Complex Chern–Simons and the first tautological class

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Complex Chern–Simons and the first tautological class

Gerard Freixas i Montplet

C.N.R.S. – Institut de Math´ematiques de Jussieu - Paris Rive Gauche

June 2021


Joint work with:

Dennis Eriksson Richard Wentworth




The first tautological class

♦Mg moduli space of curves of genus g ≥2.

♦π:Cg → Mg the universal curve.

♦First tautological class:


♦Arakelov: positivity ofκ1.


Weil–Petersson metric

♦The tangent space toMg atX is H1(X,TX).

♦X has a hyperbolic metric of constant curvature−1.

♦Ifµ, ν are harmonic representatives of classes in H1(X,TX), then

hµ, νiWP = Z


µ ν dµhyp.

♦Ahlfors: K¨ahler metric.

♦Denote the K¨ahler formωWP.


Wolpert’s theorem

♦Hyperbolic metric on fibers hermitian metric onTCg/Mg.

♦Chern–Weil theory: 2πi1 Θ(TCg/Mg,hyp) representsc1Cg/Mg).

Theorem (Wolpert ’86)

There is an equality of differential forms 1

2πi 2Z


Θ(TCg/Mg,hyp)2 = 1 2π2ωWP. In particular, 12ωWP representsκ1.



♦Wolpert is based on variational formulas in Teichm¨uller theory.

♦We propose an approach using non-abelian Hodge theory.

♦Refined “relative” characteristic classes of flat vector bundles.

♦Inspired by Deligne’s functorial conception of Arakelov geometry.


Intersection bundles


Deligne pairings of line bundles

♦LetS be a complex projective surface.

♦LetD,E be two divisors onS. Their intersection number is (D·E) =X




♦IfS is fibered over a Riemann surface R, the intersection multiplicities can be encoded in a divisor onR.


♦Formalized as Deligne pairings of line bundles.

♦f:X →S a family of compact Riemann surfaces.

♦L,M line bundles on X hL,Mi line bundle on S.

♦Cohomological interpretation:

c1(hL,Mi) =f(c1(L)∪c1(M)). (1)


I Commutes with base change.

I Equalities derived from (1) and algebraic operations with line bundles, lift to canonical isomorphisms; e.g.

hL⊗L0,Mi ' hL,Mi ⊗ hL0,Mi.


Deligne’s integral of c


♦Vector bundleE onX line bundle IC2(E) on S.

♦Cohomological interpretation:

c1(IC2(E)) =f(c2(E)). (2)


I Commutes with base change.

I Algebraic manipulations lift to canonical isomorphisms.


♦Whitney isomorphism: if 0→E0 →E →E00→0 is an exact sequence of vector bundles onX, then

IC2(E)'IC2(E0)⊗IC2(E00)⊗ hdetE0,detE00i.

♦IfE is a vector bundle of rankr andLa line bundle on X, then IC2(E⊗L)'IC2(E)⊗ hL,detEi⊗(r−1)⊗ hL,Li(2r).


♦IC2 of a rank 2 vector bundleF with a suitable section s:


♦The caseF =O(E)⊕O(D) reduces to a Deligne pairing:


Metrics on intersection bundles

♦Characteristic classes can be represented by Chern–Weil forms.

♦IfLis a line bundle on X andh a hermitian metric on L, then c1(L,h) = i

2πΘL,h, ΘL,h= curvature form.

♦IfE is a vector bundle andh a hermitian metric on E, then c2(E,h) = i

2 tr(Θ2E,h)−(tr ΘE,h)2 .


♦Hermitian vector bundles metrics on intersection bundles.

♦(L,hL),(M,hM) hermitian line bundles h(L,hL),(M,hM)i, c1(h(L,hL),(M,hM)i) =




♦(E,h) hermitian vector bundle IC2(E,h), c1(IC2(E,h)) =




♦Many simple operations lift to isometries for metrized intersection bundles.

♦But not the Whitney isomorphism.


♦Letε: 0→(E0,h0)→(E,h)→(E00,h00)→0 be an exact sequence of vector bundles with arbitrary hermitian metrics.

