Special Kähler structures, cubic differentials and hyperbolic metrics

We obtain necessary conditions for the existence of special Kähler structures with isolated singularities on compact Riemann surfaces. We prove that these conditions are also sufficient in the case of the Riemann sphere and, moreover, we determine the whole moduli space of special Kähler structures with fixed singularities. The tool we develop for this aim is a correspondence between special Kähler structures and pairs consisting of a cubic differential and a hyperbolic metric.


Introduction
For reader's convenience, let us recall the definition of the affine special Kähler structure, which is the main object of study of this article. Definition 1 [7] An (affine) special Kähler structure on a manifold is a quadruple (g, I , ω, ∇), where ( , g, I , ω) is a Kähler manifold with Riemannian metric g, complex structure I , and symplectic form ω(·, ·) = g(I ·, ·), and ∇ is a flat symplectic torsion-free connection on the tangent bundle T such that holds for all vector fields X and Y .
B Andriy Haydys andriy.haydys@math.uni-freiburg.de 1 Notice that τ depends on certain choices and, moreover, is defined locally only (or, equivalently, on the universal covering of ), howeverg is well-defined. Also,g may either degenerate or be singular at isolated points, hence, strictly speaking,g is a metric outside of some discrete subset of . This is not a concern for us, since we are interested in singular special Kähler structures, which involves singular metrics anyway.
In Definition 3 and below, a metric is said to be hyperbolic, if its Gaussian curvature is constant and equals −1. Ifg is any hyperbolic metric on , we say thatg represents a divisor n j=1 (α j − 1) p j with 0 ≤ α j = 1 if the following holds: If α j = 0, theñ g has a cusp singularity at p j ; If α j > 0,g has a conical singularity of order α j − 1, i.e.g has a conical angle of 2πα j at p j (See the precise explanation in Corollary 30).
Recall [7] that for any special Kähler structure we can also construct the associated cubic form , which is a holomorphic section of K 3 , where K is the canonical bundle of . Throughout this manuscript we assume that is non-zero. This means that we exclude special Kähler structures (g, ∇), where g is flat and ∇ is the Levi-Civita connection of g.
Thus, to any special Kähler structure on a Riemann surface we can associate a pair (g, ) as above. Our main result, Theorem 8 below, states, roughly speaking, that for any pair (g, ) consisting of a hyperbolic metric possibly singular at isolated points and a meromorphic cubic form we can construct a special Kähler structure, whose associated hyperbolic metric and associated cubic form areg and respectively.
Here ω k,∇ is the connection one-form of ∇ with respect to the trivialization (∂ x , ∂ y ), where z = x + yi is the standard coordinate on C. By [3,Thm. 5], (4) together with flat cones where β ∈ R, are local models of isolated singularities of affine special Kähler structures in complex dimension one provided the associated cubic form is meromorphic.
In other words, in the case of conical singularity the corresponding special Kähler metric is either locally conical, asymptotically cylindrical, or asymptotically conical respectively.
With this at hand, we can state our main result as follows.
Theorem 8 Let be a meromorphic cubic differential on a Riemann surface (not necessarily compact) with the divisor ( ) = p∈ ord p · p. Let alsog be a hyperbolic metric on representing a divisor D. Then there is a unique special Kähler structure (g, ∇) on whose associated hyperbolic metric and associated cubic form areg and respectively. Moreover, (g, ∇) is smooth on 0 := \ supp( )∪supp D and the following also holds: (i) A cusp singularity p ofg is a logarithmic singularity of (g, ∇) of order 1 2 (ord p + 1); (ii) A conical singularity p ofg of order α is a conical singularity of (g, ∇) of order 1 2 (ord p − α); (iii) (g, ∇) has a conical singularity of order 1 2 ord p at a point p ∈ supp( )\supp D.
A somewhat more precise version of this result is Theorem 31, which is proved in Sect. 3.
We would like to point out that the correspondence of Theorem 8 is pretty much explicit. To demonstrate this, pick any local holomorphic coordinate z and writeg = e 2v |dz| 2 and = 0 (z) dz 3 . Then the special Kähler metric of Theorem 8 is given by Using [3, (9), (11)], one can also obtain an explicit formula for a connection one-form of ∇ in terms of v and 0 . The details are provided at the end of Sect. 2.
Furthermore, pick integers k ≥ 1, ∈ [0, k], a k-tuple p = ( p 1 , . . . , p k ) of pairwise distinct points on as well as a k-tuple b = (β 1 , . . . , β k ) of real numbers. If (g, ∇) is a special Kähler structure on away from { p 1 , . . . , p k }, then each p j is an isolated singularity of the associated cubic form . It turns out that in general may have essential singularities at some of p j 's (see Example 32), however in the definition below, we assume that is meromorphic, i.e., each p j is a pole of at worst.

