Quillen metric for singular families of Riemann surfaces with cusps - - PowerPoint PPT Presentation

Quillen metric for singular families

• f Riemann surfaces with cusps

Finski Siarhei Institute Fourier Université Grenoble Alpes 15 December 2019 Taipei, Taiwan

Plan of the talk

1

Determinant line bundle and Quillen metric

2

Wolpert norm for surfaces with cusps

3

Quillen metric for surfaces with cusps

4

Continuity theorem

5

Restriction theorem

6

Applications

Determinant line bundle and Quillen metric

Family setting π :X → S proper holomorphic submersion, relative dimension 1

Family setting π :X → S proper holomorphic submersion, relative dimension 1 ωX/S = (ΛmaxT ∗(1,0)X) ⊗ (ΛmaxT ∗(1,0)S)−1 the relative canonical line bundle of π t ∈ S, Xt = π−1(t)

A family of Riemann surfaces

S X

Dolbeaut complex ξ a holomorphic vector bundle over X

Dolbeaut complex ξ a holomorphic vector bundle over X Ωi,j(Xt, ξ) = C ∞(Xt, T ∗(i,j)Xt ⊗ ξ), i, j = 0, 1

Dolbeaut complex ξ a holomorphic vector bundle over X Ωi,j(Xt, ξ) = C ∞(Xt, T ∗(i,j)Xt ⊗ ξ), i, j = 0, 1 0 − → Ω0,0(Xt, ξ) ∂ − → Ω0,1(Xt, ξ) − → 0

Dolbeaut complex ξ a holomorphic vector bundle over X Ωi,j(Xt, ξ) = C ∞(Xt, T ∗(i,j)Xt ⊗ ξ), i, j = 0, 1 0 − → Ω0,0(Xt, ξ) ∂ − → Ω0,1(Xt, ξ) − → 0 H0(Xt, ξ) = ker(∂), H1(Xt, ξ) = Ω0,1(Xt, ξ)/ Im(∂)

Grothendieck-Knudsen-Mumford construction and Quillen metric The determinant of the cohomology λ(j∗ξ)t = (ΛmaxH0(Xt, ξ|Xt))−1 ⊗ ΛmaxH1(Xt, ξ|Xt), t ∈ S family of complex lines over S

Grothendieck-Knudsen-Mumford construction and Quillen metric The determinant of the cohomology λ(j∗ξ)t = (ΛmaxH0(Xt, ξ|Xt))−1 ⊗ ΛmaxH1(Xt, ξ|Xt), t ∈ S family of complex lines over S Grothendieck-Knudsen-Mumford : λ(j∗ξ)t, t ∈ S form a holomorphic line bundle λ(j∗ξ) over S

Grothendieck-Knudsen-Mumford construction and Quillen metric The determinant of the cohomology λ(j∗ξ)t = (ΛmaxH0(Xt, ξ|Xt))−1 ⊗ ΛmaxH1(Xt, ξ|Xt), t ∈ S family of complex lines over S Grothendieck-Knudsen-Mumford : λ(j∗ξ)t, t ∈ S form a holomorphic line bundle λ(j∗ξ) over S ·ω

X/S a Hermitian norm on ωX/S, gTXt - restriction on Xt

hξ a Hermitian metric over X Quillen metric·Q (gTXt, hξ) is a natural metric on λ(j∗ξ)

L2 product and Hodge theory L2-Hermitian product. Let α, α′ ∈ Ω0,•(Xt, ξ)

• α, α′

L2 =

• Xt
• α(x), α′(x)
• hdvXt(x),

·, ·h the pointwise Hermitian product induced by hξ,·ω

X/S.

L2 product and Hodge theory L2-Hermitian product. Let α, α′ ∈ Ω0,•(Xt, ξ)

• α, α′

L2 =

• Xt
• α(x), α′(x)
• hdvXt(x),

·, ·h the pointwise Hermitian product induced by hξ,·ω

X/S.

0 − → Ω0,0(Xt, ξ) ∂ − → Ω0,1(Xt, ξ) − → 0, ξ

t = ∂ ∂ ∗ + ∂ ∗∂

L2 product and Hodge theory L2-Hermitian product. Let α, α′ ∈ Ω0,•(Xt, ξ)

• α, α′

L2 =

• Xt
• α(x), α′(x)
• hdvXt(x),

·, ·h the pointwise Hermitian product induced by hξ,·ω

X/S.

0 − → Ω0,0(Xt, ξ) ∂ − → Ω0,1(Xt, ξ) − → 0, ξ

t = ∂ ∂ ∗ + ∂ ∗∂

• ξ

t α, α

• L2 =
• ∂α, ∂α
• +
• ∂

∗α, ∂ ∗α

• ,

ker(ξ|Ω0,•(Xt,ξ)) = {s ∈ Ω0,•(Xt, ξ) | ∂s = 0, ∂

∗s = 0}

L2 product and Hodge theory L2-Hermitian product. Let α, α′ ∈ Ω0,•(Xt, ξ)

• α, α′

L2 =

• Xt
• α(x), α′(x)
• hdvXt(x),

·, ·h the pointwise Hermitian product induced by hξ,·ω

X/S.

