Holomorphic $$\text {GL}_2({\mathbb C})$$ GL 2 ( C ) -geometry on compact complex manifolds
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Indranil Biswas · Sorin Dumitrescu
Holomorphic GL2 (C)-geometry on compact complex manifolds Received: 8 April 2020 / Accepted: 11 August 2020 Abstract. We study holomorphic GL2 (C) and SL2 (C) geometries on compact complex manifolds.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2. Holomorphic GL2 (C) and SL2 (C) geometries . . . . . . . 3. GL2 (C)-structures on Kähler and Fujiki class C manifolds . 4. GL2 (C)-structures on Kähler–Einstein and Fano manifolds 5. Related open questions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction A holomorphic GL2 (C) geometric structure on a complex manifold X of complex dimension n is a holomorphic point-wise identification between the holomorphic tangent space T X and homogeneous polynomials in two variables of degree (n−1). More precisely, a GL2 (C) geometric structure on X is a pair (E, ϕ), where E is a rank two holomorphic vector bundle on X and ϕ is a holomorphic vector bundle isomorphism of T X with the (n − 1)-fold symmetric product S n−1 (E)(see Definition 2.1). If E has trivial determinant (i.e., the holomorphic line bundle 2 E is trivial), then (E, ϕ) is called a holomorphic SL2 (C) geometric structure. The above definitions are the holomorphic analogues of the concepts of GL2 (R) and SL2 (R) geometries in the real smooth category (for the study of those geometries in the real smooth category we refer the reader to [14,15,37] and references therein). This article deals with the classification of compact complex manifolds admitting holomorphic GL2 (C) and SL2 (C) geometries. I. Biswas: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] S. Dumitrescu (B): Université Côte d’Azur, CNRS, LJAD, Nice, France e-mail: [email protected] Mathematics Subject Classification: 53C07 · 53C10 · 32Q57
https://doi.org/10.1007/s00229-020-01252-9
I. Biswas, S. Dumitrescu
Holomorphic GL2 (C) and SL2 (C) geometries on X are examples of holomorphic G-structures (see [31] and Sect. 2). They correspond to the reduction of the structural group of the GLn (C)-frame bundle of X to GL2 (C) and SL2 (C) respectively. When the dimension n of X is odd, then a GL2 (C) geometry produces a holomorphic conformal structure on X (see, for example, [14], Proposition 3.2 or Sect. 2 here). Recall that a holomorphic conformal structure is defined by a holomorphic line bundle L over X and a holomorphic section of S 2 (T ∗ X ) ⊗ L, which is a Lvalued fiberwise nondegenerate holomorphic quadratic form on T X . Moreover, if n = 3, then a GL2 (C)-geometry on X is exactly a holomorphic conformal structure. The standard flat example is the smooth quadric Q 3 in CP4 defined by the equation Z 02 + Z 12 + Z 22 + Z 32 + Z 42 =
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