{"title":"贝林森-霍奇猜想的一个相对版本","authors":"Rob de Jeu, James D. Lewis, D. Patel","doi":"10.1017/CBO9781316387887.010","DOIUrl":null,"url":null,"abstract":"Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} : H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge conjecture in the case m=0. Now consider a smooth and proper map \\rho : X -> S of smooth quasi-projective varieties over k. We formulate a version of this conjecture for the generic fibre, expecting the corresponding cycle class map to be surjective. We provide some evidence in support of this in the case where X is a product, the map is the projection to one factor, and m=1.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"11 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2014-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"A relative version of the Beilinson-Hodge conjecture\",\"authors\":\"Rob de Jeu, James D. Lewis, D. Patel\",\"doi\":\"10.1017/CBO9781316387887.010\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} : H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge conjecture in the case m=0. Now consider a smooth and proper map \\\\rho : X -> S of smooth quasi-projective varieties over k. We formulate a version of this conjecture for the generic fibre, expecting the corresponding cycle class map to be surjective. We provide some evidence in support of this in the case where X is a product, the map is the projection to one factor, and m=1.\",\"PeriodicalId\":309711,\"journal\":{\"name\":\"arXiv: K-Theory and Homology\",\"volume\":\"11 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2014-04-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: K-Theory and Homology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/CBO9781316387887.010\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: K-Theory and Homology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/CBO9781316387887.010","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
摘要
设k是复数的代数闭子域,X是定义在k上的一个变种。贝林森-霍奇猜想的一个版本似乎可以通过仔细的检验,即Betti循环类映射cl_{r,m}: H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r)))是满射,在m=0的情况下等价于霍奇猜想。现在考虑k上光滑拟射影变体的光滑和适当映射\rho: X -> S。我们为一般纤维构造了这个猜想的一个版本,期望相应的循环类映射是满射的。在X是乘积的情况下,我们提供了一些证据来支持这一点,映射是对一个因子的投影,m=1。
A relative version of the Beilinson-Hodge conjecture
Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} : H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge conjecture in the case m=0. Now consider a smooth and proper map \rho : X -> S of smooth quasi-projective varieties over k. We formulate a version of this conjecture for the generic fibre, expecting the corresponding cycle class map to be surjective. We provide some evidence in support of this in the case where X is a product, the map is the projection to one factor, and m=1.