{"title":"通过超holomorphic束的超凯勒变体的D等价猜想","authors":"Davesh Maulik, Junliang Shen, Qizheng Yin","doi":"arxiv-2408.14775","DOIUrl":null,"url":null,"abstract":"We show that birational hyper-K\\\"ahler varieties of $K3^{[n]}$-type are\ntwisted derived equivalent with respect to some Brauer class. Furthermore, if a\n$K3^{[n]}$-type variety X admits a divisor class of divisibility 1 whose norm\nsatisfies a congruence condition modulo 4, we show that any hyper-K\\\"ahler\nvariety birational to X is derived equivalent to X. This verifies new cases of\nthe D-equivalence conjecture in higher dimension. The Fourier-Mukai kernels of\nour (twisted) derived equivalences are constructed from Markman's projectively\nhyperholomorphic bundles.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles\",\"authors\":\"Davesh Maulik, Junliang Shen, Qizheng Yin\",\"doi\":\"arxiv-2408.14775\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that birational hyper-K\\\\\\\"ahler varieties of $K3^{[n]}$-type are\\ntwisted derived equivalent with respect to some Brauer class. Furthermore, if a\\n$K3^{[n]}$-type variety X admits a divisor class of divisibility 1 whose norm\\nsatisfies a congruence condition modulo 4, we show that any hyper-K\\\\\\\"ahler\\nvariety birational to X is derived equivalent to X. This verifies new cases of\\nthe D-equivalence conjecture in higher dimension. The Fourier-Mukai kernels of\\nour (twisted) derived equivalences are constructed from Markman's projectively\\nhyperholomorphic bundles.\",\"PeriodicalId\":501063,\"journal\":{\"name\":\"arXiv - MATH - Algebraic Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Algebraic Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.14775\",\"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 - MATH - Algebraic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.14775","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们证明，$K3^{[n]}$型的双向超K/"ahler "综关于某个布劳尔类是扭曲派生等价的。此外，如果$K3^{[n]}$型 variety X 承认一个可分性为 1 的因子类，其规范满足 modulo 4 的全等条件，我们证明了任何与 X 双向的超（hyper-K\"ahl）ervariety 都与 X 派生等价。我们的（扭曲的）派生等价的傅里叶-穆凯核是由马克曼的投影超holomorphic束构造的。
The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles
We show that birational hyper-K\"ahler varieties of $K3^{[n]}$-type are
twisted derived equivalent with respect to some Brauer class. Furthermore, if a
$K3^{[n]}$-type variety X admits a divisor class of divisibility 1 whose norm
satisfies a congruence condition modulo 4, we show that any hyper-K\"ahler
variety birational to X is derived equivalent to X. This verifies new cases of
the D-equivalence conjecture in higher dimension. The Fourier-Mukai kernels of
our (twisted) derived equivalences are constructed from Markman's projectively
hyperholomorphic bundles.
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