Some new results on modified diagonals
Some new results on modified diagonals
复制标题
修改对角线的一些新结果
DOI：
10.2140/gt.2015.19.3307
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发表时间：
2014
影响因子：
2
通讯作者：
C. Voisin
中科院分区：
文献类型：
--
作者：
C. Voisin
奥格雷迪研究最近$m$-th修改的对角线光滑的射影品种，推广的格罗斯-Schoen修改的小对角线。这些圈$\Gamma^m（X，a）$取决于X$中参考点$a\的选择（或者更一般地说是一个度$1$零圈）。我们证明了对于任意的X，a，圈\Gamma^m（X，a）对于大的m为零。我们还证明了O 'Grady的猜想：如果$X$是$Y$的双重覆盖，并且$\Gamma^m（Y，a）$为零（其中$a$属于分支轨迹），则$\Gamma^{2 m-1}（X，a）$为零，并且我们给出了对更高次有限覆盖的推广.
最后证明了当$X=S^{[m]}$，$S$是一个K3 $曲面，$n= 2 m $时，$\Gamma^{n+1}（X，o_X）=0$，这是O 'Grady提出并证明的。
O'Grady studied recently $m$-th modified diagonals for a smooth projective variety, generalizing the Gross-Schoen modified small diagonal. These cycles $\Gamma^m(X,a)$ depend on a choice of reference point $a\in X$ (or more generally a degree $1$ zero-cycle). We prove that for any $X,a$, the cycle $\Gamma^m(X,a)$ vanishes for large $m$. We also prove the following conjecture of O'Grady: if $X$ is a double cover of $Y$ and $\Gamma^m(Y,a)$ vanishes (where $a$ belongs to the branch locus), then $\Gamma^{2m-1}(X,a)$ vanishes, and we provide a generalization to higher degree finite covers.
We finally prove the vanishing $\Gamma^{n+1}(X,o_X)=0$ when $X=S^{[m]}$, $S$ a $K3$ surface, and $n=2m$, which was conjectured by O'Grady and proved by him for $m=2,3$.