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Cohomology classes of strata of differentials
We introduce a space of stable meromorphic differentials with poles of pre-scribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplici-ties of zeros of the differential. The main goal of this chapter is to compute the Poincaré-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them.
In a second part of the last section of the chapter we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
This chapter is mostly based on the paper .
Different formulations of the problem
Stratification of the Hodge Bundle. Let g 1. Let Mg be the space of smooth curves of genus g.
The Hodge bundle, Hg ! Mg is the vector bundle whose fiber over a point [C] of Mg is the space of holomorphic differentials on C. A point of Hg is then a pair ([C]; ), where C is a curve and a differential on C. We will denote by PHg ! Mg the projectivization of the Hodge bundle.
Notation 2.1.1. Let Z (for zeros) be a vector (k1;k2; : : : ;kn) of positive integers satisfying ki = 2g – 2: i=1
We will denote by PHg(Z) the subspace of PHg composed of pairs ([C]; ) such that is a differential (defined up to a multiplicative constant) with zeros of orders k1; : : : ;kn.
The locus PHg(Z) is a smooth orbifold (or a Deligne-Mumford stack), see for instance, . However, neither PHg, nor the strata PHg(Z) are compact.
The Hodge bundle has a natural extension to the space of stable curves: Hg ! Mg:
We recall that abelian differentials over a nodal curve are allowed to have simple poles at the nodes with opposite residues on the two branches.
The space PHg is compact and smooth, and we can consider the closures PHg(Z) of the strata inside this space. Computing the Poincaré-dual cohomology classes of these strata is our motivating problem. In the present Chapter we solve this problem and present a more general computation in the case of meromorphic differentials.
On a fixed smooth curve C with one marked point x consider a family of meromorphic differentials with one pole of order p at x, such that the leading coefficient of the differential at the pole tends to 0. In order to construct a compact moduli space of meromorphic differentials we need to decide what the limit of a family like that should be. One natural idea is to include differ-entials with poles of orders less than p in the moduli space. It turns out, however, that a more convenient way to represent the limit is to allow the underlying curve to bubble at x; in other words, to allow differentials defined on semi-stable curves.
A semi-stable curve is a nodal curve with smooth marked points such that ev-ery genus 0 component of its normalization contains at least two marked points and preimages of nodes (instead of at least three for stable curves). In the example above, the limit of the family would be a meromorphic differential defined on a semi-stable curve with one unstable component and on marked point x on it. The curve maps to C under the contraction of the unstable component. The meromor-phic differential still has a pole of order exactly p at x.
Definition 2.1.2. Let n;m 2 N and let P (for poles) be a vector (p1; p2; : : : ; pm) of positive integers. A stable differential of type (g;n;P) is a tuple (C;x1; : : : ;xn+m; ) where (C;x1; : : : ;xn+m) is a semi-stable curve with n + m marked points and is a meromorphic differential on C, such that the differential has no poles outside the m last marked points and nodes; the poles at the nodes are at most simple and have opposite residues on the two branches; if pi > 1 then the pole at the marked point xn+i is of order exactly pi; if pi = 1 then xi can be a simple pole, a regular point, or a zero of any order; the group of isomorphisms of C preserving and the marked points is finite.
Definition 2.1.3. A family of stable differentials is a tuple (C ! B; 1; : : : ; n; ) where (C ! B; 1; : : : ; n) is a family of marked semi-stable curves and is a meromorphic section of the relative dualizing line bundle !C=B such that for each geometric point b of B, the tuple (Cb; 1(b); : : : ; n(b); jCb ) is a stable differential.
The stack Hg;n;P of stable differentials of type (g;n;P) is the category of fami-lies of stable differentials of type (g;n;P), fibered over the category of C-schemes.
Proposition 2.1.4. The moduli space Hg;n;P is a smooth Deligne-Mumford (DM) stack. It is of dimension 4g – 4 + pi if P is non-empty and 4g – 3 otherwise.
