1 Introduction↩︎

A \(k\)-differential \(\zeta\) on a Riemann surface \(X\) of genus \(g\) is a section of \(K^{\otimes k}\), where \(K\) is the canonical bundle of \(X\). Thus, \(k\)-differentials encode intrinsic properties of \(X\). Moreover, the integration of a \(k\)-th root of \(\zeta\neq0\) induces a flat metric with conical singularities at the zeros and poles, where the transition maps (away from the singularities) are given by translations and rotations by angles that are multiples of \(2\pi /k\).

The moduli space of \(k\)-differentials can be stratified according to the orders \(m_{i}\) of the \(n\) zeros and the \(p\) poles of \(\zeta\). These spaces, called strata and denoted by \({\Omega^{k}\mathcal{M}}_{g}(m_{1},\dots,m_{n+p})\), are complex orbifolds of dimension \(2g-2+n+p\), as proved in BCGGM3?. In this work, we mark the zeros and poles: the strata parameterize tuples \((X,\zeta, z_{1},\dots,z_{n};p_{1},\dots,p_{p})\), where \(z_{i}\) is a singularity of order \(m_{i}>-k\) and \(p_{i}\) of order \(m_{n+i}\) of the \(k\)-differential \(\zeta\). These strata play a significant role in understanding surface dynamics, as illustrated in ChenSurvey?, AMTransSurf? for \(k=1,2\) and in AAHR? for higher \(k\).

From the perspective of enumerative geometry, there have been a number of fascinating recent results concerning \(k\)-differentials and their applications. These include computing volumes and Euler characteristics of linear submanifolds in moduli spaces of flat surfaces, using cycle classes of strata of differentials to study double ramification cycles, and developing analogous structures, called \(k\)-leaky numbers, that arise in the Hurwitz counting problem of branched covers. We refer to SauvagetVolumes?, NguyenVolumes?, CMSEuler?, SchmittKdiff?, BHPSS?, CMSLeaky? for these developments.

As we recall in Section 2.1, if a singularity of \(\zeta\) is a flat-geometric zero (i.e., of order \(>-k\)), or if it is a pole whose order is not divisble by \(k\), then its \(k\)-residue is always zero. On the other hand, a pole of \(\zeta\) of order divisible by \(k\) may have a non-trivial \(k\)-residue.

A dimension count shows that for \(k \geq 2\), the only strata whose dimension equals that of their space of configurations of \(k\)-residues are of the form \(\Omega^{k}\mathcal{M}_{0}(a_{1},a_{2},-b_{1},\dots,-b_{p})\), where \(b_{1},\dots,b_{p}\in k \mathbb{N}_{>0}\) and \(a_{1},a_{2}\in{\mathbb{Z}}\) are coprime to \(k\). In this case, we define the isoresidual fibration as \[\mathop{\mathrm{Res}}\colon{\Omega^{k}\mathcal{M}}_{0}(a_1, a_2, -b_1, \ldots, -b_p) \to \mathcal{R}^{k}_{p} : \zeta \mapsto (\mathop{\mathrm{Res}}_{p_{1}}\zeta,\dots,\mathop{\mathrm{Res}}_{p_{p}}\zeta)\,,\] where the target space \(\mathcal{R}^{k}_{p}\) is the complex vector space \(\mathbb{C}^p\) that encodes the configurations of the \(k\)-residues at the poles \(p_{1},\dots,p_{p}\) of order \(-b_1,\ldots, -b_p\).

Since we label each singularity, we assume throughout the paper that \(a_1 > -k\). For \(a_2\), there are two cases: either \(a_2 > -k\), corresponding to a flat-geometric zero, or \(a_2 < -k\), corresponding to a flat-geometric pole whose order is not divisible by \(k\); in both cases, the \(k\)-residue is zero.

In these cases, when \(p\geq2\), the isoresidual fibration is a ramified cover of its image, which we call the \(k\)-isoresidual cover. The main goal of this paper is to compute the number of preimages above any configuration of residues. In particular, we determine the degree and the ramification locus of this cover. Remarkably, the degree formula involves the \(k\)-factorial function, defined by \[a!_{(k)}\mathrel{\vcenter{:}}=\prod\limits_{i=0}^{\lceil a/k \rceil}(a-ik)\,.\] More generally, we define the following function.

Note that when \(m\leq\left[\frac{a}{k}\right]+3\), the value of \(f_{k}(a,m)\) is positive, whereas for \(m>\left[\frac{a}{k}\right]+3\), the value of \(f_{k}(a,m)\) alternates in sign as \(m\) increases. Using this function, we can determine the degree of the \(k\)-isoresidual cover as follows.

The above formula takes a particularly simple form when all the poles have order \(-k\).

Now, to describe the ramification locus of the isoresidual fibration, we introduce the following polynomial expressions in the \(k\)-residues. Let \([R_1:\cdots:R_p]\) be a tuple of \(k\)-residues, defined up to simultaneous scaling by \(\mathbb{C}^{\ast}\). For any subset \(I = \{i_1,\ldots, i_d\}\) of \(\{1,\ldots, p\}\), we define the polynomial \[\label{eq:polyanul} P(R_{i_1},\ldots,R_{i_d})\mathrel{\vcenter{:}}= \prod_{\left\{(r_{i_1},\ldots,r_{i_d})\colon r_{i_j}^k = R_{i_j}\right\}}(r_{i_1}+\cdots+r_{i_d})\,,\tag{2}\] where the product runs over all tuples of \(k\)-th roots of the \(k\)-residues. This is a homogeneous polynomial of degree \(k^{d-1}\) in the ring \(\mathbb{Z}[R_1,\ldots,R_p]\) (see Lemma 8).

