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If $X:=Gr_k(V)$ denotes the Grassmannian of $k$-dimensional subspaces of an $n$-dimensional vector space $V$, it is well-known that the canonical bundle $\omega_X$ is given by $$\omega_X:= \det \Omega_X^1= \det E \otimes Q^* = \mathcal{O}(-n),$$ where $\Omega_X^1$ denotes the cotangent bundle of $X$, $E$ is the tautological subbundle of rank $k$ of the Grassmannian and $Q$ is the corresponding quotient subbundle.

This may be generalized to Grassmann bundles (e.g. https://arxiv.org/pdf/0807.3296.pdf, prop. 1.5): If $V$ is a vector bundle over some scheme $Y$ and $X=Gr_Y(k,V)$ parametrizes rank-$k$-subbundles of $V$, then similarly $$\omega_X=\det E \otimes Q^* = (\det V)^{-k} \otimes \mathcal{O}(-n).$$

Now I would like to consider Grassmannians (or, more generally, flag bundles) of other Lie types, especially in type $D$. There are notions of orthogonal flag bundles (and orthogonal Grassmann bundles), but I cannot find any references for the canonical bundle. For an $2n$-dimensional vector space $V$ equipped with a quadratic form, if $X=OG(n,V)$ parametrizes maximal isotropic subspaces of $V$, I know that the canonical divisor is given by $\omega_X = \mathcal{O}(-2n+2)$ (still I did not find any good reference).

My questions are:

  1. Is there a nice way to see $\omega_X=\mathcal{O}(-2n+2)$?
  2. How do I generalize to bundles? There should be involved some more terms coming from the base scheme, but I do not have a good intuition for this.

Many thanks in advance.

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1 Answer 1

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I figured out the following: The tangent bundle of the orthogonal Grasmann bundle over a variety $X$ is given by $$T \mathrm{OG}_X(n,V) = \wedge^2 E^* \otimes \pi^* \mathcal{L}$$ where $E$ denotes the pullback of the tautological bundle over $\mathrm{Gr}(n,2n)$ and $\mathcal{L}$ is the line bundle in which the symmetric bilinear form has its values. Taking the determinant of the dual of it we get $$\omega_{\mathrm{OG}_X(n, V)} = (E^*)^{n-1} \otimes \pi^*\mathcal{L}^{\frac{n(n-1)}{2}} = \mathcal{O}(-2n+2) \otimes \pi^*\mathcal{L}^{\frac{n(n-1)}{2}}.$$ Did I make a mistake?

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