M. S. Verbitsky, “Vanishing Theorems for Locally Conformal Hyperkähler Manifolds”, Algebraic geometry: Methods, relations, and applications, Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences, Trudy Mat. Inst. Steklova, 246, Nauka, MAIK «Nauka/Inteperiodika», M., 2004, 64–91; Proc. Steklov Inst. Math., 246 (2004), 54

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2004, Volume 246, Pages 64–91 (Mi tm146)

This article is cited in 30 scientific papers (total in 30 papers)

Vanishing Theorems for Locally Conformal Hyperkähler Manifolds

M. S. Verbitsky

University of Glasgow

Full-text PDF (355 kB) Citations (30)

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Abstract: Let

$M$ be a compact locally conformal hyperkähler manifold. We prove a version of the Kodaira–Nakano vanishing theorem for

$M$ . This is used to show that

$M$ admits no holomorphic differential forms and the cohomology of the structure sheaf

$H^i(\mathcal O_M)$ vanishes for

$i>1$ . We also prove that the first Betti number of

$M$ is

$1$ . This leads to a structure theorem for locally conformal hyperkähler manifolds that describes them in terms of

$3$ -Sasakian geometry. Similar results are proven for compact Einstein–Weyl locally conformal Kähler manifolds.

Received in February 2004

Bibliographic databases:

UDC: 514.764.226+515.179.22+515.165.4

Language: Russian

Citation: M. S. Verbitsky, “Vanishing Theorems for Locally Conformal Hyperkähler Manifolds”, Algebraic geometry: Methods, relations, and applications, Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences, Trudy Mat. Inst. Steklova, 246, Nauka, MAIK «Nauka/Inteperiodika», M., 2004, 64–91; Proc. Steklov Inst. Math., 246 (2004), 54–78

Citation in format AMSBIB

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\by M.~S.~Verbitsky

\paper Vanishing Theorems for Locally Conformal Hyperk\"ahler Manifolds

\inbook Algebraic geometry: Methods, relations, and applications

\bookinfo Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences

\serial Trudy Mat. Inst. Steklova

\yr 2004

\vol 246

\pages 64--91

\publ Nauka, MAIK «Nauka/Inteperiodika»

\publaddr M.

\mathnet{http://mi.mathnet.ru/tm146}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2101284}

\zmath{https://zbmath.org/?q=an:1101.53027}

\transl

\jour Proc. Steklov Inst. Math.

\yr 2004

\vol 246

\pages 54--78

Linking options:

https://www.mathnet.ru/eng/tm146

https://www.mathnet.ru/eng/tm/v246/p64

This publication is cited in the following 30 articles:

Ornea L., Verbitsky M., “Twisted Dolbeault Cohomology of Nilpotent Lie Algebras”, Transform. Groups, 27:1 (2022), 225–238
Ornea L., Verbitsky M., “Supersymmetry and Hodge Theory on Sasakian and Vaisman Manifolds”, Manuscr. Math., 2022
Huang T., “On Harmonic Symmetries For Locally Conformally Kahler Manifolds”, Ann. Mat. Pura Appl., 2022
Ornea L., Verbitsky M., “Closed Orbits of Reeb Fields on Sasakian Manifolds and Elliptic Curves on Vaisman Manifolds”, Math. Z., 299:3-4 (2021), 2287–2296
Andrada A., Origlia M., “Vaisman Solvmanifolds and Relations With Other Geometric Structures”, Asian J. Math., 24:1 (2020), 117–146
Ornea L., Verbitsky M., “Hopf Surfaces in Locally Conformally Kahler Manifolds With Potential”, Geod. Dedic., 207:1 (2020), 219–226
Istrati N., “Existence Criteria For Special Locally Conformally Kahler Metrics”, Ann. Mat. Pura Appl., 198:2 (2019), 335–353
Angella D., Zedda M., “Isometric Immersions of Locally Conformally Kahler Manifolds”, Ann. Glob. Anal. Geom., 56:1 (2019), 37–55
Ornea L., Verbitsky M., “Positivity of Lck Potential”, J. Geom. Anal., 29:2 (2019), 1479–1489
Andrada A., Origlia M., “Locally Conformally Kahler Solvmanifolds: a Survey”, Complex Manifolds, 6:1 (2019), 65–87
Ornea L., Otiman A., “A Characterization of Compact Locally Conformally Hyperkahler Manifolds”, Ann. Mat. Pura Appl., 198:5 (2019), 1541–1549
Ornea L., Verbitsky M., Vuletescu V., “Weighted Bott-Chern and Dolbeault Cohomology For Lck-Manifolds With Potential”, J. Math. Soc. Jpn., 70:1 (2018), 409–422
Bazzoni G., “Locally Conformally Symplectic and Kahler Geometry”, EMS Surv. Math. Sci., 5:1-2 (2018), 129–154
Ornea L., Verbitsky M., “Locally Conformally Kahler Metrics Obtained From Pseudoconvex Shells”, Proc. Amer. Math. Soc., 144:1 (2016), 325–335
Panov T., Ustinovskiy Yu., Verbitsky M., “Complex geometry of moment-angle manifolds”, Math. Z., 284:1-2 (2016), 309–333
Ornea L., Verbitsky M., “LCK rank of locally conformally K?hler manifolds with potential”, J. Geom. Phys., 107 (2016), 92–98
Ornea L., Verbitsky M., “Automorphisms of Locally Conformally Kahler Manifolds”, Int Math Res Not, 2012, no. 4, 894–903
Ornea L., Pilca M., “Remarks on the Product of Harmonic Forms”, Pacific J Math, 250:2 (2011), 353–363
Ornea L., Verbitsky M., “Oeljeklaus-Toma Manifolds Admitting No Complex Subvarieties”, Math Res Lett, 18:4 (2011), 747–754
Verbitsky M., “Hodge theory on nearly Kahler manifolds”, Geometry & Topology, 15:4 (2011), 2111–2133

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

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