Weierstrass filtration on Teichmuller curves and Lyapunov exponents

We define the Weierstrass filtration for Teichmuller curves and construct the Harder-Narasimhan filtration of the Hodge bundle of a Teichmuller curve in hyperelliptic loci and low-genus nonvarying strata. As a result we obtain the sum of Lyapunov exponents of Teichmuller curves in these strata.


Introduction
The computation of Lyapunov exponents is an important subject in the theory of Teichmüller curves. The sum for hyperelliptic connected components is known by [9]. In general it is determined by the Siegel-Veech constants which measure the boundary behavior, cf. [10] [6] [15]. In low genus cases [2] [16] [21], the Lyapunov spectrum has been worked out in some partial cases, which is based on a series of work by McMullen [17][18] [19]. Some concrete examples like square-tiled surfaces and triangle groups have been computed in [4][8] [11].
There are two types of non-varying results of the sum, one for low genus and one for hyperelliptic loci, with two completely different methods of proof [22]. One method uses a translation of the problem into algebraic geometry, in particular slope calculations [6] [5], and the other relies on the correspondence to Siegel-Veech constants [9].
In this paper, we will construct Weierstrass filtration of the Hodge bundle based on the dimension of sublinear systems of zeros of holomorphic differentials, define Weierstrass exponents according to Harder-Narasimhan filtration, and compute them in the hyperelliptic loci and low genus non-varying strata, thus give a unified method to compute the sum of Lyapunov exponents.
Let g ≥ 1 be an integer, and let (m 1 , ..., m k ) be a partition of 2g − 2. Denote by ΩM g (m 1 , ...m k ) the stratum parameterizing genus g Riemann surfaces with Abelian differentials that have k distinct zeros of order m 1 , ..., m k respectively.
Let C be a Teichmüller curve which lies in ΩM g (m 1 , ...m k ). Denote by f : S → C the universal family over a Teichmüller curve with distinct sections D 1 , .., D k . The relative canonical bundle can be computed through the formula (1): In hyperelliptic loci and low genus non-varying strata, we will construct the Harder-Narasimhan filtration of f * (ω S/C ) and show that the factors of the Jordan-Hölder filtration of each semistable graded quotient are line bundles (filtration (10)). Write 0 ⊂ V 1 ⊂ V 2 ... ⊂ V g = f * (ω S/C ) for the filtration, then i-th Weierstrass exponents are defined as deg(V i /V i−1 )/deg(L) (i = 1, · · · , g). Theorem 1.1. (Theorem 5.5)Let C be a Teichmüller curve in the hyperellitptic locus of some stratum ΩM g (m 1 , ..., m k ), and denote by (d 1 , ...d n ) the orders of singularities of underlying quadratic differentials. Then the Weierstrass exponents for C are This result can be used to regain the sum of Lyapunov exponents in hyperelliptic loci [9]. It was conjectured by Kontsevich and Zorich in [14] (for Teichmüller geodesic flow), and has been shown by M.Bainbridge in the case g = 2 [2]: Zorich communicated to D.Chen and M.Möller that, based on a limited number of computer experiments about a decade ago, Kontsevich and Zorich observed that the sum of Lyapunov exponents is non-varying among all the Teichmüller curves in a stratum roughly if the genus plus the number of zeros is less than seven, while the sum varies if this sum is greater than seven. The following two results are entirely based on the paper [6]. They have shown that the non-varying result of the sum of Lyapunov exponent when g ≤ 5 except the strata ΩM (2) A Teichmüller curve in ΩM odd g (2g − 2) has Weierstrass exponent i 2g−1 , i ∈ G p = {1, 2, 3, ..., g − 2, g − 1, 2g − 1}, and f * ω S/C splits into direct sum of line bundles.
A related work about quadratic differentials has been done in [7]. Our basic idea is to construct a filtration of f * O(ω S/C ) and then compute each graded quotient. But generally, it is difficult to compute the quotient. The quotient is a subbundle of the direct image of a bundle O aDi (dD i ). We introduce Harder-Narasimhan filtration to study the bundle. The difficulty will disappear if we assume the non-varying of the Weierstrass semigroup of fibers. The paper is organized as follows. In section 2, we introduce some background material that has appeared in [6]. In section 3, we give a basic example to define Weierstrass semigroup filtration, show that it is the Harder-Narasimhan filtration, and compute the Weierstrass exponent of such Teichmüller curves. In section 4 we define Weierstrass filtration, apply it to compute the sum of Lyapunov exponents. In section 5, we define Weierstrass exponents and compute them in the non-varying sum strata.