Definition 9 We call
is a special Kähler structure on such that is meromorphic, ≡ 0, and ord p j (g, ∇) = 1 2 β j /R >0 the moduli space of special Kähler structures with fixed singularities (or, simply the moduli space of special Kähler structures for short), where ord p j (g, ∇) is the order of (g, ∇) at p j , the first points of p are of conical type, and the remaining points are all of logarithmic type. In particular, β +1 , . . . , β k are integers, if = 0 all points are of logarithmic type, whereas for = k all points are of conical type. Notice that the group R >0 acts on the set of special Kähler structures via λ · (g, ∇) = (λg, ∇).
In addition, we call the moduli space of special Kähler metrics.
Notice that at this point both M k (p, b) and R k (p, b) are defined as sets only. We justify the name by introducing a topology on these sets in Sect. 5 below.
Theorem 10 Let be a compact Riemann surface of genus γ . If M k (p, b) is nonempty, then the following inequalities hold: where [β] is the greatest integer not exceeding β.
The proof of this theorem can be found in Sect. 5. We also establish necessary and sufficient conditions for the existence of special Kähler structures on elliptic curves as well as describe the corresponding moduli spaces in Corollary 47.
While proving our main statements we obtain also other results, which may be of some interest. In particular, as already mentioned above we construct an example of a special Kähler structure whose associated cubic form has essential singularities, see Example 32. To the best of our knowledge this is the first example of an associated cubic form with an essential singularity.
We also describe all special Kähler structures compatible with a fixed metric, see Sect. 4.
Furthermore, let (g, ∇) be a special Kähler structure on a compact Riemann surface with finitely many prescribed singularities. Then the map which assigns to (g, ∇) the associated cubic form is injective, see Theorem 37 for a more precise statement. This is surprising, since there is no reason to believe that , which a priori encodes the difference between the Levi-Civita and the flat symplectic connections only, should determine the whole special Kähler structure (with prescribed singularities). Moreover, this is a truly global statement in the sense that the corresponding local statement is clearly false.
Finally, in the last section we construct compactifications of the moduli spaces M k (p, b) and R k (p, b) in the case < k.

Preliminaries
Let ⊂ C be any domain, which is viewed as being equipped with a holomorphic coordinate z = x + yi and the flat Euclidean metric |dz| 2 = dx 2 + dy 2 . We assume that any element of H 1 ( ; R) can be represented by a co-closed 1-form.
Write a special Kähler metric g on in the form g = e −u |dz| 2 .
Using the global trivialisation of T provided by the real coordinates (x, y) the connection ∇ is described by its connection 1-form ω ∇ ∈ 1 ; gl(2, R) . A computation shows [4] that ω ∇ can be written in the form Here * denotes the Hodge star operator with respect to the flat metric, h is a smooth function, and ψ is a 1-form. These data are subject to the equation where = ∂ 2 x x + ∂ 2 yy . Moreover, given any triple (h, u, ψ) satisfying (15) the metric g = e −u |dz| 2 together with ω ∇ , which is given by (13) and (14), constitutes a special Kähler structure on (with its complex structure inherited from C).
If is the punctured disc B * 1 , any closed and co-closed 1-form can be written as aϕ, where ϕ is a generator of H 1 (B * 1 ; R). For example, we can fix Hence, a special Kähler structure on the punctured disc can be described in terms of solutions of the following equations where h, u ∈ C ∞ (B * 1 ) and a ∈ R. If (h, u, a) is a solution of (16), the associated holomorphic cubic form of the corresponding special Kähler structure is