The subset of indices \(I\) for which the above polynomial vanishes is called a resonant subset, and the locus in the space of \(k\)-residue tuples for which a resonant subset exists is called the resonant locus. Note that the resonant locus is a union of hypersurfaces, each of which generically corresponds to exactly one resonant subset.

One difference compared to the case of abelian differentials is that there is no residue theorem for \(k\)-differentials when \(k\geq2\). Hence, a subset \(I\) being resonant does not imply that \(I^c\) is resonant; in fact, we allow \(I=\{1,\ldots,p\}\). When a resonant equation occurs, some sum of \(k\)-th roots of the residues equals zero. Nevertheless, there may be more than one such sum that vanishes. This motivates the following definition.

With these preparations, we can determine the number of \(k\)-differentials whose residues satisfy a unique resonant equation.

When there are an arbitrary number of resonant subsets, the set of indices can be partitioned in several ways as \(\{1,\ldots,p\} = J_0 \sqcup J_1 \sqcup \cdots\sqcup J_s\), where \(J_1,\ldots,J_s\) are resonant subsets, and the subset \(J_0\) may be non-resonant and possibly empty. For example, the degree of the generic \(k\)-isoresidual fiber corresponds to the case \(J_0=\{1,\ldots,p\}\), while for a \(k\)-isoresidual fiber with a single resonant subset \(I\), we subtract from the degree of the generic fiber a term corresponding to \(J_1=I\) and \(J_0= I^c\). In general, the degree of the \(k\)-isoresidual fiber over an arbitrarily given \(k\)-residue tuple involves a contribution from each such partition, described as follows.

Note that Theorem 6 could theoretically be used to recover the classification of empty \(k\)-isoresidual fibers for such strata, as established in getaquad? (for quadratic differentials) and GT-k-residue? (for \(k\)-differentials with \(k \geq 3\)). However this produces delicate combinatorial identities involving the cardinality of the isoresidual fiber over each intersection of resonance hypersurfaces (see the examples at the end of Section 6.5).

Finally, we remark that the above results generalize the case \(k=1\), which was studied in GeTaIso? using flat geometry and later completed in ChPrIso? using intersection theory. Applications and alternative perspectives on this problem can be found in Sugiyama?, BuRoCoun?, OberWRS?. It seems to be unknown if these applications and perspectives can be extended to the context of \(k\geq2\).

1.0.0.1 Organization of the paper:

• In Section 2, we review the background on \(k\)-differentials, the local invariants of their singularities, their strata, and their relation to flat geometry. In particular, we introduce the residue map and the isoresidual fibration.

• In Section 3, we introduce the resonance stratification of the \(k\)-residue space and show that the number of elements in isoresidual fibers is constant over the complement of the resonance locus.

• In Section 4, we describe the multi-scale compactification of strata of \(k\)-differentials.

• In Section 5, we use intersection theory to compute the degree of the residue map above all the residue tuples, proving Theorems 2, 5, and 6.

• In Section 6, we provide alternative flat-geometric arguments to prove Corollary 3 and give various examples of computations.

• In Section 7, we deduce, from Theorem 2, a counting result for a special class of spherical metrics, stated in Theorem 32.

1.0.0.2 Acknowledgements.

Research of D.C. is supported by National Science Foundation grant DMS-2301030, Simons Travel Support for Mathematicians, and a Simons Fellowship. Research of Q.G. is supported by the grant PAAPIT UNAM-DFG DA100124 “Conectividad y conectividad simple de los estratos.” Research of M.P. is supported by the DFG-UNAM project MO 1884/3-1 and the Collaborative Research Centre TRR 326 “Geometry and Arithmetic of Uniformized Structures.” Research of G.T. is supported by the Beijing Natural Science Foundation IS23005 and the French National Research Agency under the project TIGerS (ANR-24-CE40-3604).

2 \(k\)-differentials↩︎

In this section, we review some general properties of \(k\)-differentials and their associated flat structures. The results collected here were proved in BCGGM3?.

2.1 \(k\)-differentials and local models of singularities↩︎

A \(k\)-differential on a Riemann surface \(X\) is a (holomorphic or meromorphic) section that we assume to be not identically zero of the \(k\)-th tensor power of its canonical bundle. The degree of the associated \(k\)-canonical divisor is \(k(2g-2)\).

Locally, a \(k\)-differential posses two invariants: its order and its \(k\)-residue. More precisely, given a \(k\)-differential \(\zeta\), there exists a neighborhood of any point \(P\) and a biholomorphic change of coordinates such that \(\zeta\) takes the form: \[\label{eq:standard95coordinates} \begin{cases} z^m\, (dz)^{k} &\text{if m> -k or k\nmid m,}\\ \left(\frac{r}{z}\right)^{k}(dz)^{k} &\text{if m = -k,}\\ \left(z^{m/k} + \frac{t}{z}\right)^{k}(dz)^{k} &\text{if m < -k and k\mid m,} \end{cases}\tag{4}\] where \(r \in \mathbb{C}^{\ast}\) and \(t \in \mathbb{C}\). The integer \(m\) is the order of \(\zeta\) at the point \(P\). The constants \(r\) and \(t\) are defined up to multiplication by a \(k\)-th root of unity. We define the \(k\)-residue \(\mathop{\mathrm{Res}}_{P}(\zeta)\) of \(\zeta\) at \(P\) as \(r^{k}\) in the second case, \(t^{k}\) in the third case, and zero otherwise. In particular, the \(k\)-residue can be nonzero only for poles whose order is a multiple of \(k\).