♦The Whitney isomorphism

IC2(E)'IC2(E0)⊗IC2(E00)⊗ hdetE0,detE00i has norm

exp − Z


! .

♦ec2(ε) is a Bott–Chern secondary class: functorial solution of ddcce2(ε) =c2(E0,h0) +c2(E00,h00)−c1(E0,h0)c1(E00,h00)−c2(E,h),



♦ωCg/Mg relative canonical sheaf, with the dual of the hyperbolic metric, on fibers.

♦bκ1 :=hωCg/Mg, ωCg/Mgi + induced metric.

Theorem (Reformulation of Wolpert’s formula)

c1(κb1) = 1 2π2ωWP.


Connections on intersection bundles

♦Variant for compatible connections instead of metrics.

♦∇:E →E ⊗ A1X a compatible connection: ∇0,1 =∂E.

♦Induced compatible connection onIC2(E).

♦Bott–Chern theory variant of Chern–Simons theory.

♦Similar properties as in the hermitian case.

♦Isometry parallel isomorphism.


Canonical extensions



♦f:X →S family of compact Riemann surfaces of genus g ≥2.

♦∇:E →E ⊗ A1X/S a relative compatible connection.

♦IC2(E,∇) is not defined, since ∇is just given vertically.

♦Existence of a compatible extension∇e:E →E⊗ A1X.

♦∇flat the connection onIC2(E,∇) only depends one ∇.

Notation: ∇IC2.

♦To be useful, we need some knowledge of∇.e


Harmonic and normalized extensions

♦Supposeσ:S →X is a section off :X →S.

♦∇:E →E ⊗ A1X/S compatible, flat, fiberwiseirreducible.

♦ξ:σE ' OS⊕r a rigidification.

♦∇:e E →E ⊗ A1X a compatible extension.

♦IfS is contractible:

I Xe universal cover ofX,eσ a lift ofσ.

I C trivialization ofE onXe, by vertical parallel transport ofξ.

I ∇=d

Xe/S and∇e =d

Xe+ Ξ, with



Definition IfS is contractible:

I we say that ∇e is harmonicif the coefficient matrices of Ξ(1,0) are fiberwise harmonic.

I we say that ∇e is normalizedif eσ(Ξ) is traceless.

In general,∇e is harmonic (resp. normalized) if it is harmonic (resp.

normalized) locally over contractible open subsets ofS.

♦The harmonicity condition is to be understood in the sense of non-abelian Hodge theory.


Theorem (Canonical extension)

If E is rigidifed and∇:E →E ⊗ A1X/S is compatible, flat and irreducible, then there exists a unique harmonic and normalized compatible extension∇:e E →E⊗ A1X.

♦Uses deformation theory of flat vector bundles, and non-abelian Hodge theory (harmonic representatives of cohomology classes).


Curvature of ∇


♦S contractible,X0 a fiber of f :X →S, Γ =π1(X0, σ(0)).

♦(E,∇, ξ) as above (flat, irreducible, etc.)

♦ρ: Γ→GLr(C(S)) family of holonomies of∇.

♦Classifying map

ν:S →MirB(X0,GLr).

♦dν is a section of νT(1,0)MirB(X0,GLr)⊗ A1S.

♦The fibers ofνT(1,0)MirB(X0,GLr) are H1(Γ,Ad(ρ)).



Under the running assumptions, c1(IC2(E),∇IC2) = (i/2π)FIC2 is given by

− 1 8π2





Let f:X →S be a family of compact Riemann surfaces, and E a vector bundle on X , endowed with a holomorphic connection

∇:E →E ⊗Ω1X/S. Then ∇IC2 is holomorphic.

♦Does not require∇to be irreducible.


Complex Chern–Simons



♦f:X →S family of compact Riemann surfaces of genus g ≥2.

♦M:=MirdR(X/S,SLr) relative moduli of flat vector bundles of rankr, with irreducible connections and trivial determinant.

♦X =X×S M→Mthe universal curve.

♦Locally w.r.t. Mthere is a (Eun,∇un:Eun→ Eun⊗Ω1X/M) which is universal modulo twisting byr-torsion line bundles coming from M.