Remark 17
Tracing through the description of special Kähler structures in terms of solutions of (16) as given in [4], it is easy to see that the function h is defined only up to a constant. In other words, if c is any real constant, (h, u, a) and (h + c, u, a) determine equal special Kähler structures.
A straightforward computation shows that |dh+aϕ| 2 = 16| 0 | 2 = 16| | 2 . Hence, the second equation of (16) can be written as which implies in particular that the Gaussian curvature of the special Kähler metric where A ∈ C is the residue of 0 at the origin, and denote by H a primitive of˚ 0 . Notice that H is well-defined up to a constant. Define Using The upshot of this computation is that 0 determines and is determined by h and a. Slightly more generally, let =˜ \{ p 1 , . . . , p k }, where˜ is a simply connected domain in C. Then any closed and co-closed 1-form ψ representing a non-trivial cohomology class can be written as k j=1 a k ϕ k , where ϕ k := −d arg (z − p k ) . It is easy to see that the above discussion can be repeated verbatim in this case too leading to the following result.

Proposition 20
Let =˜ \{ p 1 , . . . , p k }, where˜ is a simply connected domain in C. Any pair (u, ) satisfying (18) determines a special Kähler structure on such that the corresponding associated cubic form is . Conversely, any special Kähler structure on determines a solution of (18).
For the sake of clarity, let us spell the correspondence in the above proposition. Thus, if (u, = 0 dz 3 ) is a solution of (18), put g = e −u |dz| 2 . Also, write Then the corresponding special Kähler structure is given by (13) and (14) with ψ = j a j ϕ j .

Special Kähler structures and the period maps
Let ( , g, I , ω, ∇) be a special Kähler structure, where dim C = n. Denote by U the corresponding affine structure. This means that U is a covering of M by open sets; Moreover, each U ∈ U is equipped with a 2n-tuple of holomorphic functions (z 1 , . . . , z n ; w 1 , . . . , w n ), where (z 1 , . . . , z n ) and (w 1 , . . . , w n ) are conjugate special holomorphic coordinates on U [7]. IfŨ ∈ U is another open set equipped with (z,w), then we have a relation where P ∈ Sp(2n; R) and a, b ∈ C n are some constants. Denote Then the matrix τ := (τ jk ) is symmetric and Im τ is positive definite. In fact, ω = i 2 Im τ jk dz j ∧ dz k . In particular, we have a holomorphic map Ifτ is a map corresponding to the chartŨ , then the corresponding period maps are related by Hence, τ * g H n does not depend on the choice of an affine patch. While the pull-back metric is defined in any dimension, the case n = 1 has some special features. Indeed, in this case is a Riemann surface, H = H 1 is the upper half-plane so that τ is a local biholomorphism except perhaps at isolated points. Hence, g is non-degenerate on outside of some discrete subset. Moreover, the subset wherẽ g degenerates is easy to describe, see Proposition 26 below.
More importantly, in the case n = 1 the metric g H 1 coincides with the standard hyperbolic metric Im z −2 |dz| 2 . Hence, the pull-back metricg is also hyperbolic where it is non-degenerate.

Remark 21
Recall, that a holomorphic map τ : → H, which may be multi-valued, is called a developing map of a hyperbolic metricg, ifg = τ * g H . Hence, the very definition yields that the period map of a special Kähler structure is a developing map of the associated hyperbolic metric.
Example 22 Consider the following local example: is the punctured unit disc in C equipped with the metric g = − log |z| |dz| 2 , which is special Kähler. Then z is a special holomorphic coordinate with the conjugate given by w = 2i(z log z − z). Hence, the period map is τ = 2i log z. Of course, τ is multivalued, but all values of τ are related by translations by real numbers and therefore τ * g H is well defined and equals (|z| log |z|) −2 |dz| 2 , which is the standard Poincaré metric on the punctured disc.

Example 23
Let be the upper half-plane H equipped with the following special Kähler structure [7, Rem. 1.20] is a pair of special holomorphic conjugate coordinates. Hence, τ (z) = z, which means that τ * g H = g H .
It will be useful below to have a relation between and τ . Thus, if Z is a special holomorphic coordinate, we have Then, for an arbitrary holomorphic coordinate z we obtain Notice in particular, that we have the following statement, which will be useful below.