Note that for every \(k\geq2\) there is no residue theorem; that is, the sum of the \(k\)-residues of the poles on a compact Riemann surface need not vanish.

2.2 Strata of \(k\)-differentials↩︎

Let \(\mu\) be a partition of \(k(2g-2)\). We define the stratum \({\Omega^{k}\mathcal{M}}_{g}(\mu)\) of \(k\)-differentials of type \(\mu\) to be the space of primitive \(k\)-differentials with marked zeros and marked poles of orders prescribed by \(\mu\). Recall that a \(k\)-differential is primitive if it is not the global power of a lower-order differential. Except for a few cases (see Diaz-Marin?, getaquad?, GT-k-residue?), these strata are nonempty and form smooth complex orbifolds of dimension \(2g-2+|\mu|\).

In genus zero, it is easy to describe the partitions \(\mu\) for which the strata of primitive \(k\)-differentials \({\Omega^{k}\mathcal{M}}_{0}(\mu)\) are nonempty: they are precisely those \(\mu\) for which the greatest common divisor of its parts is coprime to \(k\).

Suppose that \(\mu=(a_{1},\dots,a_{n},-b_{1},\dots,-b_{p},-c_{1},\dots, -c_{r})\) with \(a_{i}>-k\), each \(b_{i}\) divisible by \(k\), and each \(c_{i}\) not divisible by \(k\). We define the residual space \(\mathcal{R}_{p}^{k}\) to be the complex vector space \(\mathbb{C}^{p}\). Each stratum \({\Omega^{k}\mathcal{M}}_{g}(\mu)\) is endowed with a residual map \[\label{eq:resmap} \mathop{\mathrm{Res}}\colon{\Omega^{k}\mathcal{M}}_{g}(\mu) \to \mathcal{R}^{k}_{p} : \zeta \mapsto (\mathop{\mathrm{Res}}_{p_{1}}\zeta,\dots,\mathop{\mathrm{Res}}_{p_{p}}\zeta)\tag{5}\] which assigns to each \(k\)-differential \(\zeta\) the sequence of its \(k\)-residues at the poles \(p_{i}\) of orders divisible by \(k\). This map defines the isoresidual fibration on the stratum \({\Omega^{k}\mathcal{M}}_{g}(\mu)\).

In the case of abelian differentials (i.e., \(k=1\)), it is well known (see, for example, AMTransSurf?) that the strata admit local coordinates, called period coordinates, given by integration of the differential along a basis of the first homology of the surface without the poles, relative to the zeros of the differential. A similar description holds for strata of \(k\)-differentials with \(k\geq2\) (see BCGGM3?). Roughly speaking, the coordinates are given by an eigenspace for the cyclic action on the canonical cover of the \(k\)-differentials.

2.3 \((1/k)\)-translation structures↩︎

Recall that to each \(k\)-differential \(\zeta\) there is an associated well-defined \((1/k)\)-translation structure. It is obtained by integrating a \(k\)-th root of \(\zeta\), and is well defined up to a rotation by an angle of \(2\pi/k\). This yields a structure consisting of translations together with rotations by angles that are multiples of \(2\pi/k\), with singularities whose cone angles are also multiples of \(2\pi/k\).

Note that a zero of order \(a_{i}\geq -k+1\) corresponds to a conical singularity with angle \(\frac{a_{i}+k}{k}2\pi\). A saddle connection is a geodesic arc joining two conical singularities.

3 Resonance stratification↩︎

We review the stratification of the residue space introduced for quadratic differentials in getaquad? and extended to all \(k\geq3\) in GT-k-residue?.

3.1 Periods of saddle connections↩︎

Given a \(k\)-differential \(\zeta\) in \(\Omega^{k}\mathcal{M}_{0}(a_{1},a_{2},-b_{1},\dots,-b_{p})\) with \(k\)-residues \((R_{1},\dots,R_{p})\), the lengths of its saddle connections are proportional to the norms of partial sums of \(k\)-th roots of the \(R_{i}\).

Indeed, any closed saddle connection \(\gamma\) decomposes \(\mathbb{CP}^{1}\) into two connected components, one of which is a translation surface bounded by \(\gamma\). Applying the residue theorem to this component shows that the length of \(\gamma\) coincides with the modulus of the sum of the \(1\)-residues at the poles contained in this component. Since these \(1\)-residues are \(k\)-th roots of the corresponding \(k\)-residues, the length of \(\gamma\) is proportional to a partial sum of the \(k\)-th roots of \(R_{1},\dots,R_{p}\).

The case of saddle connections joining two distinct conical singularities is slightly more involved. Since there are at most two singularities whose order is not a multiple of \(k\), cutting along such a saddle connection \(\gamma\) yields a translation surface with two boundary saddle connections of period \(z\) and \(-\epsilon z\), where \(\epsilon\) is a \(k\)-th root of unity. It then follows from the residue theorem that \((1-\epsilon)z+\sum\limits_{j=1}^{p} r_{j} =0\), where \(r_{1},\dots,r_{p}\) are again \(k\)-th roots of the \(k\)-residues \(R_{1},\dots,R_{p}\).