Background
2.1. Moduli spaces. Denote by ΩM g the moduli space of pairs (X, ω) where X is a curve of genus g and ω is a holomorphic one-form on X. It is fibred over the moduli space M g of curves. Let (m 1 , ..., m k ) be a partition of 2g − 2, and let ΩM g (m 1 , ...m k ) denote the stratum parameterizing one-forms that have k distinct zeros of order m 1 , ..., m k respectively. Denote by ΩM hyp g (m 1 , ..., m k )( resp. odd, resp. even) the hyperelliptic (resp. odd theta character, resp. even theta character) connected component. ( [15]) Let M g denote the Deligne-Mumford compactification of M g . Then ΩM g extends over M g , parameterizing sections of the dualizing sheaf or equivalently stable one-forms. We denote by ΩM g the total space of this extension.
Points in ΩM g , called flat surfaces, are also written as (X, ω) with ω a stable one-form on X.
Let C be a genus g curve and L a line bundle of degree d on C. Denote by |L| the projective space of one-dimensional subspaces of H 0 (C, L). For a (projective) r-dimension linear subspace V of |L|, we call (L, V ) a linear series of type g r d . Let ω = (ω 1 , ..., ω n ) be a tuple of integers. The generalized Brill-Noether locus BN r d,w is the locus in M g,n of pointed curves (C, p 1 , ..., p n ) with a line bundle L of degree d such that L admits a linear system g r d and h 0 (L(− w i p i )) ≥ r. We need the following generalization of Clifford's theorem for stable curves: (1) C is smooth; (2) C has at most two components; (3) C does not have separating nodes and deg(D) ≤ 4.

Teichmüller curves.
A Teichmüller curve C is an algebraic curve in M g that is totally geodesic with respect to the Teichmüller metric. After suitable base change, we can get a universal family f : S → C, which is a relatively minimal semistable model with disjoint sections D 1 , .., D k ; here D i | X is a zero of ω when restrict to each fiber X. ( [6]) Let L ⊂ f * ω S/C be the line bundle whose fiber over the point corresponding to X is Cω, the generating differential of Teichmüller curves; it is also known as the "maximal Higgs" line bundle, in the sense of [23] and [20]. Let ∆ ⊂ B be the set of points with singular fibers, then the property of being "maximal Higgs" says by definition that L ∼ = L −1 ⊗ ω C (log∆) and deg(L) = (2g(C) − 2 + |∆|)/2, together with an identification (relative canonical bundle formula): By the adjunction formula we get The variation of Hodge structures (VHS for short) over a Teichmüller curve decomposes into sub-VHS Here L i are rank-2 subsystems, maximal Higgs L 1,0 1 ≃ L for i = 1, non-unitary but not maximal Higgs for i = 1. [20] Here we collect some degeneration properties of Teichmüller curves which will be needed in the subsequent sections.
Theorem 2.2. [6] (1) The section ω of the canonical bundle of each smooth fiber over a Teichmüller curve extends to a section ω ∞ for each singular fiber X ∞ over the closure of a Teichmüller curve. The signature of zeros of ω ∞ is the same as that of ω. Moreover, X ∞ does not have separating nodes.
(2) For Teichmüller curves generated by a flat surface in ΩM g (2g − 2) the degenerating fibers are irreducible.
(3) Let C be a Teichmüller curve generated by an Abelian differential (X, ω) in ΩM g (µ). Suppose that an irreducible degenerating fiber X ∞ over a cusp of C is hyperelliptic. Then X is hyperelliptic, hence the whole Teichmüller curve lies in the locus of hyperelliptic flat surfaces.
(4) Let C be a Teichmüller curve generated by a flat surface in ΩM 5 (8) even. Then C does not intersect the Brill-Noether divisor BN 1 3 on M 5 .
where i is the maximal index such that v is in the fiber of V i over m i.e.v ∈ (V i ) m . The numbers λ i for i = 1, ..., k ≤ rank(V ) are called the Lyapunov exponents of S t . Since V is symplectic, the spectrum is symmetric in the sense that λ g+k = −λ g−k+1 . Moreover, from elementary geometric arguments it follows that one always has λ 1 = 1. Thus, the Lyapunov spectrum is defined by the remaining nonnegative Lyapunov exponents The bridge between the 'dynamical' definition of Lyapunov exponents and the 'algebraic' method applied in the sequel is given by the following result.
where W (1,0) is the (1, 0)-part of the Hodge filtration of the vector bundle associated with W. In particular, we have λ i be the sum of Lyapunov exponents, and put k µ = 1 where c area (C) is the Siegel-Veech constant corresponding to C.

