Hodge theory (after Deligne)

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In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely miraculous to specialists in Algebraic Geometry.



Can anyone explain why this is ?










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  • A while back I found this video of the lecture "Hodge Theory -- From Abel to Deligne" by Phillip Griffiths" youtube.com/watch?v=hPyDz5R5YaY rather enlightening.
    – j.c.
    10 mins ago














up vote
7
down vote

favorite












In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely miraculous to specialists in Algebraic Geometry.



Can anyone explain why this is ?










share|cite|improve this question





















  • A while back I found this video of the lecture "Hodge Theory -- From Abel to Deligne" by Phillip Griffiths" youtube.com/watch?v=hPyDz5R5YaY rather enlightening.
    – j.c.
    10 mins ago












up vote
7
down vote

favorite









up vote
7
down vote

favorite











In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely miraculous to specialists in Algebraic Geometry.



Can anyone explain why this is ?










share|cite|improve this question













In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely miraculous to specialists in Algebraic Geometry.



Can anyone explain why this is ?







ag.algebraic-geometry cohomology schemes hodge-theory zeta-functions






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asked 2 hours ago









THC

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  • A while back I found this video of the lecture "Hodge Theory -- From Abel to Deligne" by Phillip Griffiths" youtube.com/watch?v=hPyDz5R5YaY rather enlightening.
    – j.c.
    10 mins ago
















  • A while back I found this video of the lecture "Hodge Theory -- From Abel to Deligne" by Phillip Griffiths" youtube.com/watch?v=hPyDz5R5YaY rather enlightening.
    – j.c.
    10 mins ago















A while back I found this video of the lecture "Hodge Theory -- From Abel to Deligne" by Phillip Griffiths" youtube.com/watch?v=hPyDz5R5YaY rather enlightening.
– j.c.
10 mins ago




A while back I found this video of the lecture "Hodge Theory -- From Abel to Deligne" by Phillip Griffiths" youtube.com/watch?v=hPyDz5R5YaY rather enlightening.
– j.c.
10 mins ago










1 Answer
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The laudatio for the Wolf prize explains it like this:




Central to modern algebraic geometry is the theory of moduli, i.e.,
variation of algebraic or analytic structure. This theory was
traditionally mysterious and problematic. In critical special cases,
i.e., curves, it made sense, i.e., the set of curves of genus greater
than one had a natural algebraic structure. In dimensions greater than
one, there was some sort of structure locally, but globally everything
remained mysterious. [...] Building on Mumford’s and Griffiths’ work,
Pierre Deligne demonstrated how to extend the variation of Hodge
theory to singular varieties. This advance, called mixed Hodge theory,
allowed explicit calculation on the singular compactification of
moduli spaces that came up in Mumford’s geometric invariant theory.







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    1 Answer
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    1 Answer
    1






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    active

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    up vote
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    The laudatio for the Wolf prize explains it like this:




    Central to modern algebraic geometry is the theory of moduli, i.e.,
    variation of algebraic or analytic structure. This theory was
    traditionally mysterious and problematic. In critical special cases,
    i.e., curves, it made sense, i.e., the set of curves of genus greater
    than one had a natural algebraic structure. In dimensions greater than
    one, there was some sort of structure locally, but globally everything
    remained mysterious. [...] Building on Mumford’s and Griffiths’ work,
    Pierre Deligne demonstrated how to extend the variation of Hodge
    theory to singular varieties. This advance, called mixed Hodge theory,
    allowed explicit calculation on the singular compactification of
    moduli spaces that came up in Mumford’s geometric invariant theory.







    share|cite|improve this answer
























      up vote
      4
      down vote













      The laudatio for the Wolf prize explains it like this:




      Central to modern algebraic geometry is the theory of moduli, i.e.,
      variation of algebraic or analytic structure. This theory was
      traditionally mysterious and problematic. In critical special cases,
      i.e., curves, it made sense, i.e., the set of curves of genus greater
      than one had a natural algebraic structure. In dimensions greater than
      one, there was some sort of structure locally, but globally everything
      remained mysterious. [...] Building on Mumford’s and Griffiths’ work,
      Pierre Deligne demonstrated how to extend the variation of Hodge
      theory to singular varieties. This advance, called mixed Hodge theory,
      allowed explicit calculation on the singular compactification of
      moduli spaces that came up in Mumford’s geometric invariant theory.







      share|cite|improve this answer






















        up vote
        4
        down vote










        up vote
        4
        down vote









        The laudatio for the Wolf prize explains it like this:




        Central to modern algebraic geometry is the theory of moduli, i.e.,
        variation of algebraic or analytic structure. This theory was
        traditionally mysterious and problematic. In critical special cases,
        i.e., curves, it made sense, i.e., the set of curves of genus greater
        than one had a natural algebraic structure. In dimensions greater than
        one, there was some sort of structure locally, but globally everything
        remained mysterious. [...] Building on Mumford’s and Griffiths’ work,
        Pierre Deligne demonstrated how to extend the variation of Hodge
        theory to singular varieties. This advance, called mixed Hodge theory,
        allowed explicit calculation on the singular compactification of
        moduli spaces that came up in Mumford’s geometric invariant theory.







        share|cite|improve this answer












        The laudatio for the Wolf prize explains it like this:




        Central to modern algebraic geometry is the theory of moduli, i.e.,
        variation of algebraic or analytic structure. This theory was
        traditionally mysterious and problematic. In critical special cases,
        i.e., curves, it made sense, i.e., the set of curves of genus greater
        than one had a natural algebraic structure. In dimensions greater than
        one, there was some sort of structure locally, but globally everything
        remained mysterious. [...] Building on Mumford’s and Griffiths’ work,
        Pierre Deligne demonstrated how to extend the variation of Hodge
        theory to singular varieties. This advance, called mixed Hodge theory,
        allowed explicit calculation on the singular compactification of
        moduli spaces that came up in Mumford’s geometric invariant theory.








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        share|cite|improve this answer










        answered 29 mins ago









        Carlo Beenakker

        69.6k8155261




        69.6k8155261



























             

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