Therefore, the existence of scalar-flat neutral K¨ahler metrics is closely related to self-dual neutral metrics on oriented four-manifolds. The aim of this thesis is to study the existence problem of self-dual neutral K¨ahler metrics on complex surfaces. Since a neutral hyperk¨ahler metric is Ricci-flat and self-dual, non-flat neutral hyperk¨ahler metrics give us examples of non-conformally flat, self-dual neutral metrics.
Geometry of four-manifolds with neutral metric
Most typical examples of four-manifolds with self-dual neutral metric are the spaces of constant curvature R42, S24 and H24. The indefinite complex projective space CP21 and the indefinite complex hyperbolic space CH21 are also examples of four-manifolds with (anti-)self- dual (non-conformally-flat) metric (see Chapter 4). A neutral metric g on an oriented four-manifold M is self-dual (resp. anti-self-dual ) if and only if κ(σ) = 0 for any anti-self-dual (resp. self-dual ) totally null plane σ at each point of M.
Hermitian geometry of neutral metrics
In particular, on a neutral K¨ahler surface (M, g, I), its scalar curvature s vanishes everywhere on M if and only if γ ∧ ΩI ≡ 0. However, the author does not know of any self-dual neutral K¨ahler metrics on surfaces in the cases (2) and (5). In case (4), all surfaces admit a Ricci flat neutral K¨ahler (hence self-dual) metric (see Petean [82]).
K¨ ahler surfaces with time-like S 1 -symmetry
We then study a self-dual neutral K¨ahler surface with a time-like isometric S1 action in a general setting. Moreover, every self-dual neutral K¨ahler surface with a time-like isometric S1 action is obtained, at least locally, by this construction. Let (M, g) be an oriented pseudo-Riemannian four-manifold with neutral metric g admitting a time-like isometric S1 action.
Construction of self-dual K¨ ahler metrics
Then gV,θ is a neutral self-dual metric on M, and is, at least locally, conformal to a neutral scalar-flat K¨ahler metric with respect to a suitable complex structure in M. We then prove that there exists a quasi-complex structure IV onM= S2 ×S2 such that (¯gV, IV) is a neutral K¨ahler structure, if V satisfies the same assumptions as those in Proposition 2.19. For a nonconstant solution V of (2.37) satisfying the conditions in Proposition 2.19, we obtain a nonconformal flat, self-dual neutral K¨ahler metric on S2×S2.
We then construct a family of self-dual neutral K¨ahler metrics on S2×S2 based on some explicit solutions (V, θ) of (2.37). It should be noted that gG0 conforms to a restriction of the Fubini-Study type metric on the indefinite complex projective space CP21 (see Chapter 4). We thus obtain a self-dual neutral metric gV on the total space M of a trivial S1 bundle over S13.
Indeed, the additional S1-symmetry is given by the rotation about the intersection of the subspace Π and the neck sphere S2 = {ρ = 0}. The corresponding Einstein-Weyl structure on the quotient space is actually induced from the de Sitter space structure S13 and then (2.25) is equivalent to (2.37) under the substitution (2.49). For the self-dual neutral K¨ahler structure (g, I) given by (2.13), we can find the Einstein-Weyl structure on U based on Proposition 2.14.
We now prove the following result, which characterizes the self-dual neutral K¨ahler metrics constructed on CP1 ×CP1 by de Sitter ansatz.
Isometry classes
Regarding time-like killing vector fields, we first prove the following Proposition 2.32 Let be a neutral K¨ahler metric on CP1×CP1 and ξ ≡0 a time-like killing vector field on(CP1×CP1, g). Proposition 2.33 Let ξ1 and ξ2 be time-like killing vector fields on a neutral K¨ahler surface (CP1×CP1, g). In case (1) we see from proposition 2.33 that ξ and ϕ∗ξ are linearly dependent, that is ξ =kϕ∗ξ for some real constant k=0.
We first show that there exists a suitable orthonormal frame field associated with the given neutral near-hyperhermitian structure. Indeed, the non-divergence of ω is verified in the following way: Suppose that ω(X, Y) = 0 for arbitrary vector field Y tangent to E (X is also a vector field tangent to E). Conversely, if there exists a subbundle E with a symplectic structure ω of the tangent bundle T M such that TM ∼= E ⊕E, then M gives a neutral almost hyperhermitic structure (g, I,J, K) defined by (3.7 ) and ( 3.8).
Therefore, the existence of a neutral, almost hyperhermitic structure is equivalent to that of the ∆(Sp(k,R)) structure, where ∆(Sp(k,R)) denotes the image of the diagonal embedding of the real symplectic group Sp. (k,R) in Sp(k,R)× Sp(k,R). Therefore, g is compatible with (I,J, K) and therefore (g, I,J, K) is a neutral, almost hyperhermitic structure with the desired properties. Theorem 3.7 Let (g, I,J, K) be a neutral, almost hyperhermitic structure on a four-manifold M. Then I, J and K are integrable if and only if the Lee forms satisfy β1 =β2 =β3.
Thus we see that κ(σ) = 0 for any anti-self-dual total zero plane σ, and from Proposition 2.4 that g is.