2.4.
Vector bundles on curves. The readers are referred to [13] for details about sheaves on algebraic varieties. Let C be a smooth curve, V a vector bundle over A Harder-Narasimhan filtration for V is an increasing filtration: such that the graded quotients gr HN The Harder-Narasimhan filtration is unique.
A Jordan-Hölder filtration for semistable vector bundle V is a filtration: such that the graded quotients gr V i = V i /V i−1 are stable of the same slope. Jordan-Hölder filtration always exist. The graded objects gr V i = ⊕gr V i does not depend on the choice of the Jordan-Hölder filtration.

Weierstrass semigroup filtration for one section
In this section, we will consider a basic example of Weierstrass semigroup filtration and Weierstrass exponents.
A Teichmüller curve in ΩM g (2g−2) has a relative canonical bundle formula(formula (1)): By the projection formula: 3.1. Non-varying Weierstrass semigroups in ΩM g (2g − 2). We will consider the varying of the Weierstrass semigroup of p = D| F along the fiber F of universal family on the Teichmüller curve.
Definition 3.1. We define Weierstrass semigroup H p1,··· ,p k as follows and we define Weierstrass gap by the formula G p1,...,p k : More information about Weierstrass semigroup of one point p can be found in [1]. Using the Riemann-Roch theorem, we can compute the cardinality G p is equal to g. In fact: This expression is closely related to the filtration we will construct.
We define the weight w(H p ) of the Weierstrass semigroup H p to be: which satisfies the inequality with equality true if and only if 2 ∈ H p .
If moreover g ≤ 5, then: (3)A Teichmüller curve in ΩM even g is an upper-semicontinuous function, and h 0 (2D| F ) = 2 at smooth fibres. It is also known that For (2) and (3), by Theorem 2.2, the Teichmüller curve is irreducible and nonhyperelliptic. It does not have separating nodes and (g − 1) ≤ 4, so we can use Cliffold Theorem 2.1 in each of the two cases: (2) In ΩM , the equality implies that it is a hyperelliptic curve, hence a contradiction. So h 0 ((g − 1)p) = 1.
3.2. Weierstrass semigroup filtration in ΩM g (2g −2). If the Weierstrass semigroup is nonvarying, the following dimensions by Grauert semicontinuity theorem [12] (in fact, f * O(dD) is always a vector bundle even if Weierstrass semigroup is varying). Define Thus we get a filtration of the vector bundle, which we call Weierstrass filtration: Lemma 3.4. If the Weierstrass semigroup is nonvarying, then the graded quotient On the surface S we have the exact sequence: Apply f * , and use the fact that f induces an isomorphism between D and C(D is a section) By the nonvarying condition, the two sheaves O D (dD) and R 1 f * O((d − 1)D) are both locally free. Since subsheaves of a locally free sheaf are locally free, we deduce that Ker(δ) and Im(δ) are both locally free. For and then by the formula (2): We define the i-th Weierstrass exponent w i to be If moreover g ≤ 5, then: Proof. Proposition 3.2 tells us that these Teichmüller curves have non-varying Weierstrass semigroups, and for d i ∈ H p , 2g − 1 − d i ∈ G p we get the result by applying lemma 3.4.
Weight formula (5) gives the sum: It has maximal value g 2 (2g−1) by the inequality (6), where the equality is achieved if and only if 2 ∈ H p .
Even if the Weierstrass semigroup is varying, we can also bound degf * ω S/C by the sum (7) from the proof of lemma 3.4 since im(δ) is torsion.
Corollary 3.6. If the Weierstrass semigroup is non-varying, then is the Harder-Narasimhan filtration.
we get it by the uniqueness of the Harder-Narasimhan filtration.
3.3. f * O aD (dD) and Splitting lemma II. We can get more information by analysing the exact sequence (9) ), which has a good filtration.
Proof. Because f is an isomorphism between D and C (D is a section), From the exact sequence: we get the long exact sequence: By induction, we have and the exact sequence: Because D 2 < 0, we get a filtration with strictly decreasing graded quotient line bundles. By the uniqueness of the Harder-Narasimhan filtration, we get the desired result.
The filtration can be used to describe the structure of special quotients.
Proof. We have the following commutative diagram: Because h 0 (dp) = h 0 (d − m)p) + m, a similar argument as in Lemma 3.4 implies that ϑ is surjective, and h 0 ((d − m)p) = h 0 ((d − m − n + 1)p) implies that θ is an isomorphism by corollary 4.11. Thus the image of ψ is the same as the image of ϑ, that is By induction, f * O mD splits into direct sum of line bundles.
Thus by the uniqueness of Harder-Narasimhan filtration and lemma 3.7, we have Theorem 3.9. For g ≤ 5, the sheaf f * (ω S/C ) of a Teichmüller curve C in ΩM odd g (2g− 2) splits into direct sum of line bundles.
We have here the first equality is by Equation (3) and the last equality follows from Lemma 3.8.

