2014/01/31 by Valentino Tosatti
Mathematics · #Algebraic Geometry and Number Theory #Bundle #Calabi–Yau manifold #Canonical bundle #Characteristic class #Chern class #Class (philosophy) #Cohomology #Computer science #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Hyperkähler manifold #Kähler manifold #Manifold (fluid mechanics) #Mathematics #Pure mathematics #Ricci-flat manifold #Scalar curvature #Torsion (gastropod) #math.AG #math.CV #math.DG #msc:32G05 #msc:32Q20 #msc:32Q25 #msc:32W20 #msc:53C55
paper · pdf · doi:10.1090/conm/644/12770
published as Contemp. Math. 644 (2015), 261-277 · 18 pages; corrected small typos; final version to appear in Contemp. Math
openalex publication_date 2015/01/01 · arxiv created 2015/11/25 · arxiv updated 2015/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki’s class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kähler Calabi-Yau manifolds, and discuss the problem of defining and constructing canonical metrics on them.