Geometric description of a certain sphere bundle

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It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. Is there a nice geometric description of the projection to the sphere?



This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.










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    up vote
    10
    down vote

    favorite
    2












    It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. Is there a nice geometric description of the projection to the sphere?



    This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.










    share|cite|improve this question























      up vote
      10
      down vote

      favorite
      2









      up vote
      10
      down vote

      favorite
      2






      2





      It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. Is there a nice geometric description of the projection to the sphere?



      This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.










      share|cite|improve this question













      It appears to be a standard fact in topology that $mathbbCmathbbP^2#-mathbbCmathbbP^2$ has a structure of a $mathbbS^2$ bundle over $mathbbS^2$. Is there a nice geometric description of the projection to the sphere?



      This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.







      reference-request dg.differential-geometry






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      asked Aug 19 at 9:58









      Alex Gavrilov

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          Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.






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            I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):



            Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
            $S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.



            Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $mathbb Cmathbb P^n-1$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)






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              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes








              up vote
              14
              down vote



              accepted










              Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.






              share|cite|improve this answer
























                up vote
                14
                down vote



                accepted










                Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.






                share|cite|improve this answer






















                  up vote
                  14
                  down vote



                  accepted







                  up vote
                  14
                  down vote



                  accepted






                  Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.






                  share|cite|improve this answer












                  Yes. If $pinmathbbCP^2$ is a point, you can consider the blowup $X_p$ of $mathbbCP^2$ at $p$ as the space of pairs $(L,q)$ such that $LsubsetmathbbCP^2$ is a line passing through $p$ and $qin L$ is any point. Now let $MsubsetmathbbCP^2$ be any line not passing through $p$. Then one can define a map $pi:X_pto M$ by letting $pi(L,q)$ be the intersection of $L$ with $M$. Then, since $M$ is a $2$-sphere (as is every line in $mathbbCP^2$), this gives a submersion of $X_p$ onto $Msimeq S^2$ whose fibers are $2$-spheres. In fact, $pi$ is a holomorphic submersion, as is easy to verify in local coordinates.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Aug 19 at 10:10









                  Robert Bryant

                  71k5206306




                  71k5206306




















                      up vote
                      8
                      down vote













                      I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):



                      Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
                      $S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.



                      Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $mathbb Cmathbb P^n-1$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)






                      share|cite|improve this answer


























                        up vote
                        8
                        down vote













                        I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):



                        Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
                        $S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.



                        Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $mathbb Cmathbb P^n-1$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)






                        share|cite|improve this answer
























                          up vote
                          8
                          down vote










                          up vote
                          8
                          down vote









                          I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):



                          Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
                          $S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.



                          Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $mathbb Cmathbb P^n-1$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)






                          share|cite|improve this answer














                          I would like to add another geometric description of $mathbb Cmathbb P^n #overlinemathbb Cmathbb P^n$ as sphere bundle (same works for $mathbb Cmathbb P^n # mathbb Cmathbb P^n$):



                          Let $S^1 to S^2n+1 to mathbb Cmathbb P^n$ be the Hopf fibration. Then
                          $S^1$ acts on $mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $mathbb Cmathbb P^n-1$ in $mathbb Cmathbb P^n$. Thus the total space of $E$ is diffeomorphic to $mathbb Cmathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $mathbb Cmathbb P^n# overlinemathbb Cmathbb P^n$.



                          Moreover one deduces from this description that the connected sum is given as the quotient $(S^2n-1times S^2)/S^1$, where $S^1$ acts on $S^2n-1$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $mathbb Cmathbb P^n-1$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)







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                          edited Sep 10 at 10:55

























                          answered Aug 19 at 14:27









                          Panagiotis Konstantis

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