0 − → Ω0,0(Xt, ξ) ∂ − → Ω0,1(Xt, ξ) − → 0, ξ

t = ∂ ∂ ∗ + ∂ ∗∂

• ξ

t α, α

• L2 =
• ∂α, ∂α
• +
• ∂

∗α, ∂ ∗α

• ,

ker(ξ|Ω0,•(Xt,ξ)) = {s ∈ Ω0,•(Xt, ξ) | ∂s = 0, ∂

∗s = 0}

ker(ξ|Ω0,•(Xt,ξ)) → H•(Xt, ξ)

L2 product and Hodge theory L2-Hermitian product. Let α, α′ ∈ Ω0,•(Xt, ξ)

• α, α′

L2 =

• Xt
• α(x), α′(x)
• hdvXt(x),

·, ·h the pointwise Hermitian product induced by hξ,·ω

X/S.

0 − → Ω0,0(Xt, ξ) ∂ − → Ω0,1(Xt, ξ) − → 0, ξ

t = ∂ ∂ ∗ + ∂ ∗∂

• ξ

t α, α

• L2 =
• ∂α, ∂α
• +
• ∂

∗α, ∂ ∗α

• ,

ker(ξ|Ω0,•(Xt,ξ)) = {s ∈ Ω0,•(Xt, ξ) | ∂s = 0, ∂

∗s = 0}

ker(ξ|Ω0,•(Xt,ξ)) ≃ H•(Xt, ξ) Hodge theory

L2 product and Hodge theory L2-Hermitian product. Let α, α′ ∈ Ω0,•(Xt, ξ)

• α, α′

L2 =

• Xt
• α(x), α′(x)
• hdvXt(x),

·, ·h the pointwise Hermitian product induced by hξ,·ω

X/S.

0 − → Ω0,0(Xt, ξ) ∂ − → Ω0,1(Xt, ξ) − → 0, ξ

t = ∂ ∂ ∗ + ∂ ∗∂

• ξ

t α, α

• L2 =
• ∂α, ∂α
• +
• ∂

∗α, ∂ ∗α

• ,

ker(ξ|Ω0,•(Xt,ξ)) = {s ∈ Ω0,•(Xt, ξ) | ∂s = 0, ∂

∗s = 0}

ker(ξ|Ω0,•(Xt,ξ)) ≃ H•(Xt, ξ) Hodge theory induces the L2-norm·L2

• gTXt, hξ
• ver

λ(j∗ξ)t = (ΛmaxH0(Xt, ξ|Xt))−1 ⊗ ΛmaxH1(Xt, ξ|Xt)

Infinite product From now on ξ

t : = ξ|Ω0,0(Xt,ξ)

Infinite product From now on ξ

t : = ξ|Ω0,0(Xt,ξ)

ξ

t essentially self-adjoint

Infinite product From now on ξ

t : = ξ|Ω0,0(Xt,ξ)

ξ

t essentially self-adjoint

Spec(ξ

t ) = {λ1,t, λ2,t, . . .}, λi,t non decreasing, λi,t → ∞

Infinite product From now on ξ

t : = ξ|Ω0,0(Xt,ξ)

ξ

t essentially self-adjoint

Spec(ξ

t ) = {λ1,t, λ2,t, . . .}, λi,t non decreasing, λi,t → ∞

det ′ξ

t = ∞

• λi,t=0

λi,t.

Infinite product From now on ξ

t : = ξ|Ω0,0(Xt,ξ)

ξ

t essentially self-adjoint

Spec(ξ

t ) = {λ1,t, λ2,t, . . .}, λi,t non decreasing, λi,t → ∞

det ′ξ

t = ∞

• λi,t=0

λi,t. Problem : Need to make sense of the infinite product...

Zeta renormalization Weyl’s law : λi,t increase asymptotically linearly with i

Zeta renormalization Weyl’s law : λi,t increase asymptotically linearly with i ζξ,t(s) =

∞

• λi,t=0

1 (λi,t)s , for Re(s) > 1

Zeta renormalization Weyl’s law : λi,t increase asymptotically linearly with i ζξ,t(s) =

∞

• λi,t=0

1 (λi,t)s , for Re(s) > 1 Definition of the determinant. (Ray-Singer, 1973) det ′ξ

t = exp

• − ζ′

ξ,t(0)

Quillen norm Quillen norm Hermitian norm on λ(j∗ξ), given by ·Q gTXt, hξ =

• det ′ξ

t

1/2 ··L2

• gTXt, hξ

Quillen norm Quillen norm Hermitian norm on λ(j∗ξ), given by ·Q gTXt, hξ =

• det ′ξ

t

1/2 ··L2

• gTXt, hξ
• Theorem. (Bismut-Gillet-Soulé, 1988)

Hermitian norm·Q gTXt, hξ is smooth over S.

Applications of Quillen metric

• Refinement of a theorem of Riemann-Roch-Grothendieck on

the level of differential forms, aka Curvature theorem (Bismut-Gillet-Soulé).

Applications of Quillen metric

• Refinement of a theorem of Riemann-Roch-Grothendieck on

the level of differential forms, aka Curvature theorem (Bismut-Gillet-Soulé).

• Arakelov geometry : arithmetic Riemann-Roch theorem

(Gillet-Soulé, Bismut-Lebeau), arithmetic Hilbert-Samuel theorem (Gillet-Soulé, Bismut-Vasserot).

Applications of Quillen metric

• Refinement of a theorem of Riemann-Roch-Grothendieck on

the level of differential forms, aka Curvature theorem (Bismut-Gillet-Soulé).

