Outer Hodge groups of rationally connected fibrations












4














I believe the following is true and well known.



Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
$$
fcolon Xrightarrow Y
$$
be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.



I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).






ag.algebraic-geometry reference-request







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    4














    I believe the following is true and well known.



    Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
    $$
    fcolon Xrightarrow Y
    $$
    be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.



    I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).






    ag.algebraic-geometry reference-request







    share|cite|improve this question







    New contributor




    Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.























      4












      4








      4







      I believe the following is true and well known.



      Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
      $$
      fcolon Xrightarrow Y
      $$
      be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.



      I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).






      ag.algebraic-geometry reference-request







      share|cite|improve this question







      New contributor




      Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      I believe the following is true and well known.



      Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
      $$
      fcolon Xrightarrow Y
      $$
      be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.



      I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).






      ag.algebraic-geometry reference-request




      ag.algebraic-geometry reference-request






      share|cite|improve this question







      New contributor




      Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







      New contributor




      Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      share|cite|improve this question






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      asked 7 hours ago









      Duck HunterDuck Hunter

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          By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].



          Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].






          share|cite|improve this answer

















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            Thank you! I did not realize Kollár had proved this. What an excellent paper!
            – Duck Hunter
            6 hours ago











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          1 Answer
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          1 Answer
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          active

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          By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].



          Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].






          share|cite|improve this answer

















          • 1




            Thank you! I did not realize Kollár had proved this. What an excellent paper!
            – Duck Hunter
            6 hours ago
















          4














          By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].



          Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].






          share|cite|improve this answer

















          • 1




            Thank you! I did not realize Kollár had proved this. What an excellent paper!
            – Duck Hunter
            6 hours ago














          4












          4








          4






          By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].



          Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].






          share|cite|improve this answer












          By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].



          Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 7 hours ago









          Chen JiangChen Jiang

          934157




          934157








          • 1




            Thank you! I did not realize Kollár had proved this. What an excellent paper!
            – Duck Hunter
            6 hours ago














          • 1




            Thank you! I did not realize Kollár had proved this. What an excellent paper!
            – Duck Hunter
            6 hours ago








          1




          1




          Thank you! I did not realize Kollár had proved this. What an excellent paper!
          – Duck Hunter
          6 hours ago




          Thank you! I did not realize Kollár had proved this. What an excellent paper!
          – Duck Hunter
          6 hours ago










          Duck Hunter is a new contributor. Be nice, and check out our Code of Conduct.










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