Isolated singularities of affine special Kähler metrics in two dimensions
aa r X i v : . [ m a t h . DG ] M a y Isolated singularities of affine special Kählermetrics in two dimensions
Andriy HaydysUniversity of BielefeldMay 3, 2015
Abstract
We prove that there are just two types of isolated singularities of special Käh-ler metrics in real dimension two provided the associated holomorphic cubic formdoes not have essential singularities. We also construct examples of such metrics.
Special Kähler metrics attracted a lot of interest recently both in mathematics andphysics, see for instance [ACDM15, MS15, Nei14] for the most recent works. Themain source of interest to such metrics is the fact that the total space of the cotangentbundle of the underlying manifold carries a natural hyperKähler metric for which eachfiber is a Lagrangian submanifold. Such manifolds play an important role in the SYZ-conjecture [SYZ96].Soon after special Kähler metrics entered the mathematical scene [Fre99], Luproved [Lu99] that there are no complete special Kähler metrics besides flat ones. Thismotivates studying singular special Kähler metrics as the natural structure on bases ofholomorphic Lagrangian fibrations with singular fibers. In this paper we study isolatedsingularities of affine special Kähler metrics in the lowest possible dimension.Recall that a Kähler manifold ( M, g, I, ω ) is called (affine) special Kähler, if it isequipped with a symplectic, torsion-free, flat connection ∇ such that ( ∇ X I ) Y = ( ∇ Y I ) X (1)for all tangent vectors X and Y . To any special Kähler metric one can associate aholomorphic cubic form Ξ , which measures the difference between the Levi-Civitaconnection and ∇ .Throughout the rest of this paper we assume that dim R M = 2 , i.e. M is a Riemannsurface. Let m be an isolated singularity of g . Denote by n the order of Ξ at m , i.e. m is a zero of order n if n > or m is a pole of order | n | if n < or Ξ( m ) exists1nd does not vanish if n = 0 . By choosing a holomorphic coordinate z near m , wecan assume that g is a special Kähler metric on the punctured disc B ∗ = B (0) \ { } .The following is the main result of this paper. Theorem 1.1.
Let g = w | dz | be a special Kähler metric on B ∗ . Assume that Ξ isholomorhic on the punctured disc and the order of Ξ at the origin is n > −∞ . Then w = −| z | n +1 log | z | e O (1) or w = | z | β ( C + o (1)) (2) as z → , where C > and β < n + 1 .Moreover, for any n ∈ Z and β ∈ R such that β < n + 1 there is an affine specialKähler metric satisfying (2) . We prove this theorem by establishing an intimate relation between special Kählermetrics and metrics of non-positive Gaussian curvature and applying the machinerydeveloped for the latter ones. This relation allows us in particular to construct explicitexamples of special Kähler metrics from metrics of constant negative curvature.
Acknowledgements.
I would like to thank V. Cortés and R. Mazzeo for helpful dis-cussions and German Research Foundation (DFG) for financial support via CRC 701.
Let Ω ⊂ C be an open subset, which we equip with the flat metric g = | dz | = dx + dy . Denote by ∗ the corresponding Hodge operator and by ∆ = ∂ xx + ∂ yy theLaplace operator. Proposition 2.1.
For any pair ( u, η ) ∈ C ∞ (Ω) × Ω (Ω) satisfying dη = 0 , (3) ∗ d ∗ η = 2 ∗ ( ∗ η ∧ du ) − e u | η | , (4) ∆ u = | η + e − u du | e u . (5) the metric g = e − u ( dx + dy ) is special Kähler. Conversely, for any special Kählermetric on Ω there exists a solution ( u, η ) of (3) - (5) .Proof. In real coordinates ( x, y ) the connection ∇ can be represented by ω ∇ = (cid:18) ω ω ω ω (cid:19) ∈ Ω ( R ; gl ( R )) . Then (1) can be written as [ ω ( X ) , I ]( Y ) = [ ω ( Y ) , I ] X , where I = (cid:18) −
