4-dimensional symplectic contractions

Abstract

Local symplectic contractions are resolutions of singularities which admit symplectic forms. Four dimensional symplectic contractions are (relative) Mori Dream Spaces. In particular, any two such resolutions of a given singularity are connected by a sequence of Mukai flops. We discuss the cone of movable divisors on such a resolution; its faces are determined by curves whose loci are divisors, we call them essential curves. The movable cone is divided into nef chambers which are related to different resolutions; this subdivision is determined by classes of 1-cycles. We also study schemes parametrizing minimal essential curves and show that they are resolutions, possibly non-minimal, of surface Du Val singularities. Some examples, with an exhaustive description, are provided.

Appendix

1.1 Contraction to the nilpotent cone

In this section we recall known facts about flag varieties of simple Lie groups and about contractions to the nilpotent cone. This subject is classical and well documented, see e.g. [32] or [10] with the references therein. For a more recent survey see also [29]. Our point of view is somehow more geometric, related to homogeneous varieties, in the spirit of [30], and directed on understanding the picture at the level of the related root systems. We refer to [33, Ch. 18] for generalities on root systems.

Let $G$ be a complex simple algebraic group with the Lie algebra $\mathfrak{g }$. By $R$ we denote the set of roots of $\mathfrak{g }$ and consider the lattices of roots and of weights $\Lambda _R \subset \Lambda _W$ of the algebra (or group) in question; let $V=\Lambda _R\otimes \mathbb{R }$. By $B$ we denote a Borel subgroup of $G$ and $F=G/B$ is its flag variety. It is known that we have a natural isomorphism $\text{ Pic} F\simeq \Lambda _W$ under which $\text{ Nef}(F)\subset N^1(F)$ is identified with the Weyl chamber in $V$. Under this identification any irreducible representation $U_w$ of $G$ with the highest weight $w$ corresponds to the complete linear system on $F$ of a nef line bundle whose associated map, $F\rightarrow \mathbb{P }(U_w)$, maps $F$ to the unique closed orbit.

Moreover, the sum of the positive roots $\rho =\sum _{\alpha \in R^+}\alpha$ can be identified with the anticanonical class $-K_F$ and the Weyl formula, describing the dimension of irreducible representations, yields the Hilbert polynomial on $\text{ Pic} F$. That is, for every $\lambda \in \Lambda _W$, the dimension formula, or the Euler characteristic of the respective line bundle on $F$, can be written as a polynomial

$$\begin{aligned} H(\lambda )=\prod _{\alpha \in R^+} \frac{((\lambda +\rho /2),\alpha )}{(\rho /2,\alpha )} \end{aligned}$$

where $(\ \ ,\ )$ denotes the Killing form and $R^+$ is the set of positive roots. Note that the above polynomial is of degree equal to $\dim F$; $H(-\lambda -\rho )=(-1)^{\dim F}H(\lambda )$ is exactly Serre duality.

The Killing form allows to relate $V$ to its dual. For every root $\alpha \in R$ we set $V^*\ni \alpha ^\vee =(v\mapsto 2(\alpha ,v)/(\alpha ,\alpha ))$. The facets of the Weyl chamber are supported by the simple roots, that is they are hypersurfaces defined by forms $\alpha ^\vee$.

Lemma 7.1

The extremal contraction $\hat{\pi }_\alpha : F\rightarrow F_\alpha$ associated with the facet $\alpha ^\perp \cap \text{ Nef}(F)$ is a $\mathbb{P }^1$ bundle and $\alpha ^\vee$ is the class of the extremal curve in $N_1(F)$. In $\text{ Pic}(F) = \Lambda _W$ the class of the relative cotangent bundle, $\Omega (F/F_\alpha )$, is $-\alpha$.