Proposition 26 Let p be a regular point of a special Kähler structure on a Riemann surface. Then the associated hyperbolic metric degenerates at p if and only if
The next result is the key ingredient in the proof of our main result, Theorem 8.

Remark 29
We would like to point out that in the statement of Lemma 27, the domain is allowed to have no punctures, i.e., k = 0 is allowed.
Furthermore, we claim that τ * g H =g. To see this, notice that if Z is a special holomorphic coordinate (in a neighbourhood of some point), we have Here the last equality follows from (25). Hence, which yields in turng This clearly proves (i). The last part, (ii), is obtained essentially by reading the above computation backwards. That is, ifg = e 2v |dz| 2 is a metric of constant curvature −1, we have v = e 2v . Using this, it is easy to check that for any holomorhic function 0 the function u := v − log | 0 | − 2 log 2 satisfies (18). Appealing to Proposition 20, we obtain (ii).
Corollary 30 Let g be a special Kähler metric on the punctured disc B * 1 such that the associated holomorphic cubic form has order n ∈ Z at the origin. Letg be the associated hyperbolic metric. Then the following holds: (i) g is conical of order β/2 if and only ifg is conical of order n − β ∈ (−1, +∞), i.e., (ii) g has a logarithmic singularity if and only ifg has a cusp, i.e., The inequality n − β > −1 claimed in (i) has been established in [12, Thm. 1.1]. Of course, this also follows from the classifaction of isolated singularities for metrics of constant negative curvature.
Theorem 31 Let be a Riemann surface (not necessarily compact) and 0 ⊂ be an open subset. For any holomorphic cubic form and any smooth hyperbolic metric g on 0 there is a unique special Kähler structure (g, ∇) on 0 \ −1 (0) whose associated hyperbolic metric and associated cubic form areg and respectively.
If is meromorphic on with the divisor ( ) = p∈ ord p · p andg represents a divisor D, then for the special Kähler structure (g, ∇) on 0 := \ supp( ) ∪ supp D as above the following holds: (i) A cusp singularity p ofg is a logarithmic singularity of (g, ∇) of order 1 2 (ord p + 1); (ii) A conical singularity p ofg of order α is a conical singularity of (g, ∇) of order has a conical singularity of order 1 2 ord p at a point p ∈ supp( )\supp D.
Proof Pick a point p ∈ and an open set U together with a holomorphic coordinate z centered at p. If p / ∈ supp( ) ∪ supp D, we may think of U as a disc {|z| < 1}. Otherwise, U can be chosen to be the punctured disc.
Since a special Kähler metric and the associated cubic form determine the flat symplectic connection uniquely, we conclude that (g, ∇) and (ĝ,∇) coincide (more precisely, this means (g, ∇) = f * (ĝ,∇)). By the construction, (ĝ,∇) is the special Kähler structure determined by and the hyperbolic metric where the above equality follows from the definition ofû. Thus, the choice of the local coordinate used in Proposition 20 is immaterial as claimed. This proves the existence of a special Kähler structure for given andg. The uniqueness of the special Kähler structure corresponding to (g, ) follows immediately from the corresponding local statement. The other properties claimed follow directly from Corollary 30.
Example 32 (A special Kähler structure whose associated cubic form has an essential singularity) Let (z) := e 1/z dz 3 be a cubic holomorphic form on C * . may be thought of as a holomorphic cubic form on P 1 with two singularities: one essential and the other one of degree −6. Pick a hyperbolic metric singular at any 3 points w 1 , w 2 , w 3 ∈ P 1 . By Theorem 31 we obtain a special Kähler structure on P 1 with at least three and at most five singularities depending on the number of points in {w 1 , w 2 , w 3 } ∩ {0, ∞} such that is the associated cubic form. To the best of our knowledge, this is the first example of a special Kähler structure whose associated cubic form has essential singularities.
Let (g, ∇) be a special Kähler structure on the punctured disc whose associated cubic form = 0 dz 3 has an essential singularity at the origin. Assume for simplicity of exposition that the origin is a regular point for the associated hyperbolic metric g = e 2v |dz| 2 . The existence of such structures follows by Theorem 31 just like in the example above. By (28) we have Notice that e −2v has a positive limit at the origin, whereas by the great Picard Theorem | 0 | takes any positive value near the origin. Hence, in this case the behavior of g is highly irregular near the origin.