• Arakelov geometry : arithmetic Riemann-Roch theorem

(Gillet-Soulé, Bismut-Lebeau), arithmetic Hilbert-Samuel theorem (Gillet-Soulé, Bismut-Vasserot).

• Theory of automorphic forms (Yoshikawa).

Applications of Quillen metric

• Refinement of a theorem of Riemann-Roch-Grothendieck on

the level of differential forms, aka Curvature theorem (Bismut-Gillet-Soulé).

• Arakelov geometry : arithmetic Riemann-Roch theorem

(Gillet-Soulé, Bismut-Lebeau), arithmetic Hilbert-Samuel theorem (Gillet-Soulé, Bismut-Vasserot).

• Theory of automorphic forms (Yoshikawa).
• Mirror symmetry (Bershadsky-Cecotti-Ooguri-Vafa,

Fang-Lu-Yoshikawa, Eriksson-Freixas-Mourougane).

Applications of Quillen metric

• Refinement of a theorem of Riemann-Roch-Grothendieck on

the level of differential forms, aka Curvature theorem (Bismut-Gillet-Soulé).

• Arakelov geometry : arithmetic Riemann-Roch theorem

(Gillet-Soulé, Bismut-Lebeau), arithmetic Hilbert-Samuel theorem (Gillet-Soulé, Bismut-Vasserot).

• Theory of automorphic forms (Yoshikawa).
• Mirror symmetry (Bershadsky-Cecotti-Ooguri-Vafa,

Fang-Lu-Yoshikawa, Eriksson-Freixas-Mourougane).

• Explicit evaluation of some special values of Selberg zeta

function on some modular curves (Freixas, Freixas-v. Pippich).

Applications of Quillen metric

• Refinement of a theorem of Riemann-Roch-Grothendieck on

the level of differential forms, aka Curvature theorem (Bismut-Gillet-Soulé).

• Arakelov geometry : arithmetic Riemann-Roch theorem

(Gillet-Soulé, Bismut-Lebeau), arithmetic Hilbert-Samuel theorem (Gillet-Soulé, Bismut-Vasserot).

• Theory of automorphic forms (Yoshikawa).
• Mirror symmetry (Bershadsky-Cecotti-Ooguri-Vafa,

Fang-Lu-Yoshikawa, Eriksson-Freixas-Mourougane).

• Explicit evaluation of some special values of Selberg zeta

function on some modular curves (Freixas, Freixas-v. Pippich).

• Critical phenomenons of some models in statistical

mechanics (Duplandier-David, Dubédat).

What if a family has singular fibers?

Xt X0

t

What if a family has singular fibers?

Xt X0

t Natural example : Deligne-Mumford compactification M g,0.

What if a family has singular fibers?

Xt X0

t Natural example : Deligne-Mumford compactification M g,0. Studying degeneration of·Q gTXt, hξ => extension of Quillen metric theory over singular families

They used degenerating families to study non-singular spaces.

• Refinement of a theorem of Riemann-Roch-Grothendieck on

the level of differential forms, aka Curvature theorem (Bismut-Gillet-Soulé).

• Arakelov geometry : arithmetic Riemann-Roch theorem

(Gillet-Soulé, Bismut-Lebeau), arithmetic Hilbert-Samuel theorem (Gillet-Soulé, Bismut-Vasserot).

• Theory of automorphic forms (Yoshikawa).
• Mirror symmetry (Bershadsky-Cecotti-Ooguri-Vafa,

Fang-Lu-Yoshikawa, Eriksson-Freixas-Mourougane).

• Explicit evaluation of some special values of Selberg zeta

function on some modular curves (Freixas, Freixas-v. Pippich).

• Critical phenomenons of some models in statistical

mechanics (Duplandier-David, Dubédat).

Family of curves π :X → S proper holomorphic of relative dimension 1, t ∈ S, Xt = π−1(t) has at most double-point singularities (i.e. of the form {z0z1 = 0})

Family of curves π :X → S proper holomorphic of relative dimension 1, t ∈ S, Xt = π−1(t) has at most double-point singularities (i.e. of the form {z0z1 = 0}) ΣX/S singular points of the fibers, ∆ = π∗(ΣX/S)

Family of curves π :X → S proper holomorphic of relative dimension 1, t ∈ S, Xt = π−1(t) has at most double-point singularities (i.e. of the form {z0z1 = 0}) ΣX/S singular points of the fibers, ∆ = π∗(ΣX/S) ·ω

X/S a Hermitian norm on ωX/S over X \ π−1(|∆|)

gTXt - restriction on Xt, t ∈ S \ |∆| (ξ, hξ) a holomorphic Hermitian vector bundle over X

A theorem of Bismut-Bost Ln = λ(j∗ξ)12 ⊗ OS(∆)rk(ξ)

A theorem of Bismut-Bost Ln = λ(j∗ξ)12 ⊗ OS(∆)rk(ξ) ·Ln =

• ·Q

gTXt, hξ12 ⊗ (·div

∆ )rk(ξ)

·div

∆ is the canonical divisor norm on OS(∆), i.e. over S \ |∆|,

s∆div

∆ = 1 for the canonical holomoprhic section s∆.

A theorem of Bismut-Bost Ln = λ(j∗ξ)12 ⊗ OS(∆)rk(ξ) ·Ln =

• ·Q

gTXt, hξ12 ⊗ (·div

∆ )rk(ξ)

·div

∆ is the canonical divisor norm on OS(∆), i.e. over S \ |∆|,

s∆div

∆ = 1 for the canonical holomoprhic section s∆.