Hence, the degeneration of any saddle connection corresponds to the vanishing of certain sums of \(k\)-th roots of the \(k\)-residues. We will encode these conditions in the definition of the resonance locus below.

3.2 Resonance stratification of the residual space↩︎

Let \(W_{k}\) be \(\lbrace{ 0\rbrace} \cup \lbrace{ e^{\frac{2\ell i\pi}{k}}~|~\ell \in \mathbb{Z}/k\mathbb{Z} \rbrace}\). In \(\mathbb{C}^{p}\), a resonance hyperplane is the set of points \((r_{1},\dots,r_{p})\) satisfying a nontrivial equation of the form \[\sum\limits_{j=1}^{p} w_{j}r_{j}=0,\quad w_{j}\in W_{k}\,.\] The collection \(H_{k,p}\) of resonance hyperplanes defines a complex hyperplane arrangement in \(\mathbb{C}^{p}\).

We remark that each stratum of the resonance stratification is an algebraic subset of \(\mathcal{R}^{k}_{p}\).

Next, we give a description of the resonance locus by using the polynomial \(P\) introduced in Equation 2 . First let us prove that this is in fact a polynomial.

Observe that \(P(R_{i_1},\ldots,R_{i_d})\) vanishes if and only if \((R_{1},\dots,R_{p})\) belongs to the resonance locus. This observation motivates the following definition, which is used to rule out certain degenerate cases in the proof of Theorem 2.

3.3 The residual systole and flat continuation↩︎

In each resonance stratum, we define a residual systole to control deformations of \(k\)-differentials.

We now introduce a flat-geometric variant of analytic continuation.

In the isoresidual cover, a generic fiber is an isoresidual fiber lying over a point in the generic stratum of the resonance stratification; that is, over a point in the complement of the resonance locus.

4 The multi-scale compactification of isoresidual loci↩︎

In this section, we recall the basics of the multi-scale compactification for strata of \(k\)-differentials, as studied in CoMoZaArea? and based on the works BCGGM3?, BCGGM2?. This framework allows us to describe the closure of loci of \(k\)-differentials whose \(k\)-th roots of \(k\)-residues satisfy certain linear relations. Moreover, we discuss properties of multi-scale \(k\)-differentials contained in the closure of the isoresidual loci within this compactification. Since Theorem 15 in this section applies to any genus, we denote by \({\Omega^{k}\mathcal{M}}_{g}(\mu)\) a stratum of \(k\)-differentials of genus \(g\geq0\) with signature \(\mu\) when the results hold in these general cases.

4.1 The multi-scale compactification of strata↩︎

The multi-scale compactification \({\rm MS}^{k}(\mu)\) of the projectivized strata \({\mathbb{P}}{\Omega^{k}\mathcal{M}}_{g}(\mu) \mathrel{\vcenter{:}}= {\Omega^{k}\mathcal{M}}_{g}(\mu) / \mathbb{C}^*\) is constructed in CoMoZaArea?. Here, we follow chge? to give the reader an introduction to multi-scale \(k\)-differentials.

In this definition, \((X,{\boldsymbol{z}})\) is a stable pointed nodal Riemann surface of genus \(g\), and \(\zeta\) consists of a non-identically-zero \(k\)-differential \(\zeta_{i}\) on each irreducible component \(X_{i}\) of \(X\). The total order \(\preccurlyeq\) compares any two irreducible components of \(X\) and encodes information about the vanishing rates of differentials from nearby smooth surfaces as they degenerate to the sub-surfaces \(X_i\). This order induces a level structure on the dual graph of \(X\), which is referred to as a level graph.

Recall that the sum of the orders of the \(\zeta_{i}\) at the two branches of every node is equal to \(-2k\), and in the case of poles of order \(-k\), the two \(k\)-residues satisfy \(R_{1}+ (-1)^{k}R_{2}=0\). If a node has two poles of order \(k\) at its branches, the corresponding edge in the dual graph is called horizontal; otherwise, it is called vertical. At a vertical edge, if the multi-scale differential has a zero of order \(n\geq0\) at the upper nodal point, the number \(n+k\) is called the prong number of the node. This number gives the count of horizontal directions (up to a \(k\)-th root of unity) at the node.

Next, we discuss the \(k\)-prong-matching \(\sigma\) and the global \(k\)-residue condition, which differs from the abelian case for \(k\geq2\). Given a vertical edge \(e\) of the enhanced level graph \(\Gamma\), a (local) \(k\)-prong-matching \(\sigma_e\) is defined as a cyclic, order-reversing bijection between the \(k\)-prongs at the upper and lower ends of \(e\). A (global) \(k\)-prong-matching is a collection \(\sigma = (\sigma_e)_{e \in E(\Gamma)^v}\) of local \(k\)-prong-matchings at every vertical edge.

Consider a level \(L\) and a component \(Y\) of the part \(\Gamma_{>L}\) of sub-surfaces lying strictly above \(L\) in \(\Gamma\). We say that the restriction of the multi-scale \(k\)-differential \((X,\zeta)\) to \(Y\) is of abelian type if this restriction is the \(k\)-th power of a multi-scale abelian differential (see chge? for details). If \(Y\) is not of abelian type, or if it contains a pole of \(\zeta\), then there are no conditions on the residues at the poles connecting \(Y\) with the rest of \(X\). If \(Y\) is of abelian type and does not contain a pole, it must satisfy the (usual) global \(k\)-residue condition:

• \(P(\mathop{\mathrm{Res}}_{e_{i}}(\zeta)) = 0\) ,

where \(P\) is the polynomial defined in Equation 2 .

