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!