Theorem (Bismut-Bost, 1990) If the metric·ω

X/S extends smoothly over X, the induced norm

·Ln on Ln extends continuously to |∆|

A theorem of Bismut-Bost Ln = λ(j∗ξ)12 ⊗ OS(∆)rk(ξ) ·Ln =

• ·Q

gTXt, hξ12 ⊗ (·div

∆ )rk(ξ)

·div

∆ is the canonical divisor norm on OS(∆), i.e. over S \ |∆|,

s∆div

∆ = 1 for the canonical holomoprhic section s∆.

Theorem (Bismut-Bost, 1990) If the metric·ω

X/S extends smoothly over X, the induced norm

·Ln on Ln extends continuously to |∆| Problem : Typically, the metric·ω

X/S doesn’t extend smoothly.

In DM compact., there are degenerating “hyperbolic cylinders"

Continuity theorem (basic version) and the goal Ln = λ(j∗ξ)12 ⊗ OS(∆)rk(ξ) ·Ln =

• ·Q

gTXt, hξ12 ⊗ (·div

∆ )rk(ξ)

Continuity theorem (-, 2018) Under some hypothesizes on degeneration of metric near singularities (more general than in DM compactification), the norm Ln on·Ln extends continuously to |∆|

Continuity theorem (basic version) and the goal Ln = λ(j∗ξ)12 ⊗ OS(∆)rk(ξ) ·Ln =

• ·Q

gTXt, hξ12 ⊗ (·div

∆ )rk(ξ)

Continuity theorem (-, 2018) Under some hypothesizes on degeneration of metric near singularities (more general than in DM compactification), the norm Ln on·Ln extends continuously to |∆| GOAL : study the geometric meaning of this extension.

What to expect?

Xt X0

t

What to expect?

Xt X0 Y0

t

What to expect?

Xt X0 Y0

t

Belief : should be related to the Quillen metric of normalization.

What to expect?

Xt X0 Y0

t

Belief : should be related to the Quillen metric of normalization. If metric comes from total space, an analogical result has been proved by by Bismut, 1997, basing on Bismut-Lebeau, 1991.

Our case is different Problem : In many natural examples, the induced metric on the normalization is singular.

Our case is different Problem : In many natural examples, the induced metric on the normalization is singular. In Deligne-Mumford compactification and csc −1 metric on the fibers, the metric on normalization has cusp singularities,

• btained from the degeneration of “hyperbolic cylinders".

Our case is different Problem : In many natural examples, the induced metric on the normalization is singular. In Deligne-Mumford compactification and csc −1 metric on the fibers, the metric on normalization has cusp singularities,

• btained from the degeneration of “hyperbolic cylinders".

Our case is different Problem : In many natural examples, the induced metric on the normalization is singular. In Deligne-Mumford compactification and csc −1 metric on the fibers, the metric on normalization has cusp singularities,

• btained from the degeneration of “hyperbolic cylinders".

Classical Quillen metric is not defined for those spaces.

What is a surface with cusps?

What is a surface with cusps? M a compact Riemann surface DM = {P1, P2, . . . , Pm} ⊂ M, M = M \ DM

What is a surface with cusps? M a compact Riemann surface DM = {P1, P2, . . . , Pm} ⊂ M, M = M \ DM gTM is a Kähler metric on M z1, . . . , zm local holomorphic coordinates, zi(0) = {Pi} Suppose gTM over {|zi| < ǫ} is induced by √ −1dzidzi

• zi log |zi|
• 2 .

What is a surface with cusps? M a compact Riemann surface DM = {P1, P2, . . . , Pm} ⊂ M, M = M \ DM gTM is a Kähler metric on M z1, . . . , zm local holomorphic coordinates, zi(0) = {Pi} Suppose gTM over {|zi| < ǫ} is induced by √ −1dzidzi

• zi log |zi|
• 2 .

We call (M, DM, gTM) a surface with cusps

An important example Suppose 2g(M) − 2 + #DM > 0, i.e. (M, DM) is stable

An important example Suppose 2g(M) − 2 + #DM > 0, i.e. (M, DM) is stable By uniformization theorem, there is exactly one csc −1 complete metric gTM

hyp on M = M \ DM

An important example Suppose 2g(M) − 2 + #DM > 0, i.e. (M, DM) is stable By uniformization theorem, there is exactly one csc −1 complete metric gTM

hyp on M = M \ DM

The triple (M, DM, gTM

hyp) is a surface with cusps

Goals for this talk, detailed version

• 1. Define Quillen metric for surfaces with cusp singularities.

Goals for this talk, detailed version

• 1. Define Quillen metric for surfaces with cusp singularities.
• 2. Prove that this Quillen metric can be obtained when the

cusps are created by degeneration (Restriction theorem).

Xt X0 Y0

t

Goals for this talk, detailed version

• 1. Define Quillen metric for surfaces with cusp singularities.
• 2. Prove that this Quillen metric can be obtained when the

cusps are created by degeneration (Restriction theorem).

Xt X0 Y0

t

• 3. Explain some applications.

Motivation

Motivation

• 1. Generalization and analytic interpretation of

Takhtajan-Zograf analytic torsion

Motivation

• 1. Generalization and analytic interpretation of

Takhtajan-Zograf analytic torsion => Unification of curvature theorems of Bismut-Gillet-Soulé and Takhtajan-Zograf.

