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On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) Viet Cuong Do VNU University of Science FJV2018, Nha Trang Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a

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On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve)

Viet Cuong Do VNU University of Science FJV2018, Nha Trang

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 1 / 14

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Outline

1

Motivation

2

Some notations

3

Algorithm to calculate the motive

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 2 / 14

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Motivation

Some applications of Higgs bundles?

If the curve is defined over finite field, the adelic description of the stack of Higgs bundles is closely related to the space occuring in the study of the trace formula (cf. the work of Ngo on fundametal lemma for Lie algebra in 2010) If the curve is defined over complex numbers, the moduli space of Higgs bundle turns out to be diffeomorphic to the space of representations of the fundamental group of the curve (cf. the work of Hitchin in 1987, and Simpson in 1992).

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 3 / 14

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Motivation

Some applications of parabolic Higgs bundles

If the curve is defined over finite field, the parabolic Higgs bundles are used in the proof for weighted-fundametal lemma (an important generalization of the fundamental lemma proved by Ngo, it applies to the more general geometric terms in the trace formula that are

btained by truncation) of Laumon and Chaudouard in 2010 and

2012. If the curve is defined over complex numbers, the moduli space of parabolic Higgs bundles related to the space of representations of the fundamental group of punctured curve (cf. the work of Simpson in 1990). García-Prada, Heinloth, Schmitt (2014) gave an algorithm to calculate the motive of moduli spaces of Higgs bundles, but this algorithm only works with the condition that the rank and the degree

f Higgs bundles are coprime. We expected that by working with the

parabolic Higgs bundles, we can remove this condition.

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 4 / 14

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Some notations

Parabolic structure attached to a vector bundle

Let C be a smooth projective curve of genus g with ℓ marked points in the reduced divisor D = p1 + · · · + pℓ and E be a (holomorphic) bundle over C. Definition A parabolic structure on E consists of weighted flags: Ep = Ep,1 ⊃ . . . ⊃ Ep,sp ⊃ Ep,(sp+1) = 0 ≤ wp,1 < . . . < wp,mp < wp,sp+1 = 1

ver each point p ∈ D. The bundle E with this parabolic structure is

called the parabolic bundle (of a weight data type D = (s, w, m)). (Here m is the collection of all mp,i = dim(Ep,i) − dim(Ep,i+1)). A (holomorphic) map φ : E 1 → E 2 between two parabolic bundles is called strongly parabolic if w1

p,i ≥ w2 p,i′ implies that

φ(E 1

p,i) ⊂ E 2 p,i′+1 ∀p ∈ D.

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 5 / 14

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Some notations

Parabolic Higgs bundles

Definition A parabolic Higgs bundle is a pair (E, φ) consisting of a parabolic bundle E and a strongly parabolic map φ : E → E ⊗ ΩC(D) where ΩC is the sheaf of differentials on C. We denote by parµ(E) :=

deg(E)+

p∈D

sp

i=1 mp,iwp,i

rank(E)

the parabolic slope of E. Definition (Stable condition) We call the parabolic bundle E stable (resp. semi-stable) if , for every proper subbundle F of E, we have parµ(F) < parµ(E) (resp. parµ(F) ≤ parµ(E)). We call a parabolic Higgs bundle (E, φ) stable (semi-stable) if the above inequalities hold on those proper subbundles F of E, which are, in addition, φ-invariant (i.e φ(F) ⊂ F ⊗ ΩC(D)).

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 6 / 14

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Some notations

Moduli spaces of parabolic Higgs bundles

We denote by Md,st(resp. ss)

n,D

the moduli stack of stable (resp. semi-stable) parabolic Higgs bundles of fixed rank n, fixed degree d and of fixed weight data type D (compatible with n). D is called generic if Md,ss

n,D = Md,st n,D.

Theorem (Yokogawa) For the generic D, the coarse moduli space Md

n,D of Md,ss n,D can be

constructed (using GIT). Moreover, it is a smooth irreducible complex variety of dimension 2(g − 1)n2 + 2 +

p∈D
n2 −

sp

m2

p,i

The moduli space Md

n,D has an action of Gm, given by:

λ.(E, φ) = (E, λ.φ).

