I am currently reading Ed Segal’s Equivalences Between GIT Quotients of Landau-Ginzburg B-Models. In this post I want to do a computation from Section 1.1 of this paper regarding, in the notation of the paper, an isomorphism of cohomology groups \[ H^\ast_{\mathcal{X}}(\mathcal{O}(i)) \cong H^\ast_{X_\pm}(\mathcal{O}(i)),\ i \in[-1,1]. \] This is not too difficult and this post is largely for my personal understanding. Let’s recall the setup.

Let \(V= \mathbb{C}^4\) with coordinate algebra \(\mathbb{C}[V]=\mathbb{C}[x_1,x_2,y_1,y_2]\). Let \(\mathbb{C}^\ast\) act on \(\mathbb{C}[V]\) such that \(x_1,x_2\) has weight \(-1\) and \(y_1,y_2\) have weight 1. If we use the notation \( (p_1,p_2,q_1,q_2)\) for a \(\mathbb{C}\)-point of \(V\), then \[ \lambda\cdot (p_1,p_2,q_1,q_2) = (\lambda p_1,\lambda p_2,\lambda^{-1}q_1,\lambda^{-1}q_2),\ \lambda\in\mathbb{C}^\ast \] Set \(\mathcal{X} = [V/\mathbb{C}^\ast]\) to be the corresponding quotient stack. To be clear about \(\mathbb{C}\)-points vs. regular functions we will use the notation above.

Recall, to take a GIT quotient, we require a \(\mathbb{C}^\ast\)-linearized line bundle on \(V\). The collection of \(\mathbb{C}^\ast\)-linearized line bundles on \(V\) are, up to isomorphism, of the form \(\mathcal{O}(i):=\mathcal{O}_V\otimes\chi^i\), where \(\chi:\mathbb{C}^\ast\to \mathbb{C}^\ast\) is the standard primitive character. Denote the set of semi-stable points with respect to \(\mathcal{O}(i)\) by \(V_i\). Then for \(i\neq 0\) we have \[ V_i = \begin{cases} V\setminus \{p_1=p_2=0\} & i>0, \\ V\setminus\{q_1=q_2=0\} & i<0. \end{cases} \] The GIT quotients correspond to taking either \(\mathcal{O}(i)\) or \(\mathcal{O}(-i)\) as the semi-stable points with respect to any positive (resp. negative) character are the same as \(\mathcal{O}_V(1)\) (resp. \(\mathcal{O}_V(-1)\)). Define open substacks of \(\mathcal{X}\) by \(X_+ = [V_1/\mathbb{C}^\ast]\) and \(X_- = [V_{-1}/\mathbb{C}^\ast]\). We also denote the restriction of the line bundle \(\mathcal{O}(i)\) to \(X_\pm\) by \(\mathcal{O}(i)\).

The cohomology isomorphism stated above is easy to see for \(\ast = 0\) by applying Hartog’s Lemma (we have deleted a codimension 2 subvariety in each case). However, the stacks \(X_\pm\) are no longer affine. In fact, there is an isomorphism \[ X_\pm\cong \mathrm{Tot}_{\mathbb{P}^1}(\mathcal{O}(-1)^{\oplus 2}). \] Thus the line bundles \(\mathcal{O}(i)\) could have higher cohomology. Moreover, Ed’s paper generalizes this to Calabi-Yau \(\mathbb{C}^\ast\) actions on a finite dimensional vector space \(V\). So it could be that we delete a codimension 1 subvariety in which case the claim for \(\ast = 0\) is no longer clear. The paper later relies on this identification of \(X_\pm\) with an (orbifold) vector bundle over a (weighted) projective space. With that in mind, I want to explicitly do these computations using the identification of \(X_\pm\) with the total space of \(\mathcal{O}(-1)^{\oplus 2}\) over \(\mathbb{P}^1\).

It is enough to do the computation for \(X_+\). By projecting onto \( (p_1,p_2)\), we obtain a surjective mapping \(\pi:X_+\to\mathbb{P}^1\). This is well-defined since \(p_1,p_2\) cannot vanish simultaneously. There is also a \(\mathbb{C}^\ast\)-equivariant mapping \(\iota:X_+\to \mathbb{C}^2\oplus \mathbb{C}^2\) given by \[ \iota(p_1,p_2,q_1,q_2) = (p_1q_1,p_2q_1,p_1q_2,p_2q_2). \] Then \(\iota\) is a closed embedding and the image is \(\mathrm{Tot}(\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus 2})\). Indeed, the fiber over \( p=[p_1:p_2]\in\mathbb{P}^1\) consists of two copies of \(\mathbb{C}\cdot p\).

It is misleading to write \(X_+\cong\mathrm{Spec}_{\mathbb{P}^1}(\mathrm{Sym}(\mathcal{O}(1)^{\oplus 2}))\) as the restriction of the coordinate functions on \(\mathbb{C}^2\oplus \mathbb{C}^2\) are \(x_1y_1,x_2y_1,x_1y_2,x_2y_2\). Rather, we have \[ X_+\cong \mathrm{Spec}_{\mathbb{P}^1}(\mathrm{Sym}(\mathcal{O}(1)\otimes V_y^\vee)) \] where \(V_y\subset V\) consists of the points \( (0,0,q_1,q_2)\).

We can now see the claim for \(i = 0\): \begin{align*} H^\ast_{X_+}(\mathcal{O}_{X_+}) &\cong H^\ast_{\mathbb{P}^1}(\mathrm{Sym}(\mathcal{O}(1)\otimes V_y^\vee)\\ &= \mathbb{C}[x_1y_1,x_2y_1,x_1y_2,x_2y_2] \\ &= \mathbb{C}[x_1,x_2,y_1,y_2]^{\mathbb{C}^\ast} \\ &= H^\ast_{\mathcal{X}}(\mathcal{O}_{\mathcal{X}}) \end{align*}

For \(i = \pm 1\), we notice that \(\pi^\ast\mathcal{O}(i)\cong \mathcal{O}(i)\). This follows since the Picard group of \(X_+\) and \(\mathbb{P}^1\) are isomorphic via pullback. Hence, \(\pi^\ast\mathcal{O}(i)\) is either \(\mathcal{O}(i)\) or \(\mathcal{O}(-i)\). Since the coordinate functions \(x_i\) pull back to \(x_i\) they have weight 1 and so it must be that \(\pi^\ast\mathcal{O}(i)\cong\mathcal{O}(i)\). Using the projection formula, we have: \[ \pi_\ast\mathcal{O}(\pm1)\cong \mathcal{O}(\pm1)\otimes\pi_\ast\mathcal{O}\cong \mathcal{O}(\pm1)\otimes \mathrm{Sym}(\mathcal{O}(1)\otimes V_y^\vee). \] Since the terms involved have no higher cohomology, we just need to compute global sections. We compute \begin{align*} H^0(\mathcal{O}(1)\otimes\mathrm{Sym}(\mathcal{O}(1)\otimes V_y^\vee)) &\cong H^0(\mathcal{O}(1)\oplus \mathcal{O}(2)\otimes V_y^\vee\oplus \cdots) \\ &\cong \mathbb{C}[x_1,x_2,x_1^2y_1,x_2^2y_1, x_1^2y_2,x_2^2y_2] \\ &= H^0_\mathcal{X}(\mathcal{O}_\mathcal{X}(1)). \end{align*} A similar computation holds for \(\mathcal{O}(-1)\).

This is done more cleanly and precisely in Ed’s paper and I recommend you take a look!