>n'$...

added The fact stated above is clearly false since I may take the zero map. So my question as it is does not make sense.

My original question was related to the fact that asking the morphism to be compatible with the torus action semms to be a stronger condition that only asking for the filtration to be preserved. And I guess that in general this is all you can say. This reflexion was motivated by Deligne Scholie 5.1 in Hodge 1 which says the following:

Scholie 5.1 Soit $H$ et $H'$ des structures de Hodge de poids $n$ et $n'$ avec $n>n'$.Soit $f:H_{Q}\rightarrow H_Q'$ un morphisme tel que $f:H_C\rightarrow H_C'$ respecte $F$. Alors $f=0$.

Q: So how should I interpret Scholie 5.1?

Let $H_{\mathbf{Q}}$ and $H_{\mathbf{Q}}^'$ be two pure Hodge structures of weight $n$ and $n'$ respectively. How do you prove the following simple fact:

fact: If $n>n'$ and $f:H_{\mathbf{Q}}\rightarrow H_{\mathbf{Q}}^'$ is a morphism which respects the filtrations over $\mathbf{C}$, then $f=0$.

I don't quite see how to use the assumption $n>n'$...

added

My original question was related to the fact that asking the morphism to be compatible with the torus action semms to be a stronger condition that only asking for the filtration to be preserved. And I guess that in general this is all you can say. This reflexion was motivated by Deligne Scholie 5.1 in Hodge 1 which says the following:

Scholie 5.1 Soit $H$ et $H'$ des structures de Hodge de poids $n$ et $n'$ avec $n>n'$.Soit $f:H_{Q}\rightarrow H_Q'$ un morphisme tel que $f:H_C\rightarrow H_C'$ respecte $F$. Alors $f=0$.

Q: So how should I interpret Scholie 5.1?