Sunday, 23 October 2011

ag.algebraic geometry - Morphism between polarized abelian varieties

That should be true, yes.



A polarization of A is given by a bilinear form on H1(A,Z); this is equivalent to a map H1(A,Z)toH1(A,Z)vee, which is an isomorphism if the polarization is principal.



A map between two abelian varieties is given by a corresponding linear map H1(A1,Z)toH1(A2,Z). The map between the varieties is an isomorphism if the map on H1 is.



The map induced on the bilinear form then is the composition
H1(A1,Z)toH1(A2,Z)toH1(A2,Z)veetoH1(A1,Z)vee


If the form is respected by this map, then this is an isomorphism. Consequently, the left-hand map must be as well, as claimed.

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