Vanishing cohomology of line bundles on the Springer resolution

Question

My question is regarding Broer's paper "Line bundles on the cotangent bundle of the flag variety" (see http://www.springerlink.com/content/t41418q436524515/). Given the Springer resolution, and its projection to the flag variety, $p: \mathcal{T} = G \times_{B} \mathfrak{u} \rightarrow G/B$; the aim is to prove that for $\lambda \in \Lambda^+$, if $L_{T}(k_{\chi})$ denotes the pull-back of the line-bundle $L_{G/B}(k_{\chi})$ under $p$, then $H^{i}(\mathcal{T}, L_T(k_{\chi})^{*}) = 0$. Specifically, I am asking about the contents on pg $6-7$ of this paper, proving the direction $(3) \rightarrow (1)$ of Theorem $2.4$. My main question is about the hypercohomology spectral sequence techniques that Broer uses; I know the bare basics of spectral sequences, but I wasn't able to find a solid reference for hypercohomology spectral sequences in McLeary, Bott/Tu and some other pdf's I tried looking at. The wiki page on hypercohomology does mention spectral sequences but not in sufficient depth.

Question 1: Broer looks at $X = G \times_B \mathfrak{g}$, and a certain section of the the vector bundle $G \times_B (\mathfrak{g} \times \mathfrak{g}/\mathfrak{u})$ on $X$; apparently there is a Koszul resolution of $O_T$ as an $O_X$-module. How does $O_T$, and the line bundle on $T$, $L_{T}(k_{\chi})^{*}$, acquire the structure of an $O_X$-module? Of course the ideal sheaf corresponding to $T$ is an $O_X$ module, but that doesn't seem to be what he is referring to. Here I refer to the second paragraph of $2.12$. Also, given that $X \cong G/B \times \mathfrak{g} $, how do we conclude that $H^i(X, L_X(\wedge^{i}(\mathfrak{g}/\mathfrak{u}) \times k_{\chi} )^{*}) \cong k[\mathfrak{g}] \otimes H^i(G/B, L_{G/B}(\wedge^{i}(\mathfrak{g}/\mathfrak{u}) \times k_{\chi})^{*})$?

Question 2: In the last paragraph on pg $6$, he says there is a spectral sequence of graded $k[\mathfrak{g}]$-modules, with $E_1^{-j,i} = k[\mathfrak{g}] \otimes H^i(G/B, L_{G/B}(\wedge^{i}(\mathfrak{g}/\mathfrak{u}) \times k_{\chi})^{*})$. How do we construct this spectral sequence? (I am presuming this is a hypercohomology spectral sequence coming from the Koszul resolution above, but I don't understand hypercohomology spectral sequences).

Question 3: At the start of pg $7$: supposing $V_{\nu}^{*}$ occurs in $H^i(G/B, L_{G/B}(\wedge^{i}(\mathfrak{g}/\mathfrak{u}) \times k_{\chi})^{*})$, where $i-j$ is chosen to be maximal for this cohomology to not vanish; then why does a basis correspond to free generators in $E_1^{-j,i}$ of degree $j$? I do not quite understand how Nakayama is being applied here. Why do all elements in $E_1^{-j-1,i}$ have degree larger than $j$?

Question 4: After deducing that $V_{\nu}$ occurs in $H^{i-j}(T, L_T(k_{\chi}))$, he says that any weight of $S^m \mathfrak{u} \otimes k_{\chi}$ is of the form $\chi + \phi$, where $\phi \geq 0$. Why does this imply that $\nu + \rho = w(\chi + \psi + \rho) \geq \chi + \psi + \rho$, for some $\psi \geq 0$ and $l(w) = i-j$?

Answer 1

For your first question, let $i : T \hookrightarrow X$ be the inclusion; then $\mathcal O_T$ becomes a $\mathcal O_X$-module by identifying $\mathcal O_T$ with $i_* \mathcal O_T$, an $\mathcal O_X$-module. The isomorphism you ask about comes from the following general principle: For any rational $B$-modules $M$ and $N$, let $\pi : G \times^B M \to G/B$ be the projection and consider the $\mathcal O_{G \times^B M}$ module $\mathcal L_M(N) := \pi^* \mathcal L(N)$. Then, by the projection formula, $$\pi_* \pi^* \mathcal L_M(N) \cong \mathcal L(N) \otimes \pi_* \mathcal O_{G \times^B M} \cong \mathcal L(N) \otimes \mathcal L( S^\bullet( M^* ) ) .$$ This implies that $$ H^0( G \times^B M, \mathcal L_M(N) ) \cong H^0( G/B, \mathcal L(N) \otimes \mathcal L( S^\bullet( M^* ) ) .$$ Applying this principle in the particular case where $M = \mathfrak g$ and using the fact that $\mathcal L(S^\bullet(\mathfrak g^*)) \cong \mathcal L(k[ \mathfrak g ])$ is a free $ \mathcal O_{G/B} $-module gives the result.

I don't off the top of my head know the answers to your other questions, but I would second the recommendation to look at Brion-Kumar. Also, if you just want to know the characteristic 0 story, you can look at Broer's later paper, Normality of some Nilpotent Varieties and Cohomology of Line Bundles on the Cotangent Bundle of the Flag Variety. In that paper he gives a much simpler proof (using Grauert-Riemenschneider) of the cohomology vanishing result for dominant weights (Theorem 2.2); however, the full vanishing theorem that you're looking at requires more work, since it involves vanishing for some nondominant weights too. He reproves that theorem in this paper as well. I don't know exactly what results you want, but I would recommend looking at Broer's later paper; it's a lot easier to read and avoids the spectral sequence arguments. It's also worth mentioning that the method of proof used by Brion-Kumar involves Frobenius splitting and is a very different argument than the one Broer gives; their result is stronger, since it's in positive characteristic.

Answer 2

This topic (pushing forward twisted Koszul complexes) is discussed in detail in Chapter 5 of Jerzy Weyman's book Cohomology of Vector Bundles and Syzygies. In particular, you'll probably be interested in Sections 5.1-5.2.