4.2 The closure of loci of \(k\)-differentials with linear residue conditions↩︎

One can define a similar residual map from the multi-scale compactification \({\rm MS}^{k}(\mu)\) to the projective residue space \(\mathbb{P} \mathcal{R}^{k}_{p} = (\mathcal{R}^{k}_{p} \setminus \{ 0 \}) / \mathbb{C}^{*}\). In general, this map is only a rational map; for example, it is undefined on the locus of residueless differentials.

Following CGPT1?, we now describe the closure of isoresidual fibers in the moduli space of multi-scale \(k\)-differentials. This discussion applies to arbitrary genus and partitions \(\mu\). Moreover, we work in the more general setting where a linear subspace of the \(k\)-th roots of the \(k\)-residues is fixed.

Consider a stratum \({\mathbb{P}}{\Omega^{k}\mathcal{M}}_{g}(\mu)\) of meromorphic \(k\)-differentials of genus \(g\) with signature \(\mu\). Let \(\Lambda\) be a linear subspace of the cover \(\mathbb{C}^{p}\) of the residual space \(\mathcal{R}_{p}^{k}\) given in Equation 6 . Consider the subspace \(\mathcal{F}_{\Lambda}\) consisting of \(k\)-differentials whose \(k\)-residues lie in the projection of \(\Lambda\). We define \({\Lambda}^{\vee}\subset (\mathbb{C}^{p})^{\vee}\) to be the vector space of homogeneous linear equations satisfied by all residue tuples in \(\Lambda\).

We aim to characterize when a multi-scale \(k\)-differential \((X,\zeta,\sigma)\) lies in the closure of \(\mathcal{F}_\Lambda\). To this end, we introduce the generalized global \(k\)-residue condition imposed on \((X,\zeta,\sigma)\) by \(\Lambda^{\vee}\), which we denote by \(\mathcal{E}_{\Lambda}^{k}\)-GRC. Let \(q_1,\ldots, q_p\) be the marked poles in \(\Gamma\). For each \(q_i\), add a new vertex at level \(\infty\) with a marked pole \(q'_i\), and replace the original \(q_i\) with an edge connecting it to this new vertex. The new vertex can be regarded as a semistable rational component carrying a pole at \(q'_i\) of the same order as \(q_i\). Denote by \(\Gamma'\) the resulting level graph. For every finite level \(L\) of \(\Gamma'\), let \(Y_1,\ldots, Y_s\) denote the connected components of \(\Gamma'_{>L}\).

If \(q_{i}\) does not appear in any equation of \(\Lambda^{\vee}\), we say that it is a free pole. The following conditions then hold:

• For every \(Y_i\) of abelian type, one of the following conditions holds:

  1. \(Y_{i}\) contains a free pole.

  2. For all equations \(f\in \Lambda^{\vee}\) that can be written in the form \[f = \sum_{i=1}^s a_i \left(\sum_{q'_j\in Y_i} r_{q'_j}\right)\,,\] we require that there exist roots \(r_{e_{j}}\) of the residues \(R_{e_{j}}\) such that \[\sum_{i=1}^s a_i \left(\sum_{e_j \in Y_{i, L}} r_{e_j} (\zeta)\right) = 0\,,\] where the inner summation ranges over the lower endpoints of the edges in \(Y_i\) that connect to vertices at level \(L\).

We remark that, up to linear combinations, there are only finitely many non-trivial conditions imposed by the \(\mathcal{E}_{\Lambda}^{k}\)-GRC. To see this, note that if two independent relations in condition (2) involve the same subset of \(Y_i\) with nonzero coefficients, their combination can produce another relation involving a smaller subset of \(Y_i\). Repeating this process, one can reduce to a finite set of relations that form an echelon form at each level.

In what follows, we discuss examples to illustrate the above description and the \(\mathcal{E}_{\Lambda}^{k}\)-GRC.

We can now state the main result of this section as follows.

The proof is very similar to that of CGPT1?; we therefore only sketch the main idea.

4.3 The closure of isoresidual fibers↩︎

Since we are interested in isoresidual fibers, we restrict to the case \(\Lambda = \mathbb{C}^{\ast} \cdot \lambda\), where \(\lambda = (\lambda_1,\ldots, \lambda_p)\). In this case, we abuse notation and also write \(\lambda\) instead of \(\Lambda\). Note that when \(\lambda=(0,\dots,0)\), we have \(\mathcal{E}_{\lambda}^{\vee}=\mathcal{R}_{p}^{\vee}\). If \(\lambda\) has at least one nonzero entry, then \(\mathcal{E}_{\lambda}^{\vee}\) is the image by the projection 6 from \(\mathbb{C}^{p}\) to \(\mathcal{R}_{p}^{\vee}\) of the generated by equations of the form \[\label{eq:relres} f_{i,j}(q_{1},\dots,q_{p})\mathrel{\vcenter{:}}=\lambda_{i}r_{q_{j}}-\lambda_{j}r_{q_{i}}=0\,.\tag{7}\]

Analogous to the case of abelian differentials, we obtain the following results.

