Counting Fourier-Mukai partners and applications
Counting Fourier-Mukai partners and applications
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计算 Fourier-Mukai 合作伙伴和应用
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
S. Yau
中科院分区：
文献类型：
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作者：
S. Hosono;B. Lian;K. Oguiso;S. Yau
我们引入了偶非退化格T的本原嵌入到偶么模不定格的G-等价类的概念，其中G是正交群O(T)的指定子群，并给出了嵌入的G-等价类个数的一般公式。作为一般公式的特例，我们导出了K3曲面的傅里叶-Mukai对偶数的计数公式。作为一个算术应用，我们在高斯之后用K3曲面的Fm对偶重新表述了两个关于类数的问题。作为更几何的应用，我们构造了一对一维光滑投影族K3曲面，其中一系列同构的K3曲面可以在极限上分裂出两个非同构的K3曲面。正是在20年前，Mukai发现了具有等价的相干层有界派生范畴的流形的基本重要性。这种流形现在被称为Fourier-Mukai(FM)伙伴，Mukai从稳定层的模和Hodge猜想([Mu 1，2，3])的观点特别研究了交换簇和K3曲面的性质。最近，在受到同调镜像对称性[KO][Fu]和几何镜像对称性[Syz]启发的论文中，凝聚层的有界导范畴和Calabi-Yau流形的FM对偶也发挥了越来越重要的作用。例如，当我们研究镜面K3及其FM配对时，我们观察到了一些有趣和意想不到的现象。见[HLOY，定理1.17，命题5.8]。例如，对于ρ(X)=1，K3曲面族的单行群(参见[DO]，[TO])被发现与X[HLOY]的FM伙伴的数量密切相关。这一观察结果的灵感来自于[OG1]首先推导出的一个显式公式。给出并证明了任意射影K3曲面X的FM对的集合F，M(X)的基数的一个显式计数公式(计数公式3.4)。该公式是用Néron-Severi格子N S(X)和超越格子T(X)的Hodge结构(T(X)，CωX)写成的。我们强调，该公式精确地将几何量|F，M(X)|与纯算术量联系起来。作为计数公式的一个简单但显著的推论，我们看到具有ρ(X)≥3的“一般”K3曲面X没有…
We introduce a notion of G-equvalence class of primitive embed-dings of an even non-degenerate lattice T into an even unimodular indefinite lattice, where G is a prescribed subgroup of the orthogonal group O(T), and give a general formula for the number of the G-equivalence classes of embed-dings. We then derive a Counting Formula for the number of Fourier-Mukai partners of a K3 surface as a special case of the general formula. For an arithmetical application, we reformulate two questions on class numbers, after Gauss, in terms of FM partners of K3 surfaces. For a more geometrical application, we construct a pair of one-dimensional smooth projective families of K3 surfaces in which a sequence of isomorphic K3 surfaces can split off two non-isomorphic K3 surfaces in the limit. It was 20 years ago that Mukai discovered the fundamental importance of man-ifolds with equivalent bounded derived categories of coherent sheaves. Such man-ifolds are now known as Fourier-Mukai (FM) partners, whose properties Mukai studied especially in the cases of abelian varieties and K3 surfaces from the viewpoint of moduli of stable sheaves and the Hodge conjecture ([Mu 1,2,3]). More recently, among papers inspired by homological mirror symmetry [Ko][Fu] and geometric mirror symmetry [SYZ], bounded derived categories of coherent sheaves and FM partners of Calabi-Yau manifolds have also played an increasingly important role. For instance, a number of interesting and unexpected phenomena have been observed when we study mirror K3 surfaces and their FM partners. See [HLOY, Theorem 1.17, Proposition 5.8]. For ρ(X) = 1, for example, the monodromy group of a mirror family of K3 surfaces (cf. [Do], [To]) has been found to be closely related to the number of FM partners of X [HLOY]. This observation was inspired by an explicit formula first derived in [Og1]. The aim of this note is to formulate and prove an explicit counting formula for the cardinality of the set F M (X) of FM partners for an arbitrary projective K3 surface X (Counting Formula 3.4). The formula is written in terms of the Néron-Severi lattice N S(X) and the Hodge structure (T (X), Cω X) of the transcendental lattice T (X). We emphasize that the formula precisely connects the geometrical quantity |F M (X)| with a purely arithmetical quantity. As an easy but remarkable consequence of the Counting Formula, we see that a " general " K3 surface X with ρ(X) ≥ 3 has no …
K3 曲面的派生类别和单性变换的自动等价
DOI：
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发表时间：
2004
期刊：
J.Alg.Geom. 13
影响因子：
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作者：
S.Hosono;B.Lian;K.Oguiso;S.T.Yau
通讯作者：
S.T.Yau