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Poincaré polynomial of the moduli spaces of parabolic bundles

  • Yogish I. Holla
Article

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.

Keywords

Cohomology parabolic vector bundles moduli space Betti numbers Weil conjectures 

Summary of notation

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\(\bar \{{mathbb{F}}_q \).

ZX(t)

the zeta function of the curveX.

Xv

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, rj

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.)

mP

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

α

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

R

(R i P ), 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) i P =R i P -L i P .

L

a line bundle onX.

E

a vector bundle with a parabolic structure with dataR.

Jr(l)

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

α(R)

Σ P Σ i mp =1R i P α i P 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 i,k P ), the intersection type of Nitsure, which is a partition ofR.

l(I)

the length of the intersection typeI. Forjr, we have

RjI

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

R≤ jI

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

R≥ jI

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

I≥ j

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

I≥ j

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

MR,L

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

MR,LS

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′.

βR(L)

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

JR(L,I)

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

βR(L,I)

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

βR(d,I)

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

FR

P∈S Flag(n(R),m p, R i P )

fR(q)

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

C(I;d1,…,dr)

the integer defined by equation (3.8).

σk(I)

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

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

fR(t)

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

Tn(R) (t)

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

QR,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.

QR(t)

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

PR,d

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

NR(I;d1,…, dr)

the integer given by the formula (3.38).

Y

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

Nν

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,…,v2g)

a rational function given by the equation (4.2)

p(u, v1,…,v2g)

the numerator occuring in the equation (4.2).

(aJ,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

uJ

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

N

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

bJ,j

the coefficients ofh defined in (4.8).

f≥o(u,v1,…, v2g)

the function defined by the equation (4.13).

Mr

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

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).

MR′(I;d)

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

MR(I;λ)

the integer given by the formula (5.5).

QR,dλ(t)

the rational function defined in (5.7).

SR,dλ(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.

QR,dλ-

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

SR,dλ-

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

ΔQR,dλ

Q R,d λ Q R,d λ- .

ΔSR,dλ

S R,d λ S R,d λ- .

δR(L)

the integer given by the equation (5.10).

d(λ,L)

n(L)λ - α(L).

gR(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).

Pr(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.

TI

P ∈ T¦I 1,1 P =0.

ΧI

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

ψI

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

aI

1 ifd + [ψ I] is even, and

aI

0 ifd + [ψ I] is odd.

δP

α 1 P - α 2 P

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Copyright information

© Indian Academy of Sciences 2000

Authors and Affiliations

  • Yogish I. Holla
    • 1
  1. 1.School of MathematicsTata Institute of Fundamental ResearchMumbaiIndia

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