5 Counting via intersection theory↩︎

In this section, we use intersection theory to prove Theorems 2, 5, and 6, which determine the cardinality of the fibers in the isoresidual cover of strata of \(k\)-differentials on the Riemann sphere with two singularities of orders not divisible by \(k\).

5.1 Preliminaries↩︎

Given a stratum of abelian differentials \({\Omega\mathcal{M}}_{0}(\mu)\) of genus zero with a unique zero, the function \[f(a,p)\mathrel{\vcenter{:}}=\frac{a!}{(a-(p-2))!}\] represents the degree of the top intersection product \(\int_{\text{MS}(\mu)}\xi^{p-2}\), where \(\xi=c_1(\mathcal{O}_{\text{MS}(\mu)}(1))\) is the universal line bundle class of the multi-scale compactification \(\text{MS}(\mu)\mathrel{\vcenter{:}}=\mathbb{P}\Xi\overline{\mathcal{M}}_{0,p+1}(\mu)\) as proved in ChPrIso?. The purpose of this subsection is to extend this result to the setting of \(k\)-differentials and to study the properties of the counting function that generalizes \(f\). This relies on the notion of multi-scale \(k\)-differentials recalled in Section 4.1.

The function \(f_{k}(a,n)\) satisfies two identities that will be used in the proof of Theorem 2. For the first identity, let \(f_k^{(r)}(a,n)\) denote the \(r\)-th derivative of \(f_k(a,n)\) with respect to \(a\), with the convention that \(f_k^{(r)}(a,n)=0\) whenever \(r<0\) or \(r>n-2\).

The second identity is a sign-alternating sum of polynomials that equals the zero polynomial. Recall that given a subset \(I\) of \(\{1,\ldots,p\}\) the number \(c_{1,I} = a_1-\sum_{i\in I}b_i + k\) has been introduced in ??

5.2 The generic case↩︎

Now we prove Theorem 2 which computes the degree of the isoresidual cover. To do this, we replicate the method of ChPrIso?, using intersection theory on the multi-scale compactification \(\text{MS}^k(\mu)\) recalled in Section 4.

Fix a projectivized stratum \(\mathbb{P}\Omega^k(\mu)\) of primitive \(k\)-differentials on the Riemann sphere, where \(\mu=(a_1,a_2,-b_1,\ldots,-b_p)\) is a partition of \(-2k\). We denote by \(z_i\) the singularity of order \(a_i\), for \(i=1,2\), and the poles \(p_1,\ldots,p_p\), respectively. The pole orders \(b_i \geq k\), with each \(b_{i}\) divisible by \(k\) and each \(a_{i}\) relatively prime to \(k\). Since the \(b_i\) are positive multiples of \(k\), the residual map 5 induces a rational map defined by the \(k\)-residues \(\mathcal{R}=[R_1:\cdots:R_p]\) of the \(k\)-differentials.

Note that as mentioned at the beginning of the proof, the same procedure can be repeated using the relation 10 with respect to the zero \(z_2\). In that case one obtain \[\sum_{c_{2,I}>0} c_{2,I}f_k(a_2,|I|+1)f_k(a_1,p+1-|I|)\,.\]

We can now prove the special case in which all pole orders are equal to \(k\), as stated in Corollary 3.

5.3 The resonant case↩︎

Now we consider the case of \(k\)-residues lying in a resonance hyperplane. This hyperplane is defined by a \(resonant\) subset \(I\subset \{1,\ldots,p\}\), indexing \(k\)-residues such that \(P(\{R_i\}_{i\in I_1})=0\). Recall that the notation ?? is \(c_{i,I}=a_i + k -\sum\limits_{j\in I}b_j\). The following proposition recalls Theorem 5.

Before giving the proof of this result we illustrate the fact that \(\operatorname{Ab}_{\mathcal{R}}(I)\) may be \(>1\).

Finally, we prove the most general case stated in Theorem 6 recalled in the following proposition.

6 Flat geometric interpretation and examples↩︎

In this section, we first sketch certain flat geometric ideas to prove Corollary 3, computing the isoresidual degree for strata of the form \(\Omega^{k}\mathcal{M}_{0}(a_{1},a_{2},\rec[-k][p])\) with \(k \geq 2\). Then we give an alternative formula in Section 6.4 and finally we give some exemples in Section 6.5, both in the interesction theoretic setting and in the flat setting.

6.1 Some degenerations.↩︎

First, given \(\epsilon>0\), we define \[U_{\epsilon} = \left\{(R_{1},\dots,R_{p})\in (\mathbb{C}^{\ast})^{p} : |R_{j+1}| \leq \epsilon|R_{j}|\; \forall 1 \leq j \leq p-1 \right\}\,.\] Recall that the resonance locus was introduced in Definition 7. The relevance of \(U_{\epsilon}\) is established by the following result.

The local model of a pole of order \(-k\) is the end of a cylinder bounded by one or several saddle connections that meet at corners with an angle \(\pi\). We refer to this cylinder as the domain of the corresponding pole. We will show that, under generic conditions on the arguments of a configuration of \(k\)-residues lying in \(U_{\epsilon}\), each such cylinder is bounded by a unique closed saddle connection incident to a unique conical singularity.

We say that the cylinder of the \(j\)-th pole is attached to the conical singularity \(A_{i}\) (of order \(a_{i}\)) when the endpoints of its boundary saddle connection coincide with \(A_{i}\).