Special Kähler metrics versus special Kähler structures
Theorem 31 allows us to construct inequivalent special Kähler structures such that the corresponding Riemannian metrics are equal. Indeed, fix a pair (g, ) as in Theorem 31 and let g be the corresponding special Kähler metric. It is then clear from (28) that the pair (g, λ ) leads to the metric |λ| · g, where λ ∈ C * . Hence, specializing to |λ| = 1 we obtain a family of special Kähler structures parameterized by S 1 such that all corresponding Riemannian metrics are equal.

Example 33
Fix arbitrarily a hyperbolic metricg on the punctured unit disc B * 1 . Choose a holomorphic cubic differential on B * 1 such that is of order −3 at the origin. Observe that the leading coefficient ξ −3 in the expansion 0 (z) = ξ −3 z −3 +ξ −2 z −2 + · · · is independent of the choice of a local coordinate. Hence, the family { λ | |λ| = 1} consists of holomorphic cubic differentials that are pairwise inequivalent even up to a change of coordinates. Hence, for the corresponding family of special Kähler structures (g, ∇ λ ) the metric is independent of λ and the corresponding structures are pairwise inequivalent.

Proposition 34 Let be a Riemann surface.
(i) Let (g, ∇) and (ĝ,∇) be two special Kähler structures on , whose associated cubic forms are andˆ respectively. If g =ĝ, then where λ ∈ C is of absolute value 1; (ii) If (g, ∇) is a special Kähler structure on whose associated cubic form is , then for each λ ∈ S 1 there is a unique special Kähler structure (g, ∇ λ ), whose associated cubic form is λ .
Proof Clearly, to prove (i) it is enough to check (35) in a neighborhood of a regular point p ∈ . Thus, let z be a local holomorphic coordinate in a neighborhood U of p.
Claim (ii) follows from Theorem 31 by settingg to be the associated hyperbolic metric of (g, ∇).

A necessary and sufficient condition for the existence of special Kähler structures on compact Riemann surfaces
Just like in the introduction, pick integers k ≥ 1, ∈ [0, k], a k-tuple p = ( p 1 , . . . , p k ) of pairwise distinct points on as well as a k-tuple b = (β 1 , . . . , β k ) of real numbers, where β +1 , . . . , β k are integers. Denote where K is the canonical bundle of . Recall also that [β] denotes the greatest integer not exceeding β. In other words , H (p, b) consists of all non-trivial meromorphic cubic differentials which are holomorphic on \{ p 1 , . . . , p k } and satisfy For each j > choose a local holomorphic coordinate z j centered at p j and consider the holomorphic map where dots denote the higher order terms. Then H (p, b) is the subset of H 0 (L) where each f j does not vanish. Hence, H (p, b) is Zarisky open.
Theorem 37 Let be a compact Riemann surface of genus γ . Then M k (p, b) = ∅ if and only if the following two conditions hold: Moreover, the map that assigns to a special Kähler structure (g, ∇) as above its associated cubic form ∈ H (p, b) is a bijection.
Proof If M k (p, b) = ∅, then by the quantitative relationship between the special Kähler metric and the associated cubic form in Corollary 30, the associated cubic form of any special Kähler structure (g, ∇) such that [g, ∇] ∈ M k (p, b) lies in H (p, b), hence (ii) holds. Furthermore, since ord p j −β j ≥ −1, the associated hyperbolic metricg has either a conical singularity with positive angle or a cusp at each p j . Let p k+1 , . . . , p m be further points on such thatg is singular. By Corollary 30,(i) each p j with j ≥ k + 1 is conical singularity ofg of order ord p j > 0. Hence, by the Gauß-Bonnet theorem applied tog we have It remains to show that (i) and (ii) yield a special Kähler structure. Indeed, notice that (i) and (ii) imply Hence, by [13] there exists a hyperbolic metricg which has conical singularities at each zero q of of order ord q , and has either a conical singularity or a cusp at each p j for all 1 ≤ j ≤ k. The proof is finished by appealing to Theorem 31.
Denote by π : M k (p, b) → R k (p, b) the natural projection, which has been studied in Sect. 4. In particular, each fiber of π is isomorphic to the circle.
We have the commutative diagram where slightly abusing notations stays for the map, which assigns to a special Kähler structure its associated cubic form, and ξ is just the induced map. By Proportion 34 and Theorem 10 both and ξ are bijections. This can be used to define topologies on M k (p, b) and R k (p, b). Indeed, H (p, b) is naturally a subset of a vector space H 0 (L), which can be equipped with a topology by introducing a Hermitian inner product (notice that the origin is not contained in H (p, b)).

