From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives
From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives
复制标题
通过非交换动机的雅可比行列式从半正交分解到极化中间雅可比行列式
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Gonçalo Tabuada
中科院分区：
文献类型：
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作者：
M. Bernardara;Gonçalo Tabuada
设X和Y是复光滑投射簇，D^B（X）和D^B（Y）是凝聚层的有界导范畴.假设存在一个三角范畴T，它在D^B（X）和D^B（Y）中都是可容许的.利用最近的非对易动机的Jacobian理论，我们从这个范畴数据中构造了一个从X的中间代数Jacobian的乘积到Y的中间代数Jacobian的乘积的阿贝尔簇（直到Isabel）的态射t。当T在D^B（X）中的正交补有平凡雅可比矩阵时（例如当它由例外对象生成时），态射t是分裂单射的。当T在D^B（Y）中的正交补也成立时，t成为同构。此外，在的情况下，X和Y有一个单一的中间代数雅可比进行主极化，我们证明了态射t保持这个额外的结构。作为应用，我们得到了范畴Torelli定理，Kuznetsov的两个定理（一个关于Fourier-Mukai函子，另一个关于Fano三重函子）之间的不相容性，Clemens-Griffiths关于三重函子爆破的经典定理的一个新证明，以及关于二次纤维化和二次交的几个新结果.
Let X and Y be complex smooth projective varieties, and D^b(X) and D^b(Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in D^b(X) as in D^b(Y). Making use of the recent theory of Jacobians of noncommutative motives, we construct out of this categorical data a morphism t of abelian varieties (up to isogeny) from the product of the intermediate algebraic Jacobians of X to the product of the intermediate algebraic Jacobians of Y. When the orthogonal complement of T in D^b(X) has a trivial Jacobian (e.g. when it is generated by exceptional objects), the morphism t is split injective. When this also holds for the orthogonal complement of T in D^b(Y), t becomes an isomorphism. Furthermore, in the case where X and Y have a single intermediate algebraic Jacobian carrying a principal polarization, we prove that the morphism t preserves this extra structure. As an application, we obtain categorical Torelli theorems, an incompatibility between two conjectures of Kuznetsov (one concerning Fourier-Mukai functors and another one concerning Fano threefolds), a new proof of a classical theorem of Clemens-Griffiths on blow-ups of threefolds, and several new results on quadric fibrations and intersection of quadrics.
代数环和纤维化
影响因子：
0.9
作者：
Vial C
通讯作者：
Vial C