Spin and hyperelliptic structures of log twisted differentials
Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected components, due to spin and hyperelliptic structures. We prove that the spin parity can be distinguished on the boundary of the log compactification. Moreover, combining the techniques of log geometry and admissible covers, we introduce log twisted hyperelliptic differentials, and prove that their moduli stack provides a toroidal compactification of the hyperelliptic loci in the open strata.
Key words and phrases:Abelian differential, log twisted differential, spin parity, hyperelliptic structure
2010 Mathematics Subject Classification:14H10 (primary), 14D23(secondary)
Let be a partition of . Consider the stratum of abelian differentials with signature parameterizing pairs , where is a smooth genus curve and is an abelian differential on whose zero type is specified by . The study of has broad connections to many fields. For example, an abelian differential defines a flat structure with conical singularities at its zeros. The behavior of geodesics under the flat metric is closely related to billiards in polygons, which has produced abundant results in dynamics and geometry. From the algebraic viewpoint, an abelian differential is determined, up to scaling, by its underlying canonical divisor. Hence the projections of these strata to the moduli space of genus curves provide a series of special subvarieties according to properties of canonical divisors. Moreover, there is a -action on induced by varying polygonal representations of abelian differentials. The closure of any -orbit (under the analytic topology on ) is known to be algebraic by the seminal work of Eskin-Mirzakhani-Mohammadi ([EM, EMM]) and Filip ([Filip]). We refer to [Zorich, MoellerTeich, Wright, Bootcamp] for surveys on abelian differentials and relevant topics.
Just as many moduli problems, it is natural to pursue a functorial compactification for the stratum . Then one can obtain information for the interior of the stratum by analyzing the boundary of the compactification. This circle of ideas has been shown to be useful. For instance, dynamical invariants of Teichmüller curves (as dimensionally minimal -orbit closures) can be computed through their intersections with divisors of the Deligne-Mumford compactification ([CMab, CMquad]). With this as a motivation, in this paper we consider a compactification of via log geometry.
1.1. Moduli of log twisted sections
Our construction of the compactification is based on the theory of stable log maps developed in [GS, Chen, AC]. We first treat a more general situation. Let be a line bundle of relative degree over a family of genus pre-stable curves with markings , and a partition of . For each , consider a family of log curves over the underlying curve obtained by pulling back the family . See Section 2.1 for the definition of log curves. We have a projection of log schemes such that
The underlying structure is the total space of the pull-back of via ;
The log structure on is given by the zero section of the line bundle together with the log structure from , see Section LABEL:ss:log-section.
A log twisted section with signature over the log scheme is a section of the projection with contact orders at the marked points given by . Note that the underlying morphism is a section of the line bundle over . Here the contact order is a generalization of the vanishing order of sections of the line bundle . It remembers the vanishing order at the marked points, even when the underlying section vanishes completely along the components of the curve containing the marked points, see (LABEL:equ:marked-pt).
Denote by the category of log twisted sections fibered over the category of log schemes. Note that algebraic stacks are category fibered over schemes rather than log schemes. In order to prove the representability of as a Deligne-Mumford stack, we use the key observation discovered in [GS, Chen, AC] that one can view as a fibered category of minimal objects over the category of schemes. We discuss the minimality of log twisted sections in Section LABEL:ss:minimality and show that
Theorem 1.1 (Theorem LABEL:thm:log-twisted-sec).
The fibered category is represented by an algebraic stack with its minimal log structure which parameterizes minimal log twisted sections. Furthermore, the forgetful morphism
is representable and finite, where is the total space of the push-forward of along , see Section LABEL:ss:log-section.
1.2. Log twisted differentials and their spin parity
Consider the situation when is given by the universal family of genus , -marked stable curves with the (relative) dualizing line bundle. Let be a partition of , which equals the degree of . A log twisted differential with signature is a log twisted section with signature with respect to the dualizing line bundle, see Section LABEL:ss:log-twist-diff. We emphasize that log twisted differential extends the notion of abelian differential functorially to the Deligne-Mumford boundary. It naturally associates to each irreducible component of a degenerate curve a (possibly meromorphic) differential up to a -scaling controlled by the log structures, see Section LABEL:ss:fiberwise-induced-diff.
Denote by the moduli stack of log twisted differentials with signature . The open stratum is the locus parameterizing (not identically zero) abelian differentials over smooth curves. In particular, can be identified with the open locus of with the trivial log structure.
For a partition of with even entries for all , the half-canonical divisor given by an abelian differential over a smooth curve defines a theta characteristic, called the spin structure. The parity of the spin structure is a smooth deformation invariant ([Atiyah, Mumford]). It follows that the loci of abelian differentials with odd and even spin structures in the stratum are disjoint. Kontsevich and Zorich ([KZ]) classified the connected components for all strata . It turns out that can have up to three connected components, distinguished by spin and hyperelliptic structures. We next generalize the spin parity to log twisted differentials and prove the following result.
Theorem 1.2 (Theorem LABEL:thm:spin).
Consider the partition with each even. Then we have a disjoint union
where (resp. ) is the open and closed substack of parameterizing log twisted differentials with even (resp. odd) spin parity.