Weierstrass filtration for several sections
In this section we will define three kinds of filtration: Weierstrass filtration, Weierstrass semigroup filtration and Weierstrass pair filtration. The first one together with the upper bound lemma 4.10 is used to get coarse information about the upper bound of the sum. The second and the third one can be used to get more precise information about each quotient.

Weierstrass filtration. From the exact sequence
and the fact that all subsheaves of a locally free sheaf on a curve are locally free, we is a rank i vector bundle. By non varying condition, R 1 f * O(d 1 D i +...+d k D k ) is also a vector bundle.
We define the set of Weierstrass filtration as follows: is not locally free, the filtration won't give us the desired properties to compute its degree. So in many cases, we need the non-varying condition to get more information.
Denote by d the tuple (d 1 , ...d k ) Definition 4.2. We define the set of Weierstrass semigroup filtration as follows: Assuming the Weierstrass semigroup to be non-varying, we will use the Weierstrass semigroup filtration to define Weierstrass exponents in the next section. Under some weak assumptions, the following filtration is also useful for computational and theoretical reasons.
Definition 4.4. We define the set of Weierstrass pair filtration as follows: ..., 1, ..., 0)} To define these we need to verify that the exact sequence ) is a line bundle and that There is a Weierstrass pair filtration (11) in the proof of the stratum ΩM even 4 (4, 2) .

4.2.
Splitting lemma I and Upper bound lemma. The next lemma describes a splitting structure of the quotient: Proof. From the exact sequence we get the long exact sequence Because Ker(δ) and Im(δ) are both locally free and because we get We often uses it with lemma 3.8.

Proof.
Because it suffices to apply lemma 3.8 to each D j .
Corollary 4.8. Each graded quotient of a filtration in W SF resp. in W P F is a line bundle with computable degree. So the sum can be computed. All filtration in W SF have the same sum degf * ω S/C − gdeg(L).
Proof. For any filtration in W SF , we have the exact sequence Similarly for W P F we have the exact sequence , whose degree is computable.
For the second part of the claim, the sum is . Example 4.9. An explicit formula about the sum has been got in corollary 5.6 for a Teichmüller curve in the hyperelliptic locus.   F being a general fiber).
Proof. By lemma 3.7, the graded sum of the Harder-Narasimhan filtration of ⊕ In the proof of Splitting lemma 4.6 the kernel ker(δ) is a subbundle of rank equal We get the result by using the maximality of Harder-Narasimhan polygon: each rank r subbundle has degree smaller than the maximal sum of r line bundles in .

4.3.
Application to the sum of Lyapunov exponents. The existence of Weierstrass semigroup (pair) filtration is convenient for computation. In many cases, because of the absence of Weierstrass semigroup (pair) filtration, we can not get more precise information about f * ω S/C . But the partial filtration is enough to give some upper bound of it. Use this coarse filtration, we have:

1)
Proof. There is a rank g − 1 subbundle It is obvious by lemma 3.7 that grad(HN (⊕ By lemma 4.10, we want to bound the maximum of sums of g − 1 line bundles in O Di (jD i ), and the sum is (− ni(ni+1) 2(mi+1) )degL. We get there must be some n i such that n i ≥ (m i + 1)/2 if n j < (m j − 1)/2, and We can sum up the value − ni(ni+1) 2(mi+1) by changing n i to n i − 1 and n j to n j + 1 in both case.
So we assume that n i be equal to m i /2 when m i is even, and n i = (m i − 1)/2 or (m i + 1)/2 when m i is odd, with the property ♯{n i = (m i − 1)/2} = k = ♯{n i = (m i + 1)/2}.

Weierstrass exponents
This section is devoted to the construction of the Harder-Narasimhan filtration of f * O(m 1 D 1 + ... + m k D k ) and the definition of Weierstrass exponents under some additional assumptions. We also get a filtration for