Proof of Theorem 10
The proof consists of the following parts.
1. The first inequality of (11) coincides with Theorem 37 (i). The second one of (11) follows by combining the following facts: deg and R k (p, b) can be identified with S(H 0 (L)) and PH 0 (L) respectively. 3. We prove that the space M k (p, b) is non-empty for = P 1 provided (11) holds.

Remark 39
It is clear from the proof of Theorem 10 that the case = P 1 is somewhat special due to the fact that it is easy to describe when H 0 (L) is non-trivial. In general, the non-triviality of H 0 (L) depends on the complex structure on and the fixed singularities (p, b) of the special Kähler structures under consideration.
Corollary 40 Let (g, ∇) be a special Kähler structure on P 1 such that the associated cubic form is non-trivial and meromorphic. Then (g, ∇) must have at least three singularities.

Remark 41
For any n ∈ Z the metric g = r n |dz| 2 is flat on C\{0}, hence, can be thought of as a special Kähler structure on P 1 singular at most at 2 points, namely 0 and ∞. Notice that the corresponding cubic form is trivial, hence this example does not contradict Corollary 40.
Example 42 Let R 0 24 denote the moduli space of all special Kähler metrics with 24 singular points all of logarithmic type of order zero. R 0 24 fibers over Sym 24 (P 1 )\{Diagonal subset}, where each fiber is homeomorphic to a Zariski open subset of CP 18 . Hence, M 0 24 has complex dimension 42. If we also mod out by the natural action of PGL(2, C), the resulting space is of complex dimension 39. This space is of interest for elliptic K3 surfaces [11].
In what follows below we would like to describe which metrics actually appear as associated hyperbolic metrics of some special Kähler structure from M k (p, b). Thus, let R hyp be the set of all hyperbolic metrics on with isolated singularities. We have a natural map whereg is the hyperbolic metric associated with (g, ∇).

Proposition 43
For any compact Riemann surface the image of T is isomorphic to H (p, b)/C * . Moreover, if = P 1 , the image of T consists of those hyperbolic metrics g, which satisfy the following: There exist r ≥ 0 points q 1 , . . . , q r ∈ P 1 \{ p 1 , . . . , p k } as well as m ∈ Z k and n ∈ Z r >0 such that the following holds: (a)g is smooth on P 1 \{ p 1 , . . . , p k , q 1 , . . . , q r }. Hence, we obtain that the image of T is isomorphic to H (p, b)/C * .
Furthermore, think of P 1 as the affine complex line C compactified by a point at infinity. Without loss of generality we can assume that none of p j equals ∞. Then for = 0 dz 3 ∈ H (p, b) we have the following expression: Here A = 0 is a constant, m j is an integer satisfying m j > β j − 1, n j is a positive integer, and q 1 , . . . , q r are those zeros of , which are not contained in { p 1 , . . . , p k }. Moreover, (44) holds since is regular at ∞.
We note in passing that it is possible to define topologies, or even smooth structures, on M k (p, b) and R k (p, b) directly along the lines of [16]. This would then require to prove that the map is a homeomorphism, which seems to be excessive for our modest aims.