Proof

Note that the restriction of the polynomial $H(\lambda )$ to the hyperplane $\alpha ^\perp$ defined by the face $\alpha ^\vee$ is of degree $\dim F-1$ and $\alpha ^\vee (\rho )=2$, [33, 18.7.6]. This means that the extremal contraction $F\rightarrow F_\alpha$ associated with the facet $\alpha ^\perp \cap \text{ Nef}(F)$ is a $\mathbb{P }^1$ bundle and $\alpha ^\vee$ is the class of the fiber. On the other hand, $\rho -\alpha \in \alpha ^\perp$ and $H(s_\alpha (\lambda ))-\alpha )=-H(\lambda )$, which is the relative duality $\square$

Let $X$ be the total space of the cotangent bundle of $F$, that is $X=\text{ Spec}_F(\text{ Symm}(TF))$. Recall that $TF=G \times _B \mathfrak{g }/\mathfrak{b }$, where $\mathfrak{b }\subset \mathfrak{g }$ is tangent to $B$ and $B$ acts on $\mathfrak{g }/\mathfrak{b }$ via adjoint representation and the quotient $\mathfrak{g }\rightarrow \mathfrak{g }/\mathfrak{b }$. Alternatively, $T^*F=G\times _B \mathfrak{u }$ where $\mathfrak{u }\subset \mathfrak{g }$ is the nilradical of $\mathfrak{b }$. The variety $X$ is symplectic.

Since $TF$ is spanned by its global sections,which form the Lie algebra $\mathfrak{g }$, we have a map $X\rightarrow \mathfrak{g }^*$ which contracts the zero section to $0$. The image is a normal variety called the nilpotent cone, which we denote it by $Y$. The map $\pi : X\rightarrow Y$ is a symplectic contraction.

Clearly, $N^1(X/Y)=N^1(F)$, $\text{ Nef}(X/Y)=\text{ Nef}(F)$ and every extremal contraction $\hat{\pi }_\alpha : F\rightarrow F_\alpha$, which is a $\mathbb{P }^1$ bundle, extends to a divisorial contraction $\pi _\alpha : X\rightarrow X_\alpha$ with all nontrivial fibers being $\mathbb{P }^1$. Let $E_\alpha \subset X$ be the exceptional divisor of $\pi _\alpha$ and $C_\alpha$ be a general fiber of $\pi _\alpha$ restricted to $E_\alpha$.

Lemma 7.2

The class of $C_\alpha$ in $V^*=N_1(X/Y)$ is $\alpha ^\vee$. The class of $E_\alpha$ in $\text{ Pic} X = \Lambda _W$ is $-\alpha$.

Proof

We have an exact sequence of vector bundles over $F$:

$$\begin{aligned} 0\longrightarrow \hat{\pi }_\alpha ^*(\Omega F_\alpha )\longrightarrow \Omega F\longrightarrow \Omega (F/F_\alpha )\longrightarrow 0 \end{aligned}$$

and the divisor $E_\alpha$ in the total space of $\Omega F$ is the total space of the sub-bundle $\pi _\alpha ^*(\Omega F_\alpha )$. Thus, the restriction of its normal to $F$ is the line bundle $\Omega (F/F_\alpha )$ hence the lemma follows by Lemma 7.1. $\square$

Corollary 7.3

c.f. [16, (5.2)] In the above situation, the intersection matrix $E_\alpha \cdot C_\beta$ is the negative of the Cartan matrix of the respective root system.

The above observation is the key for Brieskorn-Slodowy result on the type singularity of the nilpotent cone in of codimension $2$; it can be expressed as follows:

Theorem 7.4

(Brieskorn, Slodowy) Let $\pi : X=G/B \rightarrow Y$ be the contraction to the nilpotent cone. If the root system of $G$ is of type $\mathbb{A }_n$, $\mathbb{D }_n$, $\mathbb{E }_6$, $\mathbb{E }_7$, $\mathbb{E }_8$ then in codimension 2 the contraction $\pi$ is the resolution of a surface Du Val singularity of the same $\mathbb{A }-\mathbb{D }-\mathbb{E }$ type. If $G$ is of type $\mathbb{B }_n$, $\mathbb{C }_n$, $\mathbb{F }_4$ and $\mathbb{G }_2$ then in codimension 2 the contraction $\pi$ is the resolution of singularities of type $\mathbb{A }_{2n-1}$, $\mathbb{D }_{n+1}$, $\mathbb{E }_6$ and $\mathbb{D }_4$ and the irreducible components of the exceptional set of $\pi$ are in bijection with the orbits of the action of the group of automorphisms of the Dynkin diagrams of latter type.

