LVM manifolds and lck metrics

LVM manifolds and lck metrics | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article LVM manifolds and lck metrics Bastien Faucard This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4108416/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract In this paper, we compare two type of complex non-Kähler manifolds : LVM and lck manifolds. First, lck manifolds (for locally conformally Kähler manifolds) admit a metric which is locally conforme to a Kähler metric. On the other side, LVM manifolds (for S.Lopez de Medrano, A.Verjovsky and L.Meersseman) are quotient of an open subset of C^n by an action of C*xC^m. LVM and lck manifolds have a fondamental common point : Hopf manifolds which are a specific case of LVM manifolds and which admit also lck metric. So the question of this paper is : Are LVM manifolds lck ? We provide some answers to this question. The results obtained are as follows. In the set of all LVM manifolds, there is a dense subset of LVM which are not lck. An if we consider lck manifolds with potential (whose metric derives from a potential), the diagonal Hopf manifolds are the only LVM manifolds which admit an lck metric with potential. Based on this observation, we show that there exists an lck covering with potential (non-compact) of a certain subclass of LVM manifolds. Finally, we present some examples. Mathematics Subject Classification (2000). Primary 32J27, 32L05, 32M99, 32Q15, 32Q60, 32T15, 32V99; Secondary 53A30, 53B35, 53C55, 53C56. LVM lck Kähler lck with potential p-lck structure Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 03 Jun, 2024 Reviews received at journal 06 May, 2024 Reviewers agreed at journal 21 Apr, 2024 Reviewers agreed at journal 10 Apr, 2024 Reviewers invited by journal 03 Apr, 2024 Editor assigned by journal 02 Apr, 2024 Submission checks completed at journal 15 Mar, 2024 First submitted to journal 15 Mar, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4108416","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":280196630,"identity":"4452c4ee-142e-4b3d-9f6b-023331677f74","order_by":0,"name":"Bastien Faucard","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIie3PsUrEMBjA8a8c2CXQtdKjvkJCoEe54V4lxSFLDgeXGxwy6VK4tYIPobtDIaBLwbWDQ7u0Swe73UERc2cFkcvd6pA/pLQkP/IVwGb7h+Hdg+nlyv33JBw3woPHR+LIHUH5N6HjBjWAkcAvkshTZOa+tlW1egfkFuRj+zzn6+CurjaAr0wkTgWRrGgAIUHP04Yv7x8KSlLAsTQNlgtHJrcKFiAiPZ1aPpbizEcwYOO/vLW1TD4VIK+LnCFXfFHydjsANpOSEZlITXwRTfQtDPtMvxwhcdaRjL0ohPzmOpjmnGSloMEUm8nM41W/uVEh8i6f+i6fX3gZr/tuZSY/ob8DnwI2m81mO9YXUqdW4lI5rnEAAAAASUVORK5CYII=","orcid":"","institution":"École Nationale Supérieure de l'Électronique et de ses Applications","correspondingAuthor":true,"prefix":"","firstName":"Bastien","middleName":"","lastName":"Faucard","suffix":""}],"badges":[],"createdAt":"2024-03-15 13:57:22","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4108416/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4108416/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":52942873,"identity":"8e766741-d28e-490c-97ac-e51991cf5e72","added_by":"auto","created_at":"2024-03-19 01:50:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":291287,"visible":true,"origin":"","legend":"","description":"","filename":"Manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4108416/v1_covered_b8454ff9-21f2-43df-ad68-2e180bdd5a57.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"LVM manifolds and lck metrics","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"mediterranean-journal-of-mathematics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mjom","sideBox":"Learn more about [Mediterranean Journal of Mathematics](http://link.springer.com/journal/9)","snPcode":"9","submissionUrl":"https://submission.nature.com/new-submission/9/3","title":"Mediterranean Journal of Mathematics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"LVM, lck, Kähler, lck with potential, p-lck structure","lastPublishedDoi":"10.21203/rs.3.rs-4108416/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4108416/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we compare two type of complex non-Kähler manifolds : LVM and lck manifolds. First, lck manifolds (for locally conformally Kähler manifolds) admit a metric which is locally conforme to a Kähler metric. On the other side, LVM manifolds (for S.Lopez de Medrano, A.Verjovsky and L.Meersseman) are quotient of an open subset of C^n by an action of C*xC^m. LVM and lck manifolds have a fondamental common point : Hopf manifolds which are a specific case of LVM manifolds and which admit also lck metric. So the question of this paper is :\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAre LVM manifolds lck ?\u003c/p\u003e\n\u003cp\u003eWe provide some answers to this question. The results obtained are as follows. In the set of all LVM manifolds, there is a dense subset of LVM which are not lck. An if we consider lck manifolds with potential (whose metric derives from a potential), the diagonal Hopf manifolds are the only LVM manifolds which admit an lck metric with potential. 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