Complex Chern–Simons and the first tautological class

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

Motivation

The first tautological class

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

♦π:C_g → M_g the universal curve.

♦First tautological class:

κ₁ =π∗(c₁(ω_Cg/Mg)²)∈H²(M_g,Q).

♦Arakelov: positivity ofκ1.

Weil–Petersson metric

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

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

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

hµ, νi_WP = Z

X

µ ν dµ_hyp.

♦Ahlfors: K¨ahler metric.

♦Denote the K¨ahler formω_WP.

Wolpert’s theorem

♦Hyperbolic metric on fibers hermitian metric onT_Cg/Mg.

♦Chern–Weil theory: _2πi¹ Θ(TC_g/M_g,hyp) representsc1(ωC_g/M_g).

Theorem (Wolpert ’86)

There is an equality of differential forms 1

2πi 2Z

C_g/M_g

Θ(T_Cg/Mg,hyp)² = 1 2π²ω_WP. In particular, _2π¹2ωWP representsκ1.

Goal

♦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

x∈S

multx(D∩E).

♦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)

♦Functorial:

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)

♦Functorial:

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^⊗(₂^r).

♦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

8π² tr(Θ²_E,h)−(tr Θ_E,h)² .

♦Hermitian vector bundles metrics on intersection bundles.

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

Z

X/S

c1(L,hL)∧c1(M,hM).

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

Z

X/S

c2(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

X/Sce₂(ε)

! .

♦ec₂(ε) is a Bott–Chern secondary class: functorial solution of dd^cce2(ε) =c2(E⁰,h⁰) +c2(E⁰⁰,h⁰⁰)−c1(E⁰,h⁰)c1(E⁰⁰,h⁰⁰)−c₂(E,h),

Example

♦ωC_g/M_g relative canonical sheaf, with the dual of the hyperbolic metric, on fibers.

♦bκ₁ :=hω_Cg/Mg, ω_Cg/Mgi + induced metric.

Theorem (Reformulation of Wolpert’s formula)

c₁(κb₁) = 1 2π²ω_WP.

Connections on intersection bundles

♦Variant for compatible connections instead of metrics.

♦∇:E →E ⊗ A¹_X 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

Setting

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

♦∇:E →E ⊗ A¹_X/S a relative compatible connection.

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

♦Existence of a compatible extension∇e:E →E⊗ A¹_X.

♦∇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 ⊗ A¹_X/S compatible, flat, fiberwiseirreducible.

♦ξ:σ^∗E ' O_S^⊕r a rigidification.

♦∇:e E →E ⊗ A¹_X 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

Ξ∈gl_r(C^∞(Xe))⊗A¹(S).

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 ⊗ A¹_X/S is compatible, flat and irreducible, then there exists a unique harmonic and normalized compatible extension∇:e E →E⊗ A¹_X.

♦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→GL_r(C^∞(S)) family of holonomies of∇.

♦Classifying map

ν:S →M^ir_B(X0,GLr).

♦dν is a section of ν^∗T^(1,0)M^ir_B(X₀,GL_r)⊗ A¹_S.

♦The fibers ofν^∗T^(1,0)M^ir_B(X0,GLr) are H¹(Γ,Ad(ρ)).

Theorem

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

− 1 8π²

Z

X0

(tr(dν∪dν)−tr(dν)∪tr(dν))∈A²(S).

Corollary

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 ⊗Ω¹_X/S. Then ∇^IC2 is holomorphic.

♦Does not require∇to be irreducible.

Complex Chern–Simons

Setting

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

♦M:=M^ir_dR(X/S,SL_r) 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 (E^un,∇^un:E^un→ E^un⊗Ω¹_X/M) which is universal modulo twisting byr-torsion line bundles coming from M.

The complex Chern–Simons line bundle

♦The locally defined IC2(E^un) glue into a global line bundle.

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

Definition

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

(IC2(E^un),∇^IC2)^∨. We denote it LCS(X/S).

♦Riemann–Hilbert: RH :M^ir_dR(X/S,SLr)'M^ir_B(X/S,SLr).

♦IfS ={0} is a point, then M^ir_B(X₀,SL_r) carries a natural symplectic holomorphic 2-formω_G (Atiyah–Bott–Goldman):

c₁(L_CS(X₀)) = 1 8π²ω_G.

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

p^∗LCS(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:

M^ir_B(X/S,SL₂)

P(X/S)

hol //

holf 66

M^ir_B(X/S,PSL₂).

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

♦The holomorphic vector bundle with connection K_CS(X/S) :=holf^∗L_CS(X/S)^⊗4

Theorem

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

KCS(X/S)'π^∗hω_X/S, ω_X/Si.

♦In particular, given aC^∞ / holomorphic section

σ:S → P(X/S), σ^∗K_CS(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, hω_X/S, ω_X/Si would admit a holomorphic connection, necessarily flat sinceS is one-dimensional. This contradicts Arakelov’s positivity theorem.

Fuchsian projective structures

♦T_g Teichm¨uller space of genus g.

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

♦Fuchsian uniformization:

X_t'Γ_t\H, Γ_t ⊂PSL₂(R) Fuchsian group, t ∈ T_g.

♦C^∞ section ϕ:T_g → P(X/T_g).

♦ϕ^∗KCS(X/T_g) C^∞ connection ∇_T onhω_X/Tg, ωX/T_gi.

Theorem

∇_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: κ² 'ω_X.

♦J¹(κ^∨) the first jet bundle ofκ^∨:

0→κ^∨⊗ω_X → J¹(κ^∨)→κ^∨→0.

♦Projective structures onX !flat connections onJ¹(κ^∨).

♦Equivalent to the theory of Schwarzian equations.

♦Fuchsian uniformization ofX flat connection on J¹(κ^∨).

♦J¹(κ^∨)'_C^∞ κ⊕κ^∨, with

∂=

∂_κ β 0 ∂_κ^∨

andβ∈Hom(κ^∨, κ)⊗ A^0,1_X ' A^1,1_X is the (1,1)-form induced by

dx∧dy 4y² onH.

♦Higgs field: Φ =

0 0 1 0

∈End(κ⊕κ^∨)⊗Ω¹_X.

♦The flat connection onJ¹(κ^∨) decomposes as:

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

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

Corollary (Wolpert again)

c₁(κb₁) = 1 2π²ω_WP.

♦Follows from:

I c₁(L_CS(X₀)^⊗4) = _8π⁴2ω_G, Atiyah–Bott–Goldman form.

I Goldman: the pullback ofωG toT_g ⊂“M^ir_B(X0,PSL2(R))” is

THANKS!