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Can anyone write down the kahler einstein metric for exceptional compact type hermitian symmetric spaces($\frac{E_6}{SO(10)*SO(2)}$ and $\frac{E_7}{E_6*SO(2)}$). I can find the bergmann kernel for their dual, are there anyway i can exploit this?

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  • $\begingroup$ "Write down" in terms of what? We know the Kähler structures explicitly in terms of their pullbacks as left-invariant tensors on the Lie groups $\mathrm{E}_6$ and $\mathrm{E}_7$. This is just linear algebra. I suspect, though, that you don't care for this way of "writing" them "down". $\endgroup$
    – Robert Bryant
    Sep 11, 2015 at 14:11
  • $\begingroup$ Dear Robert, what i really need is to find the volume element locally explicitly. I guess it will be similar to the bergman kernel function for their the non-compact duals, because this is the case when we consider Grassmanians or hyperquadrics. Do you know any results about this? $\endgroup$
    – user42804
    Sep 11, 2015 at 23:19
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    $\begingroup$ I don't know what you mean by 'find the volume element locally explicitly'. 'Explicitly' in terms of what? It is explicit globally as a left-invariant 32-form on $\mathrm{E}_6$ or a 54-form on $\mathrm{E}_7$, but that's trivial and surely not what you are trying to get. What about this: Take the standard embedding of $E_6$ into $\mathrm{SU}(27)$, as the subgroup that preserves Cartan's cubic $P$. Then the $16$-dimensional exceptional HSS $X$ is the singular locus of the hypersurface $P=0$ in $\mathbb{CP}^{26}$, and the metric on $X$ is just the restriction of the Fubini-Study metric to $X$. $\endgroup$
    – Robert Bryant
    Sep 12, 2015 at 11:26
  • $\begingroup$ Thanks! that's kind of what i am looking for! I want to write down the volume form in terms of local coordinates$z_1,z_2,...,z_{16}$ What about $E_7$ case? How to embed it into $CP^{55}$? $\endgroup$
    – user42804
    Sep 12, 2015 at 15:10
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    $\begingroup$ Well, for the $E_7$ case, you write down Cartan's homogeneous quartic polynomial $Q$ in $56$ variables (this gives an embedding of $E_7$ into $\mathrm{Sp}(28)\subset \mathrm{SU}(56)$), and then you look at the singular locus $Y$ of the projectivized hypersurface $Q = 0$ in $\mathbb{CP}^{55}$, which turns out to be a smooth projective variety of dimension $27$. Then the Kähler-Einstein metric on $Y$ that is invariant under $E_7$ is just the restriction of the Fubini-Study metric on $\mathbb{CP}^{55}$ to $Y$. $\endgroup$
    – Robert Bryant
    Sep 12, 2015 at 22:34

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