Weierstrass exponents. If there is a filtration
If there exists a filtration as in (10), we define i-th Weierstrass exponent w i as follows: When H p1,...,p k ∈ ∪ i W S i is non-varying, there are many Weierstrass semigroup filtration, and we can construct the Harder-Narasimhan filtration recursively.
Theorem 5.2. Assume that the Weierstrass semigroup is non-varying, then we can construct a filtration Proof. For every Weierstrass semigroup element (d 1 , ..., d k ) of a general fiber, we define the length of the set to be For any vector bundle of the form f * O(a 1 D 1 +...+a k D k ), we define the following set It is not empty because it contains the element (d 1 , ..., d k ) ∈ H p1,...,p k , for which the sum k i=1 d i reaches the minumum when (d 1 , · · · , d k ) varies in We then construct the set L i and define the number l i recursively: ..., m k ) ......
If L i is non empty, then l i is defined. If l i is defined, then L i−1 is non empty because L(d 1 , ...d j − 1..., d k ) is non empty (i ≥ 2). So the definition makes sense.
It is obvious that rk(f * O(d 1 D 1 + ... + d k D k )) = i for any (d 1 , ..., d k ) ∈ L i . For any (e 1 , ..., e k ) ∈ L i−1 , by our construction, there is a (d 1 , ..., d k ) ∈ L i , −d j /(m j + 1) = l i such that (e 1 , ..., e k ) lies in L(d 1 , ...d j − 1..., d k ). We repeat the process from L 1 , then inductively we obtain a filtration with V i /V i−1 being a line bundle. The filtration is not unique because there maybe many chooses in each step.
From the equalities and the exact sequence we get When we assume the Weierstrass semigroup to be non-varying, we get deg(V i /V i−1 )/deg(L) = l i by lemma 4.6. Therefore deg(V i /V i−1 ) is decreasing in i. Remark 5.3. From the proof, we can see that the Harder-Narasimhan filtration is constructed under the weak assumption that a subset of the Weierstrass semigroup is non-varying. This fact will be verified in hyperelliptic loci and non-varying strata of genus less than or equal five.

5.2.
Hyperelliptic loci. The square of any holomorphic 1-form ω on a hyperelliptic curve fiber F is a pullback (ω) 2 = p * q of some meromorphic quadratic differential q with simple poles on P 1 where the projection p : F → P 1 is the quotient over the hyperelliptic involution. Denote by Q(d 1 , ...d n ) the stratification by orders of zeros of the corresponding quadratic differentials (see [9] for more details).
Proposition 5.4. Let C be a Teichmüller curve in the hyperellitptic locus of some stratum ΩM g (m 1 , ..., m k ), and denote by (d 1 , ...d n ) the orders of singularities of the corresponding quadratic differentials. Then there exists a filtration Proof. Let F be the covering flat surface belonging to the stratum ΩM g (m 1 , ..., m k ), then the resulting holomorphic form ω on F has zeros of the following degrees: A zeros p j , which is an order d, of meromorphic quadratic differentials q on P 1 gives rise to zeros on F ([9]): (1) Two zeros p 1j , p 2j of degree m = d/2, when d is even. In this case, p 1j , p 2j is a g 1 2 , i.e. h 0 (p 1j + p 2j ) = 2. (2) One zero q j of degree m = d + 1, when d > 0 is odd. In this case, q j is a Weierstrass point, i.e. h 0 (2q j ) = 2.
Denote by Q j resp. P 1j resp. P 2j the section containing q j resp. p 1j resp. p 2j For each fiber F , the Weierstrass semigroup has at least a non-varying subgroup generated by the elements {{2q j } dj odd , {p 1j + p 2j } dj even }, which is equal to: We order the following g − 1 numbers {{− 2k dj+2 } 2k≤dj +1 } to {N 1 , ...N g−1 } in decreasing order.
Proof. If the curve X is smooth or irreducible, then h 0 (3p 1 + p 2 ) = 3 would imply, by Clifford's theorem, that X is hyperelliptic and 2p 1 linear equivalence to p 1 + p 2 is impossible.
For the first case, the elliptic component Y contains p 2 and h 0 (p 2 ) = 1, i.e. all the sections are given by constant functions. The other component Z contains p 1 with h 0 (3p 1 ) < 3, and a section on Z uniquely determines the constant section on Y, by its values at the nodes (assuming the same value). Hence h 0 (3p 1 + p 2 ) < 3 on X.
For the second case, h 0 (p 2 ) = 2 on the rational component Y . Then h 0 (3p 1 ) has to be 2 on Z, hence p 1 is a Weierstrass point. But in order to glue two sections on Y and Z, they need to have the same value at each of the four nodes. The four nodes form two conjugate pairs on the hyperelliptic curve Z, hence gluing two sections still imposes two conditions. Therefore h 0 (3p 1 + p 2 ) ≤ 2 + 2 − 2 < 3 on X.
Of course one has to check that the spin parity holds for nodal curves as well, i.e. h 0 (3p 1 + p 2 ) cannot be two on a degenerate X. But this should follow from the well-defined odd/even spin moduli spaces.
Proof. In all case, we have constructed a filtration (10).