Motivation

• 1. Generalization and analytic interpretation of

Takhtajan-Zograf analytic torsion => Unification of curvature theorems of Bismut-Gillet-Soulé and Takhtajan-Zograf.

• 2. Weil-Petersson form ωg,m

WP on the moduli space of m-pointed

stable curves Mg,m of genus g

Motivation

• 1. Generalization and analytic interpretation of

Takhtajan-Zograf analytic torsion => Unification of curvature theorems of Bismut-Gillet-Soulé and Takhtajan-Zograf.

• 2. Weil-Petersson form ωg,m

WP on the moduli space of m-pointed

stable curves Mg,m of genus g satisfies∗ ωg,m

WP |Mg1,m1×Mg,m2 = ωg1,m1 WP

⊕ ωg2,m2

WP

, for g1 + g2 = g, m1 + m2 − 2 = m.

Motivation

• 1. Generalization and analytic interpretation of

Takhtajan-Zograf analytic torsion => Unification of curvature theorems of Bismut-Gillet-Soulé and Takhtajan-Zograf.

• 2. Weil-Petersson form ωg,m

WP on the moduli space of m-pointed

stable curves Mg,m of genus g satisfies∗ ωg,m

WP |Mg1,m1×Mg,m2 = ωg1,m1 WP

⊕ ωg2,m2

WP

, for g1 + g2 = g, m1 + m2 − 2 = m. Similarly, in Pic(M g,m),

• λ12

g,m ⊗ ψ−1 g,m ⊗ OM g,m(∂Mg,m)

• |Mg1,m1×Mg2,m2

≃

• λ12

g1,m1 ⊗ ψ−1 g1,m1

• ⊠
• λ12

g2,m2 ⊗ ψ−1 g2,m2

• .

Motivation

• 1. Generalization and analytic interpretation of

Takhtajan-Zograf analytic torsion => Unification of curvature theorems of Bismut-Gillet-Soulé and Takhtajan-Zograf.

• 2. Weil-Petersson form ωg,m

WP on the moduli space of m-pointed

stable curves Mg,m of genus g satisfies∗ ωg,m

WP |Mg1,m1×Mg,m2 = ωg1,m1 WP

⊕ ωg2,m2

WP

, for g1 + g2 = g, m1 + m2 − 2 = m. Similarly, in Pic(M g,m),

• λ12

g,m ⊗ ψ−1 g,m ⊗ OM g,m(∂Mg,m)

• |Mg1,m1×Mg2,m2

≃

• λ12

g1,m1 ⊗ ψ−1 g1,m1

• ⊠
• λ12

g2,m2 ⊗ ψ−1 g2,m2

• .

Finally, in H2(Mg,m), we have∗ c1(λ12

g,m ⊗ ψ−1 g,m) = [ωg,m WP ]DR.

Motivation

• 1. Generalization and analytic interpretation of

Takhtajan-Zograf analytic torsion => Unification of curvature theorems of Bismut-Gillet-Soulé and Takhtajan-Zograf.

• 2. Weil-Petersson form ωg,m

WP on the moduli space of m-pointed

stable curves Mg,m of genus g satisfies∗ ωg,m

WP |Mg1,m1×Mg,m2 = ωg1,m1 WP

⊕ ωg2,m2

WP

, for g1 + g2 = g, m1 + m2 − 2 = m. Similarly, in Pic(M g,m),

• λ12

g,m ⊗ ψ−1 g,m ⊗ OM g,m(∂Mg,m)

• |Mg1,m1×Mg2,m2

≃

• λ12

g1,m1 ⊗ ψ−1 g1,m1

• ⊠
• λ12

g2,m2 ⊗ ψ−1 g2,m2

• .

Finally, in H2(Mg,m), we have∗ c1(λ12

g,m ⊗ ψ−1 g,m) = [ωg,m WP ]DR.

Are those statements form a part of one theorem?

Motivation

• 1. Generalization and analytic interpretation of

Takhtajan-Zograf analytic torsion => Unification of curvature theorems of Bismut-Gillet-Soulé and Takhtajan-Zograf.

• 2. Weil-Petersson form ωg,m

WP on the moduli space of m-pointed

stable curves Mg,m of genus g satisfies∗ ωg,m

WP |Mg1,m1×Mg,m2 = ωg1,m1 WP

⊕ ωg2,m2

WP

, for g1 + g2 = g, m1 + m2 − 2 = m. Similarly, in Pic(M g,m),

• λ12

g,m ⊗ ψ−1 g,m ⊗ OM g,m(∂Mg,m)

• |Mg1,m1×Mg2,m2

≃

• λ12

g1,m1 ⊗ ψ−1 g1,m1

• ⊠
• λ12

g2,m2 ⊗ ψ−1 g2,m2

• .

Finally, in H2(Mg,m), we have∗ c1(λ12

g,m ⊗ ψ−1 g,m) = [ωg,m WP ]DR.

Are those statements form a part of one theorem?

• 3. Related to the arithmetic Riemann-Roch theorem for pointed

stable curves, studied by Gillet-Soulé, Deligne, Freixas.