11 0 (cid:19) . X and Y , it is enough to check itsvalidity for ( X, Y ) = ( ∂∂x , ∂∂y ) , which yields ω ( ∂∂x ) − ω ( ∂∂x ) = − (cid:0) ω ( ∂∂y ) + ω ( ∂∂y ) (cid:1) ,ω ( ∂∂x ) + ω ( ∂∂x ) = ω ( ∂∂y ) − ω ( ∂∂y ) . (6)Write ω − ω = a dx + b dy . It follows from (6) that ω + ω = b dx − a dy .Denoting ω + ω = p dx + q dy , we obtain ω = p + a dx + q + b dy and ω = p − a dx + q − b dy. (7)Furthermore, the torsion of ∇ vanishes if and only if ω ( ∂∂x ) = ω ( ∂∂y ) and ω ( ∂∂y ) = ω ( ∂∂x ) . Combing this with (7) we obtain ω = q + b dx − p + a dy and ω = b − q dx + p − a dy, which yields ω ∇ = (cid:18) ω − ∗ ω ∗ ω ω (cid:19) . Then ∇ is flat if and only if dω = ω ∧ ω , d ∗ ω = − ∗ ω ∧ ω − | ω | dx ∧ dy,dω = ω ∧ ω , d ∗ ω = − ∗ ω ∧ ω − | ω | dx ∧ dy. (8)Here we used a special property of 1-forms in dimension 2, namely the identity ( ∗ α ) ∧ ( ∗ β ) = α ∧ β .Furthermore, notice that ∇ preserves the Kähler form ω = 2 e − u dx ∧ dy if and onlyif de − u = d (cid:0) ω ( ∂∂x , ∂∂y ) (cid:1) = e − u ( ω + ω ) ⇔ − du = ω + ω . Substituting this in (8) we obtain dω = du ∧ ω ,d ∗ ω = ∗ ω ∧ du − | ω | dx ∧ dy, ∆ u = − ∗ ( ∗ ω ∧ du ) + 4 | ω | + | du | = | ω + du | . Finally, substitute η = e − u ω to obtain (3)-(5).Observe that (5) implies that the Gaussian curvature of ˜ g = e u | dz | equals −| η + e − u du | and therefore is non-positive. This relation between special Kähler metricsand metrics of non-positive Gaussian curvature is central for the rest of the paper andwill be more vivid below. 3 orollary 2.2. Any pair of functions ( h, u ) satisfying ∆ h = 0 and ∆ u = | dh | e u , (9) on a domain Ω ⊂ R determines a special Kähler metric on Ω . Conversely, if H (Ω; R ) is trivial, then any special Kähler metric on Ω determines a solution of (9) .Proof. Assume η = df , which is always the case provided H (Ω; R ) is trivial. By (4)we obtain ∆ f = 2 ∗ ( ∗ df ∧ du ) − e u | df | . Denoting h = 2 f − e − u , we compute: ∆ h = 2∆ f − e − u ( − ∆ u + | du | )= 4 ∗ ( ∗ df ∧ du ) − e u | df | − e − u (cid:0) − e u | df | + 4 e u ∗ ( ∗ df ∧ du ) (cid:1) = 0 . Hence, ( h, u ) solves (9).Conversely, for any solution ( h, u ) of (9) the pair ( u, df ) solves (3)-(5), where f = ( h + e − u ) / .Observe that if h is a constant function, then (9) reduces to ∆ u = 0 , i.e., thecorresponding special Kähler metric g = e − u ( dx + dy ) is flat. Non-trivial exampleswill be constructed below. Corollary 2.3.
Assume
Ω = B ∗ . Then any triple ( h, u, a ) satisfying ∆ h = 0 and ∆ u = | dh + aϕ | e u , (10) where a ∈ R and ϕ is a 1-form generating H ( B ∗ ; R ) , determines a special Kählermetric on the punctured disc. Conversely, any special Kähler metric on the punctureddisk determines a solution of (10) .Proof. Recall that ϕ = ydx − xdyx + y is harmonic 1-form generating H ( B ∗ ; R ) . Hence, any closed η ∈ Ω( B ∗ ) can bewritten in the form η = df + a ϕ for some a ∈ R . Just like in the proof of Corollary 2.2put h = 2 f − e − u to obtain (10).Notice that the second equation of (10) (as well as (9)) is the celebrated Kazdan–Warner equation [KW74]. Remark 2.4.
Tracing through the proof one easily sees that given a solution of (10)the corresponding special Kähler structure is given by g = e − u ( dx + dy ) , ω ∇ = (cid:18) ω − ∗ ω ∗ ω ω (cid:19) ,ω = e u dh + aϕ ) − du, ω = − e u dh + aϕ ) − du. (11)4 xample 2.5. Let h be a positive harmonic function. It is straightforward to checkthat the pair ( h, − log h ) solves (9). Choosing h = − log | z | we obtain that g = − log | z | | dz | is a special Kähler metric on the punctured unit disc. Special Kählermetrics with logarithmic singularities were studied for instance in [GW00, Lof05]. Example 2.6.
