arXiv 1111.00472011-11math.AG

K3 ^[4]型全纯辛簇中的Lagrange 4-平面

Lagrangian 4-planes in holomorphic symplectic varieties of K3^[4] type

Benjamin Bakker, Andrei Jorza

备注：: An alternative analysis of the diophantine equation in section 5 has been added

摘要：

我们分类的拉格朗日4-平面$¶^4$的上同调类的光滑流形$X$变形等价于一个希尔伯特计划的4点的$K3$表面上，单值行动。经典地，$K3$曲面$S$上的有效曲线锥由非负类$C$和节点类$C$生成，其中非负类$C$为$(C,C)\geq0$，节点类$C$为$(C,C)=-2$ ; Hassett和Tschinkel猜想，全纯辛簇$X$上的有效曲线的锥类似地由“节点”类$C$控制，使得$(C,C)=-γ$，对于$(\cdot,\cdot)$现在是Beauville-Bogomolov形式，其中，$γ$对与$C$相关联的极值收缩的几何形状进行分类。特别是，他们推测，$X$变形相当于一个希尔伯特计划的$n$点的$K3$表面上，类$C=\ell$的一条线在一个光滑的拉格朗日$n$ -平面$¶^n$必须满足$(\ell,\ell)=-\frac{n+3}{2}$。通过计算$X$上的monodromy不变量环，证明了关于$n=4$的猜想，并证明了Lagrangian 4-平面存在唯一的monodromy轨道.

英文摘要：

We classify the cohomology classes of Lagrangian 4-planes $¶^4$ in a smooth manifold $X$ deformation equivalent to a Hilbert scheme of 4 points on a $K3$ surface, up to the monodromy action. Classically, the cone of effective curves on a $K3$ surface $S$ is generated by nonegative classes $C$, for which $(C,C)\geq0$, and nodal classes $C$, for which $(C,C)=-2$; Hassett and Tschinkel conjecture that the cone of effective curves on a holomorphic symplectic variety $X$ is similarly controlled by "nodal" classes $C$ such that $(C,C)=-γ$, for $(\cdot,\cdot)$ now the Beauville-Bogomolov form, where $γ$ classifies the geometry of the extremal contraction associated to $C$. In particular, they conjecture that for $X$ deformation equivalent to a Hilbert scheme of $n$ points on a $K3$ surface, the class $C=\ell$ of a line in a smooth Lagrangian $n$-plane $¶^n$ must satisfy $(\ell,\ell)=-\frac{n+3}{2}$. We prove the conjecture for $n=4$ by computing the ring of monodromy invariants on $X$, and showing there is a unique monodromy orbit of Lagrangian 4-planes.

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英文摘要：

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