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 7 / 14

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Algorithm to calculate the motive

Where do we make our calculation?

A certain completion ˆ K0(Vark) of the Grothendieck ring of varieties K0(Vark). For us the main use of this is to express the following two invariant of Md

n,D in the same terms:

If k = Fq, the map [X] → #X(Fq) defines a morphism K0(Vark) → Z. If k = C, the E-polynomial can be viewed as the map K0(Vark) → Z[u, v].

The class [GLn] is invertible in ˆ K0(Vark). This allows to define classes [X] for quotient stacks X = X/GLn by [X/GLn] = [X]/[GLn]. All stacks occurings in our calculation will admit a strastification into locally closed substacks of the form [X/GLn].

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 8 / 14

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Algorithm to calculate the motive

A variant of Hitchin’s approach to the cohomology

Theorem (Bialynicki-Birula’s decomposition) Md

n,D can be decomposed into sub-varieties Md n,D = F + i

such that the fixed point locus of the Gm- action is the disjoint union of strata Fi and the locally closed sub-varieties F +

i

f Md

n,D are affine bundles F + i

→ Fi

ver the fixed point strata.

Theorem (Simpson) The equivalence class of a stable parabolic Higgs bundles (E, φ) is fixed under the action of Gm if and only if E has a direct sum decomposition E = r

i=0 Ei as parabolic bundles, such that the restriction φi := φ|Ei of

φ is a strongly parabolic map Ei → Ei−1 ⊗ ΩC(D). Furthermore, stability implies that φ|Ei = 0 for i = 1, . . . , r.

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 9 / 14

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Algorithm to calculate the motive

Parabolic chains

Definition A (holomorphic) parabolic chain on C of length r is a collection Er

= ((Ei)i=0,...,r, (φi)i=1,...,r), where Ei are parabolic vector bundles on C

and φi : Ei → Ei−1(D) are strongly parabolic morphisms. Definition For α = (αi)i=0,...,r ∈ Rr+1 the α-slope of Er

is defined as

parµα(Er

) :=

r

rank(Ei) |rank(Er

)|(parµ(Ei) + αi).

We say Er

is α-(semi)-stable if parµα(E′r
)(≤) < parµα(Er
) for any

proper sub-chain E′r

of Er
.

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 10 / 14

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Algorithm to calculate the motive

Analogue of Atiyah -Bott’s strategy to the cohomology of moduli space of stable vector bundle

Unstable parabolic chains admit a canonical Harder-Narasimhan filtration: 0 ⊂ Er

(1) ⊂ · · · ⊂ Er
(h) = Er
, such that

parµα(Er

(1)) > · · · > parµα(Er
(h)) and the sub-quotients

Er

(i)/Er
(i−1) are α-semi-stable.

Geometric decompostion: PChainα−ss

datum = PChaindatum − ∪Harder − Narasimhan strata.

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 11 / 14

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Algorithm to calculate the motive

Some comments

The Harder-Narasimhan strata are fibered over spaces of semi-stable parabolic chains of lower rank. Mimic the work of García-Prada, Heinloth, Schmitt, for stack of any parabolic chains, we define a strastification into pieces that we can computed explicitly. A problem of convergence: the stack of parabolic chains are often very big, so that they do not defined classes in ˆ K0(Vark).

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 12 / 14

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Algorithm to calculate the motive

Wall crossing

Definition (Critical value) A parabolic chain Er

strictly α-semi-stable if and only if it has a

proper parabolic sub-chain E′r

such that parµα(E′r
) = parµα(Er
).

α is called critical value if there exist a parabolic chain Er

strictly

α-semi-stable and

rank(E ′

i )

|rank(E′r

)| =

rank(Ei) |rank(Er

)| for some i.

The set of critical points on a line of Rn+1 is discrete. We divide the line into segments such that the stacks for the parameter lying inside the same segment are isomorphic and we know how to describe the difference at the endpoints of a segment.

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 13 / 14

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Algorithm to calculate the motive

Thanks you for your attention!

Viet Cuong Do VNU University of Science On the motive of moduli spaces of parabolic Higgs bundles (on a projective curve) FJV2018, Nha Trang 14 / 14