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Poincaré polynomial of the moduli spaces of parabolic bundles

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Abstract

In this paper we use Weil conjectures (Deligne’s theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of HarderNarasimhan filtration gives us a recursive formula for the Poincaré polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincaré polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.

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Abbreviations

X :

a smooth projective geometrically irreducible curve over the finite field\({\mathbb{F}}_q \).

\(\bar X\) :

the curveX \( \otimes _{{\mathbb{F}}_q } \bar {\mathbb{F}}_q \), where\(\bar \{{mathbb{F}}_q \) is an algebraic closure of\(\bar \{{mathbb{F}}_q \).

Z X(t):

the zeta function of the curveX.

X v :

X \( \otimes _\{{mathbb{F}} _{_q } \mathbb{F}_{q^\nu } \), where\(\{{mathbb{F}}_{q^\nu } \subset \bar \{{mathbb{F}}_q \) is a finite field withq velements. For positive integersn andm and non-negative integersr 1,…,r mwithr 1 +…+r m=n,

Flag(n, m, r j :

the variety of all flagsk n=F 1 ⊃ • • • ⊃F mF m+1 = 0 of vector subspaces ink n, with dim(F j/F J+1) =r j.

\(\left( {F_j /F_{j + 1} } \right) = r_j \) :

the number of\(\mathbb{F}_q \)-rational points of the Jacobian ofX.

S :

a finite set ofk-rational points ofX. (These are the parabolic vertices.)

m P :

a fixed positive integer defined for eachPS. ForPS, and 1 ≤im P,

α :

P i ) is the set of allowed weights. ForPS, and 1 ≤im P,

R :

(R P i ), the quasi-parabolic data (or simply ‘data’).

n(R):

Σi=1/m P R i P, the rank of the dataR.

L :

a sub-data ofR and

R - L :

the complementary sub-data defined by (R - L) P i =R Pi -L Pi .

L :

a line bundle onX.

E :

a vector bundle with a parabolic structure with dataR.

J r(l):

the set of isomorphism classes of quasi-parabolic vector bundles with dataR, and determinantL.

α(R):

Σ P Σ mp i =1R P i α P i the parabolic contribution to the degree.

deg(E):

the ordinary degree ofE.

pardeg(E):

deg(E) + α(R), the parabolic degree ofE.

parμ(E):

pardeg(E)/rank(E), the parabolic slope ofE. ForP ∈ S, 1 ≤imp, and 1 ≤kr,

I :

(I Pi,k ), the intersection type of Nitsure, which is a partition ofR.

l(I):

the length of the intersection typeI. Forjr, we have

R Ij :

the sub-data defined by (R Ij ) P i =I Pi,j

R I≤ j :

the sub-data defined by (R I≤ j ) P i = Σ /≤j k I P i,k ,

R I≥ j :

the sub-data defined by (R I≥ j ) P i = Σ k/≥ j I Pi,k .

I ≥ j :

the partition ofR I≤ j defined by (I ≤ j)P Pi,k =I Pi,k wherekj.

I ≥ j :

the partition ofR I≥j defined byI (I ≥ j) P i,k =I Pi,k wherekj.

M R,L :

the moduli space of parabolic semi-stable bundles with the dataR and determinantL.

M SR,L :

the open sub variety ofM R,Lcorresponding to the parabolic stable bundles. For parabolic bundlesE′, E andE″ with dataR′, R andR″, we denote by

[E]:

(E, i,j), a parabolic extension ofE″ byE′.

ParExt(E″, E′):

the set of equivalence classes of parabolic extensions ofE″ byE′.

β R (L):

Σ(l/ParAut(E)¦) where summation is over allE ∈ J R(L) such thatE is parabolic semistable.

J R(L,I):

the set of isomorphism classes of parabolic bundles with weightsα, of intersection typeI, and determinantL.

β R (L,I):

Σ(l/¦ParAut(E)¦), where the summation is over allE inJ R(L,I). We also write

β R (d,I):

β R (L,I), since β R (L,I) depends onL only via its degreed = deg(L).

F R :

P∈S Flag(n(R),m p, R Pi )

f R(q):

the number of234-001-valued points of the varietyF R.

C(I;d 1,…,d r):

the integer defined by equation (3.8).

σ k(I):

\(\sum _{P \in S} \sum _{1 > t} \sum _{l< r - k + 1} I_{i,r - k + 1}^P I_{t,l}^P \cdot \)

σ R (I):

\(\sum _k \sigma _k (I)\). For a vector bundleF

x(F):

the Euler characteristic.

\(x\left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\nu _1 } \\ {\delta _1 } \\ \end{array} } & {\begin{array}{*{20}c} {...} \\ {...} \\ \end{array} } & {\begin{array}{*{20}c} {\nu _r } \\ {\delta _r } \\ \end{array} } \\ \end{array} } \right)\) :

the numerical function of Desale-Ramanan defined by the equation(3.20).

