Characterizations of complex projective spaces and hyperquadrics

Kobayashi, Shôshichi; Ochiai, Tadashi

doi:10.1215/kjm/1250523432

Cited by 273 publications

(202 citation statements)

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“…In case (iii) Kobayashi and Ochiai [13] have a similar result for quadrics which allows us to conclude that X ss Q3. Remark.…”

Section: Corollarysupporting

confidence: 58%

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Compactifications of 𝐶ⁿ

Brenton¹,

Morrow²

1978

Trans. Amer. Math. Soc.

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Abstract.Let X be a compactification of C. We assume that AT is a compact complex manifold and that A = X -C is a proper subvariety of X. If we suppose that A is a Kahler manifold, then we prove that X is projective algebraic, H*(A, Z) » H*(V"-\ Z), and H*(X, Z) = tf*(P"', Z). Various additional conditions are shown to imply that X = P". It is known that no additional conditions are needed to imply X = P" in the cases n = 1, 2. In this paper we prove that if n = 3, X = P3. 0. Introduction. In this paper we continue our study of compactifications of [19]. This introduction will give some definitions and state two of our most important results. More detailed statements will come in later sections. We will also state here some theorems which will be useful in our proofs.0.1. Definitions. Let A' be a (connected) compact complex manifold and let A be a (closed) subvariety of X. We say that X is a compactification of C if X -A is biholomorphic to C-we will see later that A is then necessarily a connected subvariety of pure dimension n -1. By a complex homology n-cell we mean a noncompact complex manifold M of complex dimension n with HJC(M, Z) = 0, for/ « 0, . . . , 2« -1, where HJC(M, Z) is integral cohomology with compact support.First, let us make some remarks about dimensions < 2. For n = 1, we notice that the only complex homology 1-cell is the disk D = [z: \z\ < 1}, or the line C1. However, D is not compactifiable-if X -A = D, then any nonconstant, bounded, holomorphic function on D would extend to a nonconstant (bounded) holomorphic function on the compact manifold X. Thus the only compactifiable homology 1-cell is C1. Now, we will see later that A = X -C1 is a point, where X is a compactification of C1 (A is a connected 0-dimensional set-see Theorem 1.1). We define a continuous map a: ^-»P1 = C1 u {oo} by a(z) = z if z e C1, a(A) = oo. By Rado's theorem a is holomorphic. Since a is one-one, a ~ ' is holomorphic and X is biholomorphic to P1. For n = 2 we have the results of [2], [19]. In particular, tí A = X -C2 is a manifold, then A = P1 and X = P2. Already in the case n = 2, compacti-

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“…In case (iii) Kobayashi and Ochiai [13] have a similar result for quadrics which allows us to conclude that X ss Q3. Remark.…”

Section: Corollarysupporting

confidence: 58%

“…Such results are discussed in [11] and [18]. However, the paper of Kobayashi and Ochiai [13] gives the most convenient hypotheses. We will now state those results of theirs that we shall use.…”

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confidence: 95%

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Compactifications of 𝐶ⁿ

Brenton¹,

Morrow²

1978

Trans. Amer. Math. Soc.

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“…Now (34.2) is immediate from (iv) in §22. There are also further necessary conditions, stemming from a theorem of Kobayashi and Ochiai [12]. See [9] for details.…”

Section: One Of Its Consequences Ismentioning

confidence: 99%

Special Kähler-Ricci potentials on compact Kähler manifolds

Derdziński¹,

Maschler²

2006

Journal Fur Die Reine Und Angewandte Mathematik (Crelles Journal)

100

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Abstract. A special Kähler-Ricci potential on a Kähler manifold is any nonconstant C ∞ function τ ι such that J(∇τ ι) is a Killing vector field and, at every point with dτ ι = 0, all nonzero tangent vectors orthogonal to ∇τ ι and J(∇τ ι) are eigenvectors of both ∇dτ ι and the Ricci tensor. For instance, this is always the case if τ ι is a nonconstant C ∞ function on a Kähler manifold (M, g) of complex dimension m > 2 and the metricg = g/τ ι 2 , defined wherever τ ι = 0, is Einstein. (When such τ ι exists, (M, g) may be called almost-everywhere conformally Einstein.) We provide a complete classification of compact Kähler manifolds with special Kähler-Ricci potentials and use it to prove a structure theorem for compact Kähler manifolds of any complex dimension m > 2 which are almost-everywhere conformally Einstein. §0. Introduction This paper, although self-contained, can also be viewed as the second in a series of three papers that starts with [8] and ends with [9].We call τ ι a special Kähler-Ricci potential on a Kähler manifold (M, g) if τ ι is a nonconstant Killing potential on (M, g) and, at every point (0.1) with dτ ι = 0, all nonzero tangent vectors orthogonal to v = ∇τ ι and to u = Jv are eigenvectors of both ∇dτ ι and the Ricci tensor r.(Cf.[8], §7; for more on Killing potentials, see §4 below.) The word 'potential' reflects the fact that (0.1) is closely related, although not equivalent, to the requirement that ∇dτ ι + χ r = σg for some C ∞ functions χ, σ (see [8], beginning of §7). This requirement is reminiscent of Kähler-Ricci solitons (some of which, in fact, do satisfy (0.1), cf.[10] and Remark 10.1 below); while, in complex dimensions m > 2, it implies that τ ι arises from a Hamiltonian 2-form on the underlying Kähler manifold ([2], §1.4). See also [5]. What further sparked our interest in (0.1) was its being, in cases such as (0.4) below, a consequence of the following assumption:(M, g) is a Kähler manifold of complex dimension m and τ ι is (0.2) a nonconstant C ∞ function on M such that the conformally related metricg = g/τ ι 2 , defined wherever τ ι = 0, is Einstein.When m > 2, (0.2) implies the seemingly stronger condition (0.3) M, g, m, τ ι satisfy (0.2) and dτ ι ∧ d∆τ ι = 0 everywhere in M 1991 Mathematics Subject Classification. Primary 53C55, 53C21; Secondary 53C25.

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“…Now either M is a linear CP" i.e. M is holomorphically isometric to CP"(\) or c, = nü which implies M is biholomorphic to Q" as a hypersurface of a linear subspace CPn+1(l) of CP"+p(l) (S. Kobayashi and T. Ochiai [3]). And since M is Einsteinian, M has to be also isometric to Q" (B. Smyth [5]).…”

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confidence: 99%