Algebraic cycles and very special cubic fourfolds

Informed by the Bloch–Beilinson conjectures, Voisin has made a conjecture about 0-cycles on self-products of Calabi–Yau varieties. In this note, we consider variant versions of Voisin’s conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.     

红外导数光谱-统计学法区别鉴定麦冬及其伪品的研究

为了建立中药材麦冬及其伪品的直接鉴定的理论和方法 ,采用OMNI采样器直接测定法测定了样品的红外光谱 ,并经过二阶导数光谱转换后进行峰位一致率检验。结果发现麦冬表皮以内部分和外表皮的二阶导数红外光谱峰位基本无差别 ,而且不同产地的道地药材具有较好的相关性。研究中发现浙麦冬与川麦冬表皮以内部分的二阶导数FTIR峰值 (χ2c =891 882 ,P =0 0 0 0 )的差异性极显著 ;与土麦冬 (χ2c=4 2 9 2 17,P =0 0 0 0 )、与湖北麦冬 (χ2c=6 92 70 6 ,P =0 0 0 0 )、与短葶山麦冬 (χ2c=981 970 ,P =0 0 0 0 )、与阔叶山麦冬 (χ2c=716 2 17,P =0 0 0 0 )、与竹叶麦冬 (χ2c=70 1 393,P =0 0 0 0 )的差异性都极显著。实验结果表明二阶导数红外光谱结合统计学法可用于中药材麦冬的真伪鉴别 ,本法简便、快速和准确     

Factorization properties of birational mappings

We analyse birational mappings generated by transformations on q × q matrices which correspond respectively to two kinds of transformations: the matrix inversion and a permutation of the entries of the q × q matrix. Remarkable factorization properties emerge for quite general involutive permutations.

It is shown that factorization properties do exist, even for birational transformations associated with noninvolutive permutations of entries of q × q matrices, and even for more general transformation which are rational transformations but no longer birational. The existence of factorization relations independent of q, the size of the matrices, is underlined.

The relations between the polynomial growth of the complexity of the iterations, the existence of recursions in a single variable and the integrability of the mappings, are sketched for the permutations yielding these properties.

All these results show that permutations of the entries of the matrix yielding factorization properties are not so rare. In contrast, the occurrence of recursions in a single variable, or of the polynomial growth of the complexity are, of course, less frequent but not completely exceptional.     

Higher dimensional Shimura varieties in the Prym loci of ramified double covers

In this paper, we construct Shimura subvarieties of dimension bigger than one of the moduli space ${\mathsf{A}}_p^{\delta }$ of δ-polarized abelian varieties of dimension p, which are generically contained in the Prym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely, to use families of Galois covers of ${\mathbb{P}}^1$. The case of abelian covers is treated in detail, since in this case, it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield a Shimura subvariety of ${\mathsf{A}}_p^{\delta }$.     

Stability conditions on Kuznetsov components of Gushel–Mukai threefolds and Serre functor

We show that the stability conditions on the Kuznetsov component of a Gushel–Mukai threefold, constructed by Bayer, Lahoz, Macrì and Stellari, are preserved by the Serre functor, up to the action of the universal cover of ${\text{GL}}_2^{+}(\mathbb{R})$. As application, we construct stability conditions on the Kuznetsov component of special Gushel–Mukai fourfolds.