We have the following immediate consequence of Lemma 7.1 and Lemma 7.2.

Corollary 7.5

In the above case $\text{ Mov}(X/Y)=\text{ Nef}(X/Y)$ coincides with the Weyl chamber.

1.2 Resolving $\mathbb{C }^4/\sigma _3$

In this last section we will give an explicit description of the symplectic resolution of the quotient $\mathbb{C }^4 / \sigma _3$ introduced in Sect. 6.3 (see also [3]). We will constantly refer to the following commutative diagram, which comes from the presentation of $\sigma _3=D_6$ in terms of a semisimple product, $\sigma _3 = \mathbb{Z }_3\rtimes \mathbb{Z }_2$:

(7.1)

The vertical map $q: T \rightarrow \mathbb{C }^4/\mathbb{Z }_3$ is the toric resolution of $\mathbb{C }^4/ \mathbb{Z }_3$ which can be described as follows. Let $N_0$ be a lattice with the basis $e_1,e_2,f_1,f_2$ and in $N_0\otimes \mathbb{R }$ take the standard cone $\langle e_1,e_2,f_1,f_2 \rangle$ representing $\mathbb{C }^4$. The toric singularity $\mathbb{C }^4/\mathbb{Z }_{3}$ is obtained by extending $N_0$ to an overlattice $N$ (keeping the same cone) generated by adding to $N_0$ an extra generator $v_1=(e_1+e_2)/3+2(f_1+f_2)/3$. If $v_2=2(e_1+e_2)/3+(f_1+f_2)/3$ then the rays generated by $e_i$’s, $f_i$’s and $v_i$’s are in the fan of the toric resolution of $\mathbb{C }^4/\mathbb{Z }_3$ which is presented in the following picture by taking a affine hyperplane section of the cone $\langle e_1,e_2,f_1,f_2 \rangle$. The solid edges are the boundary of the cone while its division is marked by dotted line segments.

The exceptional set of this resolution consists of two divisors, $E_1, E_2$, both isomorphic to a $\mathbb{P }^2$- bundle over $\mathbb{P }^{1}$, namely $\mathbb{P }(\mathcal{O }(2)\oplus \mathcal{O }\oplus \mathcal{O })$. They intersect along a smooth quadric $\mathbb{P }^{1}\times \mathbb{P }^{1}$.

The action of $\mathbb{Z }_2$ on $\mathbb{C }^4/\mathbb{Z }_3$ can be lifted up to an action on $T$. This action, which is induced by the reflections in $\sigma _3=D_6$, identifies the two divisors by identifying the $\mathbb{P }^2$ ruling of $E_1$ with that of $E_2$; on the intersection the action interchanges the coordinates on $\mathbb{P }^{1}\times \mathbb{P }^{1}$.

Going back to the Diagram 7.1, $p_2$ is the resolution of the quotient $T/\mathbb{Z }_2$ obtained by blowing up the surface which is the locus of $A_1$-singularities. The morphism $p_1$ is the blow-up along the fixed point set of the $\mathbb{Z }_2$-action. We denote by $\Delta _ W$ and $\Delta _Z$ the exceptional divisors. Then $\nu$ is a $2:1$ cover ramified along $\Delta _W$.

The divisor $\Delta _W$ is irreducible and its intersection with the fiber over the special point, which is the strict transformof $E_1\cup E_2$, call it $E_1^{\prime }\cup E_2^{\prime }$, is equal to the 3rd Hirzebruch surface $F_3$. This follows from computing the normal of the curve which is the fixed point set of the $\mathbb{Z }_2$ action in the exceptional locus of $T$. Indeed, the normal of the intersection $E_1\cap E_2=\mathbb{P }^{1}\times \mathbb{P }^{1}$ is $\mathcal{O }(1,-2)+\mathcal{O }(-2,1)$ and the normal of the diagonal in the intersection is $\mathcal{O }(2)$. Thus the normal of the diagonal of $\mathbb{P }^{1}\times \mathbb{P }^{1}$ in $T$ is $\mathcal{O }(-1)\oplus \mathcal{O }(-1)\oplus \mathcal{O }(2)$ and, since its normal in the fixed point set is $\mathcal{O }(-1)$, it follows that the normal of the fixed point set over the diagonal is $\mathcal{O }(-1)\oplus \mathcal{O }(2)$. Finally, let us note that the intersection in $W$ of the $F_3$ surface with the strict transform of $\mathbb{P }^{1}\times \mathbb{P }^{1}$ is the exceptional curve in the surface $F_3$ and the diagonal in $\mathbb{P }^{1}\times \mathbb{P }^{1}$.