The \(k\)-differentials in the same isoresidual fiber over \(U_{\epsilon}\) can thus be described in terms of the way these cylinders are attached to the conical singularities. We distinguish two cases, depending on whether the corresponding flat surface has one or two conical singularities.

6.2 The case \(a_{2}<-k\): one conical singularity↩︎

For such strata, Corollary 3 states that for a generic configuration of \(k\)-residues, the number of \(k\)-differentials in the corresponding isoresidual fiber is \[\frac{a_{1}!_{(k)}}{(a_{1}+k-kp)!_{(k)}}\,.\] The following sketch of the proof parallels the one given in the abelian case in GeTaIso?.

We consider a fiber over a configuration of \(k\)-residues in \(U_{\epsilon}\) and proceed by induction on \(p\). In the case \(p=1\), the isoresidual fiber contains only a single \(k\)-differential. The flat surface associated with this \(k\)-differential consists of a cone of angle \(2\pi\frac{a_{1}+k}{k}\) to which a polar region with the given residue is attached.

Now suppose the formula has been proved for \(p-1\) poles. We then add a new pole of order \(-k\), whose residue is real and has magnitude at most \(\epsilon\) times smaller than the other residues. We place the new pole at a separatrix of slope a multiple of \(\frac{2\pi}{k}\) emanating from the singularity \(z_1\) of order \(a_{1}\) on the flat surfaces of the isoresidual fiber in the stratum \({\Omega^{k}\mathcal{M}}_{0}(a_{1},a_{2};-b_{1},\dots,-b_{p-1})\) as follows.

In terms of flat geometry, consider a separatrix \(v\) of length \(r_{p}\) starting at \(z_1\). Note that the points lying on the line orthogonal to \(v\) form a band that contains no conical singularities in its interior and intersects \(z_{1}\) only at the starting point of \(v\). We remove the upper half-infinite portion of this band and glue the two half-infinite boundary rays together by translation. Finally, we attach a half-infinite cylinder along \(v\) to obtain the desired surface. In the analytic picture, this corresponds to smoothing a twisted \(k\)-differential obtained by gluing a \(k\)-differential in \({\Omega^{k}\mathcal{M}}_{0}(a_{1}+k;-k,-a_{1}-2k)\) to a differential in \({\Omega^{k}\mathcal{M}}_{0}(a_{1},a_{2};-b_{1},\dots,-b_{p-1})\) in such a way that the residue at the pole of order \(-k\) is real.

On each flat surface to which we attach the horizontal pole, there are \(a_{1}+k\) horizontal directions where we can place the new pole of order \(-k\). Hence, the total number of possibilities is \[\begin{align} (a_{1}+k) \frac{a_{1}!_{(k)}}{(a_{1}+k-k(p-1))!_{(k)}} &=& \frac{(a_{1}+k)!_{(k)}}{(a_{1}+k-k(p-1))!_{(k)}}\\ &=& \frac{(a_{1}+k)!_{(k)}}{((a_{1}+k)+k-kp)!_{(k)}}\,. \end{align}\]

6.3 The case \(a_{1},a_{2}>-k\): two conical singularities↩︎

We now sketch two flat-geometric proofs.

6.3.1 The first proof↩︎

We perform induction on the number \(p\) of poles. One has that the degree is \(1\) in any stratum \(\Omega^{k}\mathcal{M}_{0}(a_{1},a_{2},-k)\) with \(0>a_{1},a_{2}>-k\). Suppose that Corollary 3 is proved for \(\Omega^{k}\mathcal{M}_{0}(a_{1},a_{2},\rec[-k][p])\). Consider an isoresidual fiber with the last residue \(R_{p}\) real and such that the residues belong to \(U_{\epsilon}\) for sufficiently small \(\epsilon\). Let \(R_{p}\) tend to zero. Note that this does not change the number of differentials in the fiber since we are in \(U_{\epsilon}\).

Now, either the pole was attached to \(a_{1}\), in which case there are \(\binom{p-2}{s-1} (a_{1}-k)!_{(k)} \cdot a_{2}!_{(k)}\) limits, or to \(a_{2}\), in which case there are \(\binom{p-2}{s} a_{1}!_{(k)} \cdot (a_{2}-k)!_{(k)}\) limits. In the first case, we glue the pole of order \(k\) at the zero of order \(a_{1}-k\) in one of the \(a_{1}\) horizontal directions. Similarly, in the second case, we glue it at the zero of order \(a_{2}-k\).

Since there are \(a_{i}\) horizontal directions at a zero of order \(a_{i}-k\), the total number of such differentials is \[\binom{p-2}{s-1} \cdot a_{1}\cdot (a_{1}-k)!_{(k)} \cdot a_{2}!_{(k)} + \binom{p-2}{s} \cdot a_{2}\cdot a_{1}!_{(k)} \cdot (a_{2}-k)!_{(k)} = a_{1}!_{(k)} \cdot a_{2}!_{(k)} \cdot \binom{p-1}{s}\,.\]

6.3.2 The second proof↩︎

We introduce the notations \(a_{1}=k \cdot d_{1}+\overline{a}_{1}\) and \(a_{2}=k \cdot d_{2}+\overline{a}_{2}\), where \(-k<\overline{a}_{1}<-\frac{k}{2}\) and \(-\frac{k}{2}<\overline{a}_{2}<0\). We have \(d_{1}+d_{2}=p-1\). The degree of the isoresidual cover is then \[\binom{p-1}{d_{1}} \cdot \prod\limits_{1 \leq i \leq d_{1}} (\overline{a}_{1}+k\cdot i) \cdot \prod\limits_{1 \leq j \leq d_{2}} (\overline{a}_{2}+k\cdot j)\,.\]