The space Hg;n;P carries a natural C -action given by the multiplication of the differential by non-zero scalars. Besides, there exists a forgetful map Hg;n;P ! Mg;n+m that maps a family stable differentials to the stabilization of its underlying family of semi-stable curves. However, the space Hg;n;P does not have a natural vector bundle structure because there is no natural definition of the sum of two differentials with fixed orders of poles.
We will construct a partial coarsification of Hg;n;P that has the structure of an orbifold cone over Mg;n+m.
Proposition 2.1.5. There exists a unique DM stack Hg;n;P fitting in the following commutative triangle Hg;n;P /Hg;n;P
Definition 2.1.6. The space Hg;n;P is the called the space of stable differentials.
Proposition 2.1.7. The space of stable differentials is an orbifold cone over Mg;n+m. Besides the space Hg;n;P and its projectivization are normal.
We prove these propositions in Section 2.2, where we will also give a definition of an orbifold cone. At present it suffices to note that the cone structure on Hg;n;P allows one to define a projectivization PHg;n;P, a tautological line bundle over the projectivization, and the Segre class. Besides, the morphism Hg;n;P ! Hg;n;P is C -equivariant.
Remark 2.1.8. The stack Hg;n;P can be endowed with the structure of an orbifold cone over a different moduli space Mg;n;P. The space Mg;n;P is a Z=(pi – 1)Z -i=1 gerb over Mg;n+m. The fibers of Hg;n;P ! Mg;n;P are vector spaces, but the C – action on these spaces has nontrivial weights.
One can define the projectivization of Hg;n;P and the tautological line bundle over this projectivization. Then we have a map PHg;n;P ! PHg;n;P which is a bijection between the geometric points of these two stacks.
Therefore we have natural isomorphisms H (PHg;n;P;Q) ’ H (PHg;n;P;Q) and A (PHg;n;P;Q) ’ A (PHg;n;P;Q). Thus, all the results of this text are valid for both spaces.
While the space Hg;n;P is the more natural choice for the moduli space of dif-ferentials, in the present Chapter we prefer to work with Hg;n;P in order to have Mg;n+m as the base of our cone.
Notation 2.1.9. Let P = (p1; : : : ; pm) be a vector of positive integers and Z = (k1; : : : ;kn) a vector of nonnegative integers. We denote by Ag;Z;P Hg;n;P, the locus of stable differentials (C;x1; : : : ;xn+m; ) such that C is smooth and has ze-ros exactly of orders prescribed by Z at the first n marked points. The locus Ag;Z;P is invariant under the C -action. We denote by PAg;Z;P the projectivization of Ag;Z;P.
Table of contents :
Chapter 1. Introduction
1.1. Les surfaces de translation
1.2. Espaces des modules de courbes
1.3. Anneaux tautologiques
1.4. Stratification des espaces des modules de courbes stables
1.5. Stratification des espaces de différentielles
1.6. Différentielles d’ordres supérieurs et classes de Prym-Tyurin
1.7. Nombres d’Hurwitz
1.8. Cycles de double ramification
Chapter 2. Cohomology classes of strata of differentials
2.1. Different formulations of the problem
2.2. Stable differentials
2.3. The induction formula
2.4. Examples of computation
2.5. Relations in the Picard group of the strata
Chapter 3. Prym-Tyurin classes and loci of degenerate differentials
3.1. Prym-Tyurin classes
3.2. Space of admissible n-differentials
3.3. Bergman tau function and Hodge class on PM
3.4. An alternative computation of deg
3.5. Prym-Tyurin differentials on b C and holomorphic n-differentials on C
Chapter 4. Hurwitz numbers and intersection in spaces of differentials
4.1. Some families of Hurwitz numbers
4.2. From stable differentials to stable maps
4.3. End of the proof of Theorem 4.1.8
4.4. Completed cycles
Chapter 5. Double ramification cycles and strata of differentials
5.1. Moduli space of r-spin structures
5.2. Double Ramification cycles
5.3. Twisted canonical divisors
Appendix A. Algebraic Stacks
A.1. Sites and sheaves
A.3. Moduli spaces of curves