The fibers of the ruling $E_i \rightarrow \mathbb{P }^{1}$, for $i=1,\ 2$, are blown up in $E_i^{\prime }$ to ruled surfaces (1st Hirzebruch) and the map $E_i^{\prime }\rightarrow \mathbb{P }^{1}$ can be factored either by blow down $E_i^{\prime }\rightarrow E_i$ or by a $\mathbb{P }^{1}$-bundle $E_i^{\prime }\rightarrow F_3$.

The strict transform of the surface $E_1\cap E_2=\mathbb{P }^{1}\times \mathbb{P }^{1}$ is mapped via the quotient map $W\rightarrow Z$ to $\mathbb{P }^2$, and this is a double covering ramified over the diagonal in $\mathbb{P }^{1}\times \mathbb{P }^{1}$. The exceptional curve of $F_3$, which is the diagonal in $\mathbb{P }^{1}\times \mathbb{P }^{1}$, becomes a conic in $\mathbb{P }^2$. Thus, eventually, we see that $E_1^{\prime }$ is identified with $E_2^{\prime }$ and via $\nu$ they are sent to a (non-normal) divisor $E_Z$ in $Z$. The divisors $\Delta _W$ and $E_Z$ generate $Pic Z$ and $K_Z=E_Z$.

From the computation of the intersection of curves and divisors we see that the divisor $E_Z$ is not numerically effective, hence $Z$ admits a birational Fano-Mori contraction $Z\rightarrow X$ with exceptional divisor $E_Z$. We describe the contraction by looking at the normalization of $E_Z$. Namely, by looking at the numerical classes of curves, we conclude that the resulting map is a composition $E_1^{\prime }\rightarrow F_3\rightarrow S_3$, where the latter map is the contraction of the exceptional curve in $F_3$ to the vertex of the cubic cone $S_3$. Therefore a general fiber of $Z\rightarrow X$ over $E_Z$ is a $\mathbb{P }^{1}$—-that is, generally this is a blow-down of the divisor $E_Z$ to a surface—while the special fiber is a $\mathbb{P }^2$. Such a contraction was discussed in [2] where it was proved that the image $X$ is a smooth $4$-fold and the divisor $E_Z \subset Z$ is blow-down to the rational cubic cone $S_3 \subset X$. Moreover $K_X = \mathcal{O }_X$.

Let us finally consider the induced map $\pi : X \rightarrow Y:= \mathbb{C }^4 / \sigma _3$. It is a crepant contraction which contracts the divisor $\Delta _W$ to a surface $S$ which, outside the point $0$, is a smooth surface of $A_1$ singularities (coming from the $\mathbb{Z }_2$-action); moreover it contracts $S_3$ to $0$. The surface $S$ is non-normal in $0$. This is a crepant, hence symplectic, resolution of $\mathbb{C }^4 / \sigma _3$.

Note that $Pic(X/Y) = \mathbb{Z }$, therefore $\text{ Mov}(X/Y)$ is one dimensional. This is the only SQM model over $Y$.

We conclude with the description of the family of rational curves (of $X$ over $Y$). Let $C$ be the essential curve of the symplectic resolution $\pi : X \rightarrow Y:= \mathbb{C }^4 / \sigma _3$ and let $\mathcal{V }\subset RatCurves^n(X/Y)$ be a family containing $C$. Then $\mathcal{V }$ is a smooth surface which contains a $(-1)$-curve, which parametrizes the lines in the ruling of $S_3$. The normalization of $S$ is a smooth surface and $\mathcal{V }$ is obtained by blowing up the point of the normalization which stays over $0$.