The complex Chern–Simons line bundle

♦The locally defined IC2(Eun) glue into a global line bundle.

♦The locally defined ∇IC2 attached to∇un glue into a global holomorphic connection.


Thecomplex Chern–Simons line bundleonMirdR(X/S,SLr) is the holomorphic line bundle with connection defined by

(IC2(Eun),∇IC2). We denote it LCS(X/S).

♦Riemann–Hilbert: RH :MirdR(X/S,SLr)'MirB(X/S,SLr).


♦IfS ={0} is a point, then MirB(X0,SLr) carries a natural symplectic holomorphic 2-formωG (Atiyah–Bott–Goldman):

c1(LCS(X0)) = 1 8π2ωG.

♦If (S,0) is simply connected complex analytic retraction p:MirB(X/S,SLr)→MirB(X0,SLr). There is a natural parallel isomorphism

pLCS(X0)' LCS(X/S) except forg =r = 2.

♦We sayLCS(X/S) is crystalline.


Projective structures


Projective structures and L


♦X a compact Riemann surface of genus g ≥2.

♦A projective structure subordinate to the complex structure of X is a maximal atlas of charts with:

I values in P1(C).

I changes of charts induced by the action of PSL2(C).

♦Notion of relative projective structures for familiesf :X →S.

♦Moduli space of relative projective structuresP(X/S)→S.


♦Projective structures give flat projective vector bundles.

♦Relative holonomy map and its lift:



hol //

holf 66


♦The lifthol exists only locally wrtf S. It can be obtained by introducing relative theta characteristics.

♦The holomorphic vector bundle with connection KCS(X/S) :=holfLCS(X/S)⊗4



Letπ:P(X/S)→S be the structure map. There is a canonical isomorphism of line bundles

KCS(X/S)'πX/S, ωX/Si.

♦In particular, given aC / holomorphic section

σ:S → P(X/S), σKCS(X/S) induces aC / holomorphic connection on the Deligne pairing.


Corollary (S. Zhao)

If S is a compact Riemann surface and X →S is non-isotrivial, then there are no relative holomorphic families of projective structures for X →S .

♦Otherwise, hωX/S, ωX/Si would admit a holomorphic connection, necessarily flat sinceS is one-dimensional. This contradicts Arakelov’s positivity theorem.


Fuchsian projective structures

♦Tg Teichm¨uller space of genus g.

♦X → Tg =T(X0) universal Teichm¨uller curve (Bers).

♦Fuchsian uniformization:

Xtt\H, Γt ⊂PSL2(R) Fuchsian group, t ∈ Tg.

♦C section ϕ:Tg → P(X/Tg).

♦ϕKCS(X/Tg) C connection ∇T onhωX/Tg, ωX/Tgi.


T is the Chern connection on the Deligne pairing, induced by the choice of the hyperbolic metric on fibers.


♦Relative version of uniformizing Higgs bundles.

♦X compact Riemann surface of genus g ≥2.

♦Fix a theta characteristic: κ2X.

♦J1) the first jet bundle ofκ:

0→κ⊗ωX → J1)→κ→0.

♦Projective structures onX !flat connections onJ1).

♦Equivalent to the theory of Schwarzian equations.


♦Fuchsian uniformization ofX flat connection on J1).

♦J1)'C κ⊕κ, with


κ β 0 ∂κ

andβ∈Hom(κ, κ)⊗ A0,1X ' A1,1X is the (1,1)-form induced by

dx∧dy 4y2 onH.

♦Higgs field: Φ =

0 0 1 0


♦The flat connection onJ1) decomposes as:

Chern connection on κ⊕κ+ Φ + Φ, whereκ andκ are endowed with the metric induced by the hyperbolic metric.


♦Recallbκ1 =hωCg/Mg, ωCg/Mgi + metric induced by the hyperbolic metric on fibers.

Corollary (Wolpert again)

c1(κb1) = 1 2π2ωWP.

♦Follows from:

I c1(LCS(X0)⊗4) = 42ωG, Atiyah–Bott–Goldman form.

I Goldman: the pullback ofωG toTg ⊂“MirB(X0,PSL2(R))” is






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