We remark that the moduli space of log twisted differentials maps onto the Farkas-Pandharipande moduli space of twisted canonical divisors ([FP]), hence it may contain components that are entirely over the Deligne-Mumford boundary. Nevertheless, the spin parity defined above is not only for the main component of , but also for all boundary components that contain non-smoothable log twisted differentials. Therefore, and are in general reducible, consisting of other components besides the main (spin) components. In addition, depending on the spin parity, one of and contains the hyperelliptic component if it exists.
1.3. The hyperelliptic structure
An abelian differential over a smooth curve is called hyperelliptic if is hyperelliptic and , where is the hyperelliptic involution. In order to remember the involution structure over the boundary, we combine the techniques of log geometry and admissible covers, and define log twisted hyperelliptic differentials, see Section LABEL:ss:hyp-diff. Given a partition of compatible with the hyperelliptic involution, we introduce the category of log twisted hyperelliptic differentials fibered over the category of log schemes, denoted by , and prove that
Theorem 1.4 (Theorem LABEL:thm:hyp).
The fibered category is represented by a separated, log smooth Deligne-Mumford stack with its universal minimal log structure. Furthermore, the forgetful morphism to the Hodge bundle is representable and finite.
For the convenience of the construction, in the hyperelliptic case we mark all the involution fixed points, hence zero contact orders are allowed in the partition , see Section LABEL:ss:hyp-moduli.
The compactification treats not only the hyperelliptic components of , but also the hyperelliptic loci in for any partition compatible with . The log smoothness of is equivalent to the statement that the boundary of is toroidal. In particular, all log twisted hyperelliptic differentials are smoothable.
1.4. Related works and comparison
There are several recent attempts to study degeneration of abelian differentials from the classical viewpoint without using log structures, i.e., compactifying the strata in the Hodge bundle over (resp. in ) using twisted differentials (resp. twisted canonical divisors). Gendron ([Gendron]) proved the smoothability of twisted differentials when they have zero residues at all nodes. The first author ([ChenDiff]) studied twisted canonical divisors on curves of pseudo-compact type. Farkas and Pandharipande ([FP]) studied systematically the space of twisted canonical divisors and showed that it is in general reducible, which contains, besides the closure of the stratum, a number of boundary components that have dimension one less than the dimension of the stratum. Finally in [BCGGM1] a crucial global residue condition was found and used to isolate the closure of the stratum. As mentioned before, the moduli space of log twisted differentials maps onto the Farkas-Pandharipande space of twisted canonical divisors, hence it also contains extra boundary components. Nevertheless, there is a way to implant the global residue condition under the log setting to isolate the main component, which will be addressed in our future work.
Without log structures, the odd and even spin components can intersect in the boundary of the aforementioned compactifications. This phenomenon has been already observed in [EMZ, Gendron, ChenDiff].
We mention that the log twisted differential considered in this paper is closely related to the strata compactification studied by Guéré ([Guere]), whose construction also relies on the theory of stable log maps, but is motivated from the viewpoint of Gromov-Witten theory.
If one replaces the canonical line bundle by its -th power, then it is natural to consider the strata of -differentials. Setting to be the -th power of the relative canonical line bundle in Theorem LABEL:thm:log-twisted-sec, our result gives also a log compactification for the strata of -differentials parameterizing log twisted -differentials (similarly see [Guere]). The space of twisted -differentials is in general reducible and a modified global residue condition via canonical -covers can isolate the main component ([BCGGM3]).
1.5. Conventions and notations
We use capital letters such as , , and to denote log schemes or log stacks. Their underlying schemes or stacks are denoted by , , and respectively. Given a log structure over a scheme , denote by the characteristic sheaf of . All log structures considered here are fine and saturated. We refer to [KKato, log-survey] for the basics of log geometry.
This paper is heavily built on the theory of stable log maps, and we refer to [AC, Chen, GS] for general developments of stable log maps. Throughout this paper, we work over an algebraically closed field of characteristic zero.
The authors thank Dan Abramovich, Matt Bainbridge, Gavril Farkas, Quentin Gendron, Samuel Grushevsky, Jérémy Guéré, Martin Möller, Rahul Pandharipande and Jonathan Wise for stimulating discussions on related topics.
2. Log twisted sections
In this section, we introduce the general set-up of log twisted sections of a given line bundle, and study their moduli stacks.
2.1. Log curves
First recall the canonical log structure on pre-stable curves ([FKato, LogCurve]). Denote by the universal family of pre-stable curves. Note that the boundary parameterizing singular underlying curves is a normal crossings divisor in . Thus we denote by the log stack, with the underlying structure given by and the log structure given by the divisorial log structure associated to , see [KKato, (1.5)]
Similarly, we consider the boundary consisting of singular fibers with markings. The boundary is again a normal crossings divisor in . Denote by the log stack with underlying structure , and by the log structure of given by the divisorial log structure associated to the boundary . Since the morphism of the pairs
is toroidal, it induces a morphism of log stacks
Now consider a family of genus , -marked pre-stable curves . Such a family is obtained from the following cartesian diagram