Existence of special Kahler metrics on Riemann surfaces with positive genera
Corollary 46 Let be a compact Riemann surface with genus γ > 0. Then M k (p, b) is non-empty if and only if the following three conditions hold: Proof Suppose that these three conditions hold. By the last two ones, there exists a meromorphic cubic differential which is holomorphic outside { p 1 , . . . , p k } and satisfies ord p j ≥ [β j ] for 1 ≤ j ≤ and ord p j = β j − 1 for + 1 ≤ j ≤ k. The conclusion follows from the first condition and Theorem 37. Suppose that M k (p, b) is non-empty. The first two conditions follows from Theorem 37. We show the third one by contradiction. Suppose that there exist +1 ≤ j ≤ k such that Then we find H 0 (L − p j ) = H 0 (L). We pick a special Kähler structure (g, ∇) in M k (p, b) with associated cubic differential . By Theorem 37, we know that has order β j − 1 at p j . On the other hand, since belongs to H 0 (L) = H 0 (L − p j ), the order of at p j should be greater than or equal β j . This is a contradiction.
In the case is an elliptic curve (compact Riemann surface of genus one), the statement of the above corollary can be made more explicit. M k (p, b) is non-empty if and only if the following three conditions hold: Proof While a proof of this corollary could be obtained from Corollary 46, we prefer a more direct approach. Thus, suppose there exists a special Kähler structure (g, ∇) as in the statement of this corollary. The first two condtions follow from Theorem 37 directly. Suppose deg L = 1. Then H 0 (L) has dimension one by the Riemann-Roch theorem. If there exists + 1 ≤ j ≤ k such that the divisor −D + p j := −D(p, b) + p j is equivalent to zero, then there exists an elliptic function f such that ( f ) = −D + p j and H 0 (L) is generated by f dz 3 , where dz is a nonwhere vanishing holomorphic one-form on E. Furthermore, the associated cubic differential equals Const. f dz 3 and has order β j at p j . This is a contradiction, which finishes the proof of the "only if" part.

Corollary 47 For an elliptic curve E the space
Assume that (i)-(iii) hold. We divide the proof of the "if" part into the following two cases. Case 1. Suppose deg L = 0. Then L is trivial, in particular L has a non-trivial holomorphic section, and there exists an elliptic function f on E such that ( f ) = −D. By Theorem 37, there exists a special Kähler structure (g, ∇) whose associated cubic differential is f dz 3 . Case 2. Suppose d := deg L > 0. By the Abel-Jacobi theorem, we can find a point q ∈ E = C/ and an elliptic function f on E such that ( f ) = dq − D. If q / ∈ {p +1 , . . . , p k }, then by Theorem 37 there exists such a special Kähler structure (g, ∇) whose associated cubic differential is f dz 3 , where dz is a non-where vanishing holomorphic 1-form on E. Hence, it remains to consider the case q ∈ {p +1 , . . . , p k }. Subcase 2.1. Suppose d ≥ 2. Since there exist q 1 , . . . , q d in E\{ p +1 , . . . , p k } such that q 1 + · · · + q d ≡ dq (mod ), we can find an elliptic function f on E such that ( f ) = q 1 + · · · + q d − D. We are done. Subcase 2.2. Suppose d = 1. Then there exists + 1 ≤ j ≤ k such that −D + p j ∼ 0. However, this possibility is excluded by (iii).
We conclude this section by proposing a conjecture about the existence of certain cubic differentials and special Kähler structures on compact Riemann surfaces of genera greater than one.
with the top stratum being M k (p, b). Moreover, the map extends as a bijective map : M k (p, b) → H 0 (L)\{0}/R >0 such that each M k (p, b m ) is mapped bijectively to H (p, b m ). In particular, (50) consists of a finite number of strata. Just like in the case of M k (p, b), we use¯ to endow M k (p, b) with a topology.
Clearly, we can also construct a compactification of R k (p, b) in a similar manner. Namely, defining we obtain a bijectionξ fitting into the commutative diagram cf. (38). Here S(H 0 (L)) denotes the sphere of unite radius in H 0 (L) with respect to some norm. Summarizing, we obtain the following.

Proposition 51
The map¯ establishes a natural bijective correspondence between M k (p, b) and the unit sphere in H 0 (L). Likely,ξ establishes a natural bijective correspondence between R k (p, b) and PH 0 (L).