Isomorphism on the level of line bundles

Isomorphism on the level of line bundles

Xt X0 Y0

t

Family of curves and normalization Let π :X → S same family ΣX/S singular points of the fibers, ∆ = π∗(ΣX/S)

Family of curves and normalization Let π :X → S same family ΣX/S singular points of the fibers, ∆ = π∗(ΣX/S) For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0}

Family of curves and normalization Let π :X → S same family ΣX/S singular points of the fibers, ∆ = π∗(ΣX/S) For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0} Let ρ :Y → X0 be the normalisation of the singular fiber Let DY : = ρ−1(ΣX/S), here ΣX/S singular points

Family of curves and normalization Let π :X → S same family ΣX/S singular points of the fibers, ∆ = π∗(ΣX/S) For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0} Let ρ :Y → X0 be the normalisation of the singular fiber Let DY : = ρ−1(ΣX/S), here ΣX/S singular points ωY(D) : = ωY ⊗ OY(DY)

Family of curves and normalization Let π :X → S same family ΣX/S singular points of the fibers, ∆ = π∗(ΣX/S) For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0} Let ρ :Y → X0 be the normalisation of the singular fiber Let DY : = ρ−1(ΣX/S), here ΣX/S singular points ωY(D) : = ωY ⊗ OY(DY) ωY(D) ≃ ρ∗(ωX/S)

Restriction of the line bundle to singular locus We want to describe the restriction of the line bundle Ln = λ

• j∗(ξ ⊗ ωn

X/S)

12 ⊗ OS(∆)rk(ξ) to ∆ as some natural line bundle on the normalization.

Restriction of the divisor line bundle to the singular locus

Ln = λ

• j∗(ξ ⊗ ωn

X/S)

12 ⊗ OS(∆)rk(ξ)

Poincaré residue morphism We denote by k : = #ΣX/S, then

• ωk

S ⊗ OS(∆)

• ||∆| → O|∆|.

Poincaré residue morphism We denote by k : = #ΣX/S, then

• ωk

S ⊗ OS(∆)

• ||∆| → O|∆|.

Double-point singularities give a natural isomorphism ωk

S||∆| → ⊗P∈ρ−1(ΣX/S)ωY0|P

Poincaré residue morphism We denote by k : = #ΣX/S, then

• ωk

S ⊗ OS(∆)

• ||∆| → O|∆|.

Double-point singularities give a natural isomorphism ωk

S||∆| → ⊗P∈ρ−1(ΣX/S)ωY0|P

By combining, we have a natural isomorphism OS(∆)||∆| →

• ⊗P∈ρ−1(ΣX/S) ωY0|P

−1

Restriction of the determinant to the singular locus

Ln = λ

• j∗(ξ ⊗ ωn

X/S)

12 ⊗ OS(∆)rk(ξ)

Restriction of the determinant to the singular locus Short exact sequence 0 → OX0

• ξ ⊗ ωn

X/S

• → ρ∗OY
• ρ∗(ξ) ⊗ ωY(D)n

→ OΣX/S

• ξ|ΣX/S
• → 0,

Restriction of the determinant to the singular locus Short exact sequence 0 → OX0

• ξ ⊗ ωn

X/S

• → ρ∗OY
• ρ∗(ξ) ⊗ ωY(D)n

→ OΣX/S

• ξ|ΣX/S
• → 0,

Induces isomorphism λ

• j∗(ξ ⊗ ωn

X/S)

• ||∆|

≃ λ

• ρ∗(ξ) ⊗ ωY(D)n

⊗ det

• π∗(ξ|ΣX/S)
• .

A combination is the answer For the line bundle Ln, defined by Ln = λ

• j∗(ξ ⊗ ωn

X/S)

12 ⊗ OS(∆)rk(ξ), we have the following isomorphism of line bundles on |∆| Ln||∆| ≃ λ

• ρ∗(ξ) ⊗ ωY(D)n12

⊗ det

• π∗(ξ|ΣX/S)

12 ⊗

• ⊗P∈ρ−1(ΣX/S) ωY0|P

−rk(ξ) .

Wolpert norm for surfaces with cusps

The Wolpert norm (M, DM, gTM), DM = {P1, . . . , Pm} surface with cusps

The Wolpert norm (M, DM, gTM), DM = {P1, . . . , Pm} surface with cusps z1, . . . , zm local holomorphic coordinates, zi(0) = {Pi} gTM over {|zi| < ǫ} is induced by √ −1dzidzi

• zi log |zi|
• 2

The Wolpert norm (M, DM, gTM), DM = {P1, . . . , Pm} surface with cusps z1, . . . , zm local holomorphic coordinates, zi(0) = {Pi} gTM over {|zi| < ǫ} is induced by √ −1dzidzi

• zi log |zi|
• 2

Wolpert norm ·W on ⊗m

i=1ωM|Pi is defined by

• ⊗i dzi|Pi
• W = 1.