Consider a configuration \((R_{1},\dots,R_{p})\) of \(k\)-residues that belong to \(U_{\epsilon}\) for some small \(\epsilon\). We consider the path in \(U_{\epsilon}\) defined by \((R_{1},tR_{2},\dots,tR_{p})\) with \(t \in ]0,1]\). In the limit \(t \rightarrow 0\), the \(k\)-differentials of the corresponding isoresidual fibers degenerate to multi-scale differentials whose dual graph is a cherry graph (each irreducible component is a complex line):

• A top component contains the pole of order \(-k\) with \(k\)-residue \(R_{1}\) and two nodal singularities of orders \(\overline{a}_{1}\) and \(\overline{a}_{2}\).

• A bottom component contains \(z_{1}\), \(d_{1}\) poles of order \(-k\), and a nodal singularity of order \(-2k-\overline{a}_{1}\);

• Another bottom component contains \(z_{2}\), \(d_{2}\) poles of order \(-k\), and a nodal singularity of order \(-2k-\overline{a}_{2}\).

There is only one possible shape for the top component. We then apply the formula proved in Section 6.2 to count the possible configurations for the two bottom components. There are \(\binom{p-1}{d_{1}}\) ways to split the \(p-1\) poles of order \(-k\) corresponding to the \(k\)-residues \(tR_{2},\dots,tR_{p}\). Multiplying these factors yields the degree formula.

6.4 An alternative formula↩︎

Given that a singularity of order \(a>-k\) of a \(k\)-differential can be interpreted as a conical singularity of angle \(\frac{a+k}{k}2\pi\) in the corresponding flat metric, the degree formula can be written as follows.

6.5 Examples↩︎

We will describe the \(k\)-differentials in certain isoresidual fibers to illustrate the constructions and computations discussed in the preceding section.

In the quadratic case (\(k=2\)), we consider quadratic residues in \({\mathbb{R}}_{>0}\) to describe the differentials in the fiber using the incidence graphs introduced in getaquad?. These graphs are embedded in the sphere, with vertices corresponding to the components of the differential obtained by cutting along its saddle connections. For each saddle connection, there is an edge connecting the two corresponding vertices. In the case of two zeros, getaquad? shows that the graph is a cycle of odd length with trees attached at its vertices. Each face corresponds to a zero; when both zeros have the same order, we assume that \(a_{1}\) corresponds to the compact face.

In the following examples, we illustrate the formulas from Theorem 5 and 6 on specific configurations of residues for cubic and quartic differentials. For each of these configurations, the isoresidual fiber is known to be empty according to the classification of obstructions GT-k-residue?. These examples thus provide a consistency check for our formula in the presence of known obstructions to the residue realization problem (and vice versa).

7 Applications to spherical geometry↩︎

The generalization of the uniformization theorem to surfaces endowed with metrics having conical singularities of prescribed angles remains an open problem in the case of metrics with constant positive curvature (see Er1? for a general survey). This problem has been approached using various methods, including PDEs (see, for example, CL?), hypergeometric functions (see EGT?), and algebraic geometry (see LSX?). Recently, progress has been made by considering specific subclasses of cone spherical metrics with restricted monodromy. In particular, when the monodromy is coaxial—meaning that the monodromy group, a subgroup of \(\text{SO}(3)\), is confined to a one-parameter family of rotations around a single axis—Eremenko Erm? obtained a complete characterization of the configurations of conical angles that can be realized by such metrics.

A slightly larger class of cone spherical metrics has also been considered. A metric is said to have dihedral monodromy if its monodromy preserves a great circle of the model sphere. It was shown in SongSpher? that these metrics are defined by (multi-valued) developing maps of the form \(z \mapsto \exp\left(\int^{z} \sqrt{q}\right)\), where \(q\) is a meromorphic quadratic differential with at most double poles and such that every period of \(\sqrt{q}\) is real. In particular, the quadratic residue at each double pole must be a positive real number. The corresponding spherical metric is obtained as the pullback of \(\frac{4|dw|^{2}}{(1+|w|^{2})^{2}}\). The interpretation of the singularities of \(q\) is as follows:

• a zero of order \(a\) corresponds to a conical singularity of angle \((2+a)\pi\);

• a simple pole corresponds to a conical singularity of angle \(\pi\);

• a double pole with quadratic residue \(\frac{\theta^{2}}{4\pi^{2}}\) corresponds to a conical singularity of angle \(\theta\).

A complete characterization of the configurations of conical angles realized by metrics with dihedral monodromy was obtained in getasphere?. More recently, the topology of the moduli space of such metrics with constrained monodromy has been studied in LX?.

For metrics on the sphere with dihedral monodromy, certain configurations of conical angles can be realized by only finitely many distinct cone spherical metrics. In this case, the counting problem reduces to enumerating the number of elements in an isoresidual fiber. Specifically, for quadratic differentials with at most double poles, Theorem 2 provides such a count.

Note that, since \(a+b=2n\), the formula in the theorem remains unchanged if \(a\) is replaced by \(b\). Moreover, the hypothesis that \(a\) and \(b\) are odd implies that all these metrics are strictly dihedral; that is, they are not coaxial.