Can distinct morphisms between curves induce the same morphism on singular cohomology? The...



Can distinct morphisms between curves induce the same morphism on singular cohomology?



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Suppose $f,g:X rightarrow Y$ are finite morphisms between connected smooth curves over $mathbb{C}$, with $Y$ of genus at least $2$.



If $f$ and $g$ induce the same morphism $H^*(Y,mathbb{C}) rightarrow H^*(X,mathbb{C})$, does $f=g$?






ag.algebraic-geometry







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$endgroup$

















    1












    $begingroup$


    Suppose $f,g:X rightarrow Y$ are finite morphisms between connected smooth curves over $mathbb{C}$, with $Y$ of genus at least $2$.



    If $f$ and $g$ induce the same morphism $H^*(Y,mathbb{C}) rightarrow H^*(X,mathbb{C})$, does $f=g$?






    ag.algebraic-geometry







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Suppose $f,g:X rightarrow Y$ are finite morphisms between connected smooth curves over $mathbb{C}$, with $Y$ of genus at least $2$.



      If $f$ and $g$ induce the same morphism $H^*(Y,mathbb{C}) rightarrow H^*(X,mathbb{C})$, does $f=g$?






      ag.algebraic-geometry







      share|cite|improve this question









      $endgroup$




      Suppose $f,g:X rightarrow Y$ are finite morphisms between connected smooth curves over $mathbb{C}$, with $Y$ of genus at least $2$.



      If $f$ and $g$ induce the same morphism $H^*(Y,mathbb{C}) rightarrow H^*(X,mathbb{C})$, does $f=g$?






      ag.algebraic-geometry




      ag.algebraic-geometry






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      share|cite|improve this question











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      asked 1 hour ago









      rj7k8rj7k8

      180117




      180117






















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          $begingroup$

          Yes. Since $Y$ embeds into its Jacobian $B$, it is enough to prove the statement for pairs of maps to an abelian variety $f, gcolon Xto B$ sending a base point $xin X$ to $0in B$. Every such map factors uniquely through the Albanese variety $A$ of $X$, so we reduce further to the case of pairs of maps $f, gcolon Ato B$ between abelian varieties (sending $0$ to $0$). Every such map is necessarily a group homomorphism, and is uniquely determined by what it does on $pi_1 = H_1$, or on $H^1(-, mathbf{C})$.






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            $begingroup$

            Yes. Since $Y$ embeds into its Jacobian $B$, it is enough to prove the statement for pairs of maps to an abelian variety $f, gcolon Xto B$ sending a base point $xin X$ to $0in B$. Every such map factors uniquely through the Albanese variety $A$ of $X$, so we reduce further to the case of pairs of maps $f, gcolon Ato B$ between abelian varieties (sending $0$ to $0$). Every such map is necessarily a group homomorphism, and is uniquely determined by what it does on $pi_1 = H_1$, or on $H^1(-, mathbf{C})$.






            share|cite|improve this answer









            $endgroup$


















              4












              $begingroup$

              Yes. Since $Y$ embeds into its Jacobian $B$, it is enough to prove the statement for pairs of maps to an abelian variety $f, gcolon Xto B$ sending a base point $xin X$ to $0in B$. Every such map factors uniquely through the Albanese variety $A$ of $X$, so we reduce further to the case of pairs of maps $f, gcolon Ato B$ between abelian varieties (sending $0$ to $0$). Every such map is necessarily a group homomorphism, and is uniquely determined by what it does on $pi_1 = H_1$, or on $H^1(-, mathbf{C})$.






              share|cite|improve this answer









              $endgroup$
















                4












                4








                4





                $begingroup$

                Yes. Since $Y$ embeds into its Jacobian $B$, it is enough to prove the statement for pairs of maps to an abelian variety $f, gcolon Xto B$ sending a base point $xin X$ to $0in B$. Every such map factors uniquely through the Albanese variety $A$ of $X$, so we reduce further to the case of pairs of maps $f, gcolon Ato B$ between abelian varieties (sending $0$ to $0$). Every such map is necessarily a group homomorphism, and is uniquely determined by what it does on $pi_1 = H_1$, or on $H^1(-, mathbf{C})$.






                share|cite|improve this answer









                $endgroup$



                Yes. Since $Y$ embeds into its Jacobian $B$, it is enough to prove the statement for pairs of maps to an abelian variety $f, gcolon Xto B$ sending a base point $xin X$ to $0in B$. Every such map factors uniquely through the Albanese variety $A$ of $X$, so we reduce further to the case of pairs of maps $f, gcolon Ato B$ between abelian varieties (sending $0$ to $0$). Every such map is necessarily a group homomorphism, and is uniquely determined by what it does on $pi_1 = H_1$, or on $H^1(-, mathbf{C})$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 1 hour ago









                Piotr AchingerPiotr Achinger

                8,49712854




                8,49712854






























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