The Wolpert norm (M, DM, gTM), DM = {P1, . . . , Pm} surface with cusps z1, . . . , zm local holomorphic coordinates, zi(0) = {Pi} gTM over {|zi| < ǫ} is induced by √ −1dzidzi

• zi log |zi|
• 2

Wolpert norm ·W on ⊗m

i=1ωM|Pi is defined by

• ⊗i dzi|Pi
• W = 1.
• n

D∗ √ −1dzdz

• z log |z|
• 2
• dz|0
• W = 1

The Wolpert norm (M, DM, gTM), DM = {P1, . . . , Pm} surface with cusps z1, . . . , zm local holomorphic coordinates, zi(0) = {Pi} gTM over {|zi| < ǫ} is induced by √ −1dzidzi

• zi log |zi|
• 2

Wolpert norm ·W on ⊗m

i=1ωM|Pi is defined by

• ⊗i dzi|Pi
• W = 1.
• n

D∗ √ −1dzdz

• z log |2z|
• 2
• dz|0
• W = 1

2

Quillen metric for surfaces with cusps

The L2-norm

·Q =

• det ′

1/2 ··L2

The L2-norm Let (M, DM, gTM) be a surface with cusps ·ω

M the induced Hermitian norm on ωM over M

The L2-norm Let (M, DM, gTM) be a surface with cusps ·ω

M the induced Hermitian norm on ωM over M

ωM(D) = ωM ⊗ OM(DM) the twisted canonical line bundle

The L2-norm Let (M, DM, gTM) be a surface with cusps ·ω

M the induced Hermitian norm on ωM over M

ωM(D) = ωM ⊗ OM(DM) the twisted canonical line bundle ωM(D) ≃ ωM,

• ver M

induces the Hermitian norm·M on ωM(D) over M

The L2-norm Let (M, DM, gTM) be a surface with cusps ·ω

M the induced Hermitian norm on ωM over M

ωM(D) = ωM ⊗ OM(DM) the twisted canonical line bundle ωM(D) ≃ ωM,

• ver M

induces the Hermitian norm·M on ωM(D) over M This norm has log singularity

• dzi ⊗ sDM/zi
• M = | log |zi||

The L2-norm Let (M, DM, gTM) be a surface with cusps ·ω

M the induced Hermitian norm on ωM over M

ωM(D) = ωM ⊗ OM(DM) the twisted canonical line bundle ωM(D) ≃ ωM,

• ver M

induces the Hermitian norm·M on ωM(D) over M This norm has log singularity

• dzi ⊗ sDM/zi
• M = | log |zi||

(ξ, hξ) a holomorphic Hermitian vector bundle over M

The L2-norm Let (M, DM, gTM) be a surface with cusps ·ω

M the induced Hermitian norm on ωM over M

ωM(D) = ωM ⊗ OM(DM) the twisted canonical line bundle ωM(D) ≃ ωM,

• ver M

induces the Hermitian norm·M on ωM(D) over M This norm has log singularity

• dzi ⊗ sDM/zi
• M = | log |zi||

(ξ, hξ) a holomorphic Hermitian vector bundle over M Eξ

n = ξ ⊗ ωM(D)n,

hξ ⊗ (·M)2n

The L2-norm Let (M, DM, gTM) be a surface with cusps ·ω

M the induced Hermitian norm on ωM over M

ωM(D) = ωM ⊗ OM(DM) the twisted canonical line bundle ωM(D) ≃ ωM,

• ver M

induces the Hermitian norm·M on ωM(D) over M This norm has log singularity

• dzi ⊗ sDM/zi
• M = | log |zi||

(ξ, hξ) a holomorphic Hermitian vector bundle over M Eξ

n = ξ ⊗ ωM(D)n,

hξ ⊗ (·M)2n For n ≤ 0, by Hodge theory∗

• ·, ·
• L2 induces the L2-norm·L2 on

λ(Eξ

n) = (ΛmaxH0(M, Eξ n))−1 ⊗ ΛmaxH1(M, Eξ n)

The determinant

·Q =

• det ′

1/2 ··L2

Problem with the determinant Eξ

n :Ω0,0(M, Eξ

n) → Ω0,0(M, Eξ n)

Problem with the determinant Eξ

n :Ω0,0(M, Eξ

n) → Ω0,0(M, Eξ n)

As M is non-compact, in general Spec(Eξ

n ) is not discrete

Problem with the determinant Eξ

n :Ω0,0(M, Eξ

n) → Ω0,0(M, Eξ n)

As M is non-compact, in general Spec(Eξ

n ) is not discrete

det ′Eξ

n =

∞

• λi=0

λi.

Takhtajan-Zograf approach { Length of closed geodesics } ↔ Spec(Eξ

n )

Takhtajan-Zograf approach { Length of closed geodesics } ↔ Spec(Eξ

n )

Suppose (ξ, hξ) trivial, (M, DM, gTM

hyp) has csc −1

Takhtajan-Zograf approach { Length of closed geodesics } ↔ Spec(Eξ

n )

Suppose (ξ, hξ) trivial, (M, DM, gTM

hyp) has csc −1

then the set of primitive closed geodesics is discrete

Takhtajan-Zograf approach { Length of closed geodesics } ↔ Spec(Eξ

n )

Suppose (ξ, hξ) trivial, (M, DM, gTM

hyp) has csc −1

then the set of primitive closed geodesics is discrete Z(M,DM)(s) =

• γ

∞

• k=0

(1 − e−(s+k)l(γ)) γ primitive closed geodesics on M ; l(γ) is the length of γ.

Takhtajan-Zograf approach { Length of closed geodesics } ↔ Spec(Eξ

n )

Suppose (ξ, hξ) trivial, (M, DM, gTM

hyp) has csc −1

then the set of primitive closed geodesics is discrete Z(M,DM)(s) =

• γ

∞

• k=0

(1 − e−(s+k)l(γ)) γ primitive closed geodesics on M ; l(γ) is the length of γ. Takhtajan-Zograf definition using Selberg zeta-function, 1991 det ′

TZEξ

n =

   Z ′

(M,DM)(1),

for n = 0, Z(M,DM)(−n + 1), for n < 0.