Neutral hyperk¨ ahler structures
Taking into account the above remark and statement 3.6, we obtain the following statement 3.14. Let (Ω1,Ω2,Ω3) be a triplet of symplectic structures on a four-manifold M that satisfies the relation (3.10). From Proposition 3.7 we see that I,J, K are integrable, and therefore (g, I,J, K) is a neutral hyperkähler structure on M that satisfies the required conditions. 1ΩK is a non-vanishing closed (2,0)-form (i.e. a holomorphic biform) on (M, I), which trivializes the canonical bundle K(M,I), as a holomorphic vector bundle.
We can also prove that any neutral hyperk¨ahler metric is Ricci-flat and self-dual. On the other hand, for a neutral K¨ahler surface (M, I), the Ricci form of (M, g, I) is determined by the curvature form R∇ of the connection on Λ2− induced by the Levi- Civita connection∇of (M, g). Theorem 3.15 Any neutral hyperk¨ahler surface (M, g, I,J, K) is Ricci-flat and self-dual, and possesses a non-destructive holomorphic two-form ΩJ.
To conclude this section, we note the following conditions on a compact neutral hyperk¨ahler surface (M, g, I,J, K). Clearly, any complex torus has the standard flat neutral hyperk¨ahler structure induced from the complex plane C2. Moreover, we will also discuss the existence of non-flat neutral hyperk¨ahler structures on a primary Kodaira surface and a complex torus.
Primary Kodaira surfaces
Conversely, under suitable complex coordinates (w1, w2) of C2, the fundamental form of any neutral hyperkähler structure opX is expressed as (3.15). Then φ:=dz1 gives rise to a non-vanishing holomorphic one-form on X, generating the cohomology group H0(X; Ω1X) ∼= H1,0. 1Ω3 is a non-vanishing holomorphic biform onX, and therefore defines a global section of the canonical bundle KX.
Note that the cohomology classes of Ω2,Ω3,Ω−2,Ω−3 generate the cohomology groupH2(X;R) and satisfy relations similar to those in (3.17). Integrating the above equation, we get a2 +b2 = 1, so we can set a = cos and b= sin for some real constants. Using expression (3.15), we can give a characterization of planar neutral hyperk¨ahler structures on a primary Kodaira surface in terms of the potential function ϕ, which shows that any nonconstant function ϕ on the base torus of every primal Kodaira surface defines a non-flat neutral hyperk¨ahler metric gϕ (cf. Petean [82]).
From the mean value property of the operator ∂∂ we then conclude that F must be constant.
Complex tori
Therefore, the self-dual W+ part of the Weyl conformal curvature tensor vanishes everywhere, that is, the metric is anti-self-dual. However, we consider here another description of the Fubini-Study type metric from a de Sitter ansatz perspective. Then ¯I is integrable and moreover (gFS,I) defines a neutral K¨¯ ahler structure in CP21. σ1 is the bound form of the Hopf fibration S3 → S2.
From Proposition 2.17 it follows that gFS is a self-dual neutral metric with respect to the orientation defined by I. Note that gFS is an anti-self-dual neutral metric with respect to the orientation defined by ¯I.). Taking its quotient with Zk+1, we can consider gLB as a neutral metric on the total space of the complex line bundle L⊗H(k+1)2 →H2, where LH2 →H2 denotes a complex line bundle induced from infinite Hopf bundle H13 = SL2(R) → H2. When k = 1, we can show that gLB is a Ricci-flat neutral K¨ahler (i.e. self-dual) metric on T∗H2 or T∗(H2/Γ), which is an infinite analogue of the Eguchi-Hanson metric on the cotangent bundle T∗CP1 (cf. Eguchi-Hanson [23]).
Note that Riemannian analogues of these self-dual neutral metrics of Bianchi type VIII are obtained as Riemannian metrics of Bianchi type IX. Theorem 5.1 Letπ: (M, g) → (N, gˇ) be a pseudo-Riemannian immersion with fully geodesic fibers, where M is an (n+ 1)-dimensional manifold and N is an n-dimensional manifold, and ξ is a Killing vector field on (M, g) tangent to the fibers with g(ξ, ξ)≡ −1. It should be noted that our definition of the curvature tensor R has a different sign than that in Besse [7].
The ratio between the Ricci curvatures rD and ˇr of the Weyl connection D and the Levi-Civita connection ∇ of ˇg is given by.
Hirzebruch signature and Euler characteristic
Let∇be the Levi-Civita connection of (M, g) and Ωjk the components of the curvature form of∇with respect to {e1, e2, e3, e4}. If g is anti-self-dual with respect to the complex orientation, then the signature τ(M) is non-positive, and τ(M) = 0 only if g is conformally planar. Let (M, g) be a compactly oriented pseudo-Riemannian manifold with neutral metric g and ∇ its Levi-Civita connection.
Note that Petean [82] observed this result by considering c1(M, I) = (1/2π)[γ] = (sg/8π)[ΩI] and [ΩI]2 < 0, and an interesting result obtained for the existence of neutral K¨ahler Einstein metrics on compact, complex surfaces.
Liouville’s theorem
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No.4 Masami Fujimori: Integral and rational points on algebraic curves of certain types and their Jacobian variants over number fields, 1997. No.6 Setsuro Fujii'e: Solutions ramifi´ees des probl`emes de Cauchy caract´eristiques et fonctions hyper ´ eom´etriques `a deux variables, 1997. No.7 Miho Tanigaki: Saturation of the approximation by spectral decompositions associated with the Schr¨odinger operator, 1998.
No.14 Tetsuya Taniguchi: Nonisotropic Harmonic Tori in Complex Projective Spaces and Configurations of Points on Riemann Surfaces, 1999.