Pick any integer n and consider the harmonic function h ( x, y ) = ( Re z n +1 if n = − , log | z | if n = − . Clearly, there are some positive constants C and C such that − C | z | n ≤ −| dh | ≤− C | z | n . Choose a point z ∈ B (0) \ B (0) and a non-positive smooth function ˜ K satisfying ˜ K | B (0) = −| dh | , ˜ K | C \ B (0) = − , − C ′ | z − z | − n ≤ ˜ K ≤ − C ′ | z − z | − n , where C ′ and C ′ are some positive constants. Clearly, ˜ K can be extended as a smoothfunction to CP \ { z , } , where we think of CP as C ∪ { ∞ } .Let g be a Riemannian metric on CP such that g = | dz | on B (0) . Denote by K the Gaussian curvature of g . By [McO93, Thm.II] for any β ≤ n + 1 there existsa solution u of ∆ g u + ˜ Ke u = K such that u = ( − β log | z | + c + o (1) , if β < n + 1 , − ( n + 1) log | z | − log (cid:12)(cid:12) log | z | (cid:12)(cid:12) + O (1) if β = n + 1 , as z → . ( u has also a similar behaviuor near z .) Thus, on B (0) the pair ( h, u ) solves (9). Inparticular, g = w | dz | = e − u | dz | is a special Kähler metric satisfying (2). Proof of Theorem 1.1.
Let π (1 , ∈ Ω , ( T C B ∗ ) be the projection onto T , B ∗ . Recallthe definition of the holomorphic cubic form: Ξ = − ω (cid:0) π (1 , , ∇ π (1 , (cid:1) . Then, a direct computation yields
Ξ = e − u (cid:0) ∗ ( ω − ω ) − i ( ω − ω ) (cid:1) dz . Therefore, substituting (11) we obtain
Ξ = Ξ dz , where Ξ = 12 (cid:16) a z − ∂h∂z i (cid:17) . Notice that | Ξ | = | dh + aϕ | =: − ˜ K .5enoting w = e − u , we obtain ∆ u = − ˜ Ke u on the punctured disc. Under thehypothesis of this theorem there exist positive constants C and C such that the in-equalities − C | z | n ≤ ˜ K ≤ − C | z | n (12)hold on a (possibly smaller) punctured disc.By [McO93, Appendix B] we obtain that u = − β log | z | + c + o (1) or u = − ( n + 1) log | z | − log | log | z || + O (1) , where β < n + 1 . Since w = e − u , we obtain (2).The existence of special Kähler metrics satisfying (2) has been established in Ex-ample 2.6.We remark in passing that starting from a different perspective, Loftin [Lof05]utilized an equation equivalent to (5) to study special Kähler metrics on CP . In this section we construct more examples — in particular explicit — of special Käh-ler metrics from metrics of constant negative Gaussian curvature.
Proposition 3.1.
Let ˜ g = e u | dz | be a metric of constant negative Gaussian curvatureon a domain Ω ⊂ R . Then g = e − u | dz | is special Kähler.Proof. Since ˜ g has a constant negative scalar curvature, say − , we have ∆ u = e u .Put h ( x, y ) = x and observe that the pair ( h, u ) satisfies (9). Hence, g = e − u | dz | is aspecial Kähler metric. Example 3.2.
Different incarnations of the Poicaré metric lead to differently lookingspecial Kähler metrics, which are gathered in the following table:Constant negative curvature metric Domain Special Kähler metric ˜ g = (Im z ) − | dz | upper half-plane g = Im z | dz | ˜ g = 4(1 − | z | ) − | dz | unit disc g = (1 − | z | ) | dz | ˜ g = ( | z | log | z | ) − | dz | punctured disc g = −| z | log | z | | dz | Table 1: Poincaré metrics and corresponding special Kähler metricsThe special Kähler metric appearing in the first row can be found in [Fre99, Rem. 1.20].Particularly interesting for us is the one appearing in the last row, as this yields an ex-ample of special Kähler metric with an isolated singularity.6 xample 3.3.
Explicit local examples of metrics of constant negative Gaussian cur-vature with conical singularities can be found in [KR08, Ex.2.1]. For instance, if α ∈ (0 , , then ˜ g = 1 | z | α (cid:16) − α − | z | − α ) (cid:17) | dz | is such a metric. Hence, g = 11 − α | z | α (cid:0) − | z | − α ) (cid:1) | dz | is a special Kähler metric with conical singularity. Example 3.4.
It is a classical result of Picard [Pic93] that for any given n ≥ pairwisedistinct points ( z , . . . , z n ) in C and any n real numbers ( α , . . . , α n ) such that α j < and P α j > there exists a metric of constant negative curvature ˜ g on C \{ z , . . . , z n } satisfying ˜ g = | z − z j | − α j ( c + o (1)) | dz | near z j . Hence, the corresponding specialKähler metric g has a conical singularity near z j : g = | z − z j | α j ( c + o (1)) | dz | . Explicit examples of constant negative curvature — hence special Kähler — met-rics on three times punctured complex plane can be found in [KRS11] and referencestherein.Notice also that it is possible to allow α i = 1 changing the asymptotic behaviourcorrespondingly and to replace C by P . References [ACDM15] D.V. Alekseevsky, V. Cortés, M. Dyckmanns, and T. Mohaupt,
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