Takhtajan-Zograf approach { Length of closed geodesics } ↔ Spec(Eξ

n )

Suppose (ξ, hξ) trivial, (M, DM, gTM

hyp) has csc −1

then the set of primitive closed geodesics is discrete Z(M,DM)(s) =

• γ

∞

• k=0

(1 − e−(s+k)l(γ)) γ primitive closed geodesics on M ; l(γ) is the length of γ. Takhtajan-Zograf definition using Selberg zeta-function, 1991 det ′

TZEξ

n =

   Z ′

(M,DM)(1),

for n = 0, Z(M,DM)(−n + 1), for n < 0. Motivated by a theorem of Phong-D’Hoker, 1986, which says that when m = 0, two sides of the previous equation coincide∗

Takhtajan-Zograf version of curvature theorem With this definition, Takhtajan-Zograf, 1991, proved a curvature-type theorem for surfaces with cusps. This was the first curvature-type theorem for holomorphic families with non-compact fibers.

Drawbacks of this definition

• 1. Only for csc −1 metrics.

Drawbacks of this definition

• 1. Only for csc −1 metrics.
• 2. No liberty∗ in choosing (ξ, hξ).

Drawbacks of this definition

• 1. Only for csc −1 metrics.
• 2. No liberty∗ in choosing (ξ, hξ).
• 3. Not clear how to put it in the framework of Bismut,

Bismut-Gillet-Soulé, Bismut-Lebeau, which study problems beyond the curvature theorem.

Analytic approach to the determinant λ−s = 1 Γ(s) +∞ exp(−λt)ts−1dt

Analytic approach to the determinant λ−s = 1 Γ(s) +∞ exp(−λt)ts−1dt If M is compact, i.e. m = 0 ζEξ

n (s) =

• λ∈Spec(Eξ

n )\{0}

λ−s (⋆) = 1 Γ(s) +∞ Tr

• exp⊥(−tEξ

n )

• ts−1dt

(⋆⋆)

Analytic approach to the determinant λ−s = 1 Γ(s) +∞ exp(−λt)ts−1dt If M is compact, i.e. m = 0 ζEξ

n (s) =

• λ∈Spec(Eξ

n )\{0}

λ−s (⋆) = 1 Γ(s) +∞ Tr

• exp⊥(−tEξ

n )

• ts−1dt

(⋆⋆) For m > 0? Idea : define ζEξ

n (s) for m > 0 using (⋆⋆) and not (⋆)

Analytic approach to the determinant λ−s = 1 Γ(s) +∞ exp(−λt)ts−1dt If M is compact, i.e. m = 0 ζEξ

n (s) =

• λ∈Spec(Eξ

n )\{0}

λ−s (⋆) = 1 Γ(s) +∞ Tr

• exp⊥(−tEξ

n )

• ts−1dt

(⋆⋆) For m > 0? Idea : define ζEξ

n (s) for m > 0 using (⋆⋆) and not (⋆)

Problem : exp⊥(−tEξ

n ) is not of trace class for m > 0

Regularizing trace, I The operator exp(−tEξ

n ) has a smooth Schwartz kernel

exp(−tEξ

n )(x, y) ∈ (Eξ

n)x ⊗ (Eξ n)∗ y,

x, y ∈ M exp(−tEξ

n )s =

• M
• exp(−tEξ

n )(x, y), s(y)

• dvM(y).

Regularizing trace, I The operator exp(−tEξ

n ) has a smooth Schwartz kernel

exp(−tEξ

n )(x, y) ∈ (Eξ

n)x ⊗ (Eξ n)∗ y,

x, y ∈ M exp(−tEξ

n )s =

• M
• exp(−tEξ

n )(x, y), s(y)

• dvM(y).

If m = 0, Tr

• exp(−tEξ

n )

• =
• M

Tr

• exp(−tEξ

n )(x, x)

• dvM(x).

Regularizing trace, I The operator exp(−tEξ

n ) has a smooth Schwartz kernel

exp(−tEξ

n )(x, y) ∈ (Eξ

n)x ⊗ (Eξ n)∗ y,

x, y ∈ M exp(−tEξ

n )s =

• M
• exp(−tEξ

n )(x, y), s(y)

• dvM(y).

If m = 0, Tr

• exp(−tEξ

n )

• =
• M

Tr

• exp(−tEξ

n )(x, x)

• dvM(x).

Idea : define Trr exp(−tEξ

n )

• by taking the finite part of
• Mr

Tr

• exp(−tEξ

n )(x, x)

• dvM(x)

as r → 0, where Mr is the non-striped region

|Zi|<r

Regularizing trace, II P = CP1 \ {0, 1, ∞}, gTP hyperbolic metric csc −1 over P

Regularizing trace, II P = CP1 \ {0, 1, ∞}, gTP hyperbolic metric csc −1 over P We fix n ≤ 0 gn(r, t) = 1 3

• Pr

exp(−tωP(D)n)(x, x)dvP(x), where Pr is the non-striped region

Regularizing trace, III

• Theorem. (-, 2018)

For any (M, DM, gTM), (ξ, hξ), t > 0, the function R>0 ∋ r →

• Mr

Tr

• exp(−tEξ

n )(x, x)

• dvM(x) − rk(ξ) · m · gn(r, t)