Σo :

denotes the summation over all (d 1,…,d r) ∈ ℤr with\(\sum _i d_i = d\) and satisfying equation (3.7).

\(\tau _{n(R)} (q)\)% MathType!End!2!1!:

T n(R)(q)\(\frac{{q^{\left( {n\left( R \right)^2 - 1} \right)\left( {g - 1} \right)} }}{{q - 1}}Z_X \left( {q^{ - 2} } \right)...Z_X \left( {q^{ - n\left( R \right)} } \right)\)

f R(t):

the rational function corresponding tof Rgiven by the equation (3.30).

T n(R) (t):

the rational function corresponding toT n(R) given by the equation (3.31).

Q R,d(t):

\(t^{n\left( R \right)^2 \left( {g - 1} \right)} \left( {1 + t^{ - 1} } \right)^{2g} \tilde \beta _R \left( d \right)\), this is the main function for the recursion.

Q R(t):

\(t^{n\left( R \right)^2 \left( {g - 1} \right)} \tilde f_R \left( t \right)\tilde \tau _{n\left( R \right)} \left( t \right)\).

P R,d :

the power series whose coefficients compute the Betti numbers of the moduli space of parabolic stable bundles with dataR and degreed.

N R(I;d 1,…, dr):

the integer given by the formula (3.38).

Y :

a smooth projective variety over\(\mathbb{F}_q \).

N ν :

the number of\(\mathbb{F}_{q^\nu } \)% MathType!End!2!1!-rational points ofY. Fori = 1,…,2g, we have

i :

a fixed algebraic integer of normq 1/2.

h(u, v1,…,v 2g):

a rational function given by the equation (4.2)

p(u, v1,…,v 2g):

the numerator occuring in the equation (4.2).

(a J,j):

the coefficients occuring in (4.3).

J :

the multi-indexJ = (i 1, i2,…,i 2g),

¦J¦ :

\(\sum _{r = 1}^{2g} i_r \)% MathType!End!2!1!, and

u J :

\(v_1^{i1} v_2^{i2} ...v_{2g}^{i2g} \cdot \).

N :

the ‘weighted degree’ ofp(u,v 1,v 2, …,v 2g).

b J,j :

the coefficients ofh defined in (4.8).

f≥o(u,v 1,…, v2g):

the function defined by the equation (4.13).

M r :

\(f_{ \geqslant 0} \left( {q^r ,\omega _1^r ,...,\omega _{2g}^r } \right)\).

Z1(t):

the formal power series defined in (4.14).

Z2(t):

the formal power series defined in (4.15).

Z(t):

Z1 (t)Z2 (t). For a meromorphic functionh on a disc in\(\mathbb{C}\), and α >0,

μ(h, α):

the number of zeros minus the number of poles counted with multiplicities ofh with norm α.

P(T):

the polynomial defined by the equation (4.19).

M R′(I;d):

the integer given by the formula (5.4). For a real number A,

M R(I;λ):

the integer given by the formula (5.5).

Q λR,d (t):

the rational function defined in (5.7).

S λR,d (t):

the rational function defined in (5.8).

\(\sum _{o\lambda } \) :

denotes the summation over (d 1,…,d r) \(\mathbb{Z}^r \) such that Σi d i =d and the equation (5.9) holds.

Q λ-R,d :

Q λ- ∈R,d for ∈ small enough such that the functionQ λR,d has no jumps in the interval [λ - ∈, λ).

S λ-R,d :

S λ- ∈R,d for e small enough such that the functionS λR,d has no jumps in the interval [λ - ∈, λ).

ΔQ λR,d :

Q λR,d Q λ-R,d .

ΔS λR,d :

S λR,d S λ-R,d .

δ R (L):

the integer given by the equation (5.10).

d(λ,L):

n(L)λ - α(L).

g R(I;d):

the rational function given by (5.17).

σR′(I):

\(\sum _{P \in S} \sum k_{ > l,i< t} I_{i,k}^P I_{t,l}^P \).

Mg(I;λ):

the integer given by the formula (5.25).

P r(t):

the rational function (polynomial) defined by the equation (5.26). For a dataR with rankn(R) = 2,

T :

the subset ofS consisting of parabolic vertices where the parabolic filtration is non-trivial.

T I :

P ∈ T¦I P1,1 =0.

Χ I :

a characteristic function onT, defined by the equation (6.3).

ψ I :

\(\sum _{P \in T} X_I \left( P \right)\left( {\alpha _{1,1}^P = 0} \right)\).

a I :

1 ifd + [ψ I] is even, and

a I :

0 ifd + [ψ I] is odd.

δ P :

α P1 - α P2

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Holla, Y.I. Poincaré polynomial of the moduli spaces of parabolic bundles. Proc Math Sci 110, 233–261 (2000). https://doi.org/10.1007/BF02878682

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