Documenta Math. 653 Asymptotics of Complete Kähler-Einstein Metrics – Negativity of the Holomorphic Sectional Curvature

We consider complete Kahler-Einstein metrics on the complements of smooth divisors in projective manifolds. The estimates proven earlier by the author [5] imply that in directions parallel to the divisor at infinity the metric tensor converges to the Kahler-Einstein metric on the divisor. Here we show that the holomorphic sectional curvature is bounded from above by a negative constant near infinity.


Introduction and Statement of the Result
Complete Kähler-Einstein metrics ds 2 X of constant Ricci curvature on quasiprojective varieties are of interest in various geometric situations.The existence of a unique complete Kähler-Einstein metric ds 2 X of constant Ricci curvature −1 on a manifold X of the form X\C, where X is a compact complex manifold and C a smooth divisor is guaranteed under the condition [2,3,7]).
In [5], asymptotic properties of ds 2 X (with Kähler form ω X ) were investigated.In the special asymptotic situation, it is possible to prove estimates for the curvature tensor based on constant negative Ricci curvature.We obtain the following theorem.
Theorem 1.Let X be a compact complex surface, C ⊂ X a smooth divisor satisfying (N ).Then the holomorphic sectional curvature of the complete Kähler-Einstein metric on X = X\C is bounded from above by a negative constant near the compactifying divisor.The sectional and holomorphic bisectional curvatures are bounded on X.
For X = P 2 , the assumption of the theorem is satisfied, if the degree of the curve C is at least four.So the statement of the above theorem appears to be related to the Kobayashi conjecture about the algebraic degeneracy of entire holomorphic curves and the hyperbolicity of complements of plane curves.

Properties of the Complete Kähler-Einstein Metric
The above condition (N ) also implies the existence of a Kähler-Einstein metric ω C on C. Using a canonical section σ of [C] as a local coordinate function on X one can restrict the complete Kähler-Einstein metric ω X to the locally defined sets C σ0 = {σ = σ 0 }.The notion of locally uniform convergence of ω X |C σ0 , where σ 0 → 0, makes sense.Theorem 2 ( [5]).The Kähler-Einstein metric ω X converges to the Kähler-Einstein metric ω C , when restricted to directions parallel to C.
The result is a precise analytic version of the adjunction formula: Let h be an hermitian metric on [C] and Ω ∞ a volume form of class C ∞ on X, such that the restriction of Ω ∞ /h to C is the Kähler-Einstein volume form on C. In terms of the Hölder spaces C k,λ (X) and C k,λ (W ) for open subsets W ⊂ X depending on quasi-coordinates used in [1] by Cheng and Yau (cf.[2,3]), the above statement follows from the estimate below [5]: There exists a number 0 < α ≤ 1 such that for all k ∈ N and all 0 < λ < 1 the volume form of the complete Kähler-Einstein metric is of the form For any given ν ∈C k,λ (W ) according to [5,2] there exists some .
From now on we assume that dim C (X) = 2.
where g 0 σσ , g 0 σw , g 0 wσ , g 0 ww are in C k,λ (W ), whereas g ∞ ww is of class C ∞ , and We observe that in the determinant of the components of the metric tensor the diagonal terms dominate the rest.Moreover: where p ∼ q denotes the existence of some C > 1 such that (1/C)•p ≤ q ≤ C •p.

Asymptotics of the Curvature Tensor
The order of the arguments is critical.We begin with the off-diagonal terms of the curvature tensor, which require special attention.In the sequel we need the volume form Ω X in our local coordinates (σ, w), and we set D := g σσ • g ww − g σw • g wσ , i.e.
Next, we claim (11) Step 1.The following estimates hold for the components of the curvature tensor: We consider (13): In the same way we arrive at the estimates ( 14), (15).Some of these estimates need to be improved in a second step.
Proof.Concerning (16) we consider the equation

According to
Step 1 and Proposition 1 the term R σσσσ g σσ can be estimated from above and below by 1/|σ| We need to do (16) again.
We consider once again −g 2 σσ = R σσσσ g σσ g σσ + R σσσw g wσ g σσ + R σσwσ g σw g σσ + R σσww g ww g σσ .The last three terms on the right-hand side are at least in . This shows (18), and moreover −R σσσσ ∼ g 2 σσ .
Let us conclude with a refinement of (15) Step Proof.We regard −g ww = R wwww g ww + R wσww g σw + R wwσw g wσ + R wσσw g σσ .The first summand is in O(1), whereas the remaining three terms are at least in O 1/ log α (1/|σ| 2 ) .
We summarize our estimates in the following way.
Proposition 2. In a neighborhood of the divisor at infinity we have Proof of Theorem 1.The above Proposition 2 implies that the curvature tensor is dominated by the diagonal terms (19) and (20).We determine a domain of negative holomorphic sectional curvature.Let Now we pick a small upper bound for σ 2 which yields a negative upper bound for the holomorphic sectional curvature.The sectional and holomorphic bisectional curvatures are dealt with in a similar way.