2015/12/31 by Aleksandra Borówka, Aleksandra W. Borowka, David M. J. Calderbank
Mathematics · #Complex manifold #Complex projective space #Connection (principal bundle) #Cotangent bundle #Curvature #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Holomorphic function #Hyperkähler manifold #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Normal bundle #Projective space #Projective test #Pure mathematics #Quaternionic projective space #Scalar curvature #Sectional curvature #Submanifold #Trigonometric functions #Twistor space #Twistor theory #Vector bundle #math.DG #msc:32L25 #msc:53A20 #msc:53B10 #msc:53C26 #msc:53C28
paper · pdf · doi:10.1090/tran/7719
published as Trans. Amer. Math. Soc. 372 (2019), 4729-4760 · 28 pages, (v2) added material on Swann bundles, quaternionic Kaehler metrics and the Haydys-Hitchin correspondence, (v3) refereed version, restructured content, to appear in TAMS
openalex publication_date 2018/10/15 · arxiv created 2019/01/04 · openalex created_date 2019/07/30 · arxiv updated 2020/12/17 · openalex updated_date 2026/08/05
Starting from a complex manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a real-analytic c-projective structure whose curvature has type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(1,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and a complex line bundle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L right-arrow upper S"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr L→ S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a connection whose curvature has type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(1,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we construct the twistor space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a quaternionic manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a quaternionic circle action which contains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkähler metrics on cotangent bundles, obtained independently by Feix and Kaledin. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Riemann surface, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a self-dual conformal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -manifold and the quotient of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by the circle action is an Einstein–Weyl manifold with an asymptotically hyperbolic end, and our construction coincides with the construction presented by Borówka. The extension also applies to quaternionic Kähler manifolds with circle actions, as studied by Haydys and Hitchin.