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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

Outline Motivation 1 Some notations 2 Algorithm to calculate the motive 3 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

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

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 obtained 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 of 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

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 = p 1 + · · · + p ℓ and E be a (holomorphic) bundle over C . Definition A parabolic structure on E consists of weighted flags: = ⊃ . . . ⊃ ⊃ E p , ( s p + 1 ) = 0 E p E p , 1 E p , s p 0 ≤ w p , 1 < . . . < w p , m p < w p , s p + 1 = 1 over 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 m p , i = dim( E p , i ) − dim( E p , i + 1 ) ). A (holomorphic) map φ : E 1 → E 2 between two parabolic bundles is called strongly parabolic if w 1 p , i ≥ w 2 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

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 ⊗ Ω C ( D ) where Ω C is the sheaf of differentials on C . � sp deg( E )+ � i = 1 m p , i w p , i p ∈ D We denote by par µ ( E ) := the parabolic slope of rank ( E ) E . Definition (Stable condition) We call the parabolic bundle E stable (resp. semi-stable) if , for every proper subbundle F of E , we have par µ ( F ) < par µ ( E ) (resp. par µ ( F ) ≤ par µ ( 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 ⊗ Ω C ( 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

Some notations Moduli spaces of parabolic Higgs bundles We denote by M d , st ( resp. ss ) the moduli stack of stable (resp. n , D 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 M d , ss n , D = M d , st n , D . Theorem (Yokogawa) n , D of M d , ss For the generic D , the coarse moduli space M d n , D can be constructed (using GIT). Moreover, it is a smooth irreducible complex variety of dimension s p � � 2 ( g − 1 ) n 2 + 2 + n 2 − � � m 2 . p , i p ∈ D i = 1 The moduli space M d n , D has an action of G m , 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

Algorithm to calculate the motive Where do we make our calculation? A certain completion ˆ K 0 ( Var k ) of the Grothendieck ring of varieties K 0 ( Var k ) . For us the main use of this is to express the following two invariant of M d n , D in the same terms: If k = F q , the map [ X ] �→ # X ( F q ) defines a morphism K 0 ( Var k ) → Z . If k = C , the E -polynomial can be viewed as the map K 0 ( Var k ) → Z [ u , v ] . The class [ GL n ] is invertible in ˆ K 0 ( Var k ) . This allows to define classes [ X ] for quotient stacks X = X / GL n by [ X / GL n ] = [ X ] / [ GL n ] . All stacks occurings in our calculation will admit a strastification into locally closed substacks of the form [ X / GL n ] . 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

Algorithm to calculate the motive A variant of Hitchin’s approach to the cohomology Theorem (Bialynicki-Birula’s decomposition) n , D = � F + M d n , D can be decomposed into sub-varieties M d such that the i fixed point locus of the G m - action is the disjoint union of strata F i and the locally closed sub-varieties F + n , D are affine bundles F + of M d → F i i i over the fixed point strata. Theorem (Simpson) The equivalence class of a stable parabolic Higgs bundles ( E , φ ) is fixed under the action of G m if and only if E has a direct sum decomposition E = � r i = 0 E i as parabolic bundles, such that the restriction φ i := φ | E i of φ is a strongly parabolic map E i → E i − 1 ⊗ Ω C ( D ) . Furthermore, stability implies that φ | E i � = 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

Algorithm to calculate the motive Parabolic chains Definition A (holomorphic) parabolic chain on C of length r is a collection E r • = (( E i ) i = 0 ,..., r , ( φ i ) i = 1 ,..., r ) , where E i are parabolic vector bundles on C and φ i : E i → E i − 1 ( D ) are strongly parabolic morphisms. Definition For α = ( α i ) i = 0 ,..., r ∈ R r + 1 the α -slope of E r • is defined as r rank ( E i ) � par µ α ( E r • ) := • ) | ( par µ ( E i ) + α i ) . | rank ( E r i = 0 • is α -(semi)-stable if par µ α ( E ′ r We say E r • )( ≤ ) < par µ α ( E r • ) for any proper sub-chain E ′ r • of E 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 10 / 14

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 ( 1 ) ⊂ · · · ⊂ E r ( h ) = E r filtration: 0 ⊂ E r • , such that • • ( 1 ) ) > · · · > par µ α ( E r ( h ) ) and the sub-quotients par µ α ( E r • • ( i − 1 ) are α -semi-stable. ( i ) / E r E r • • Geometric decompostion: PChain α − ss datum = PChain datum − ∪ 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

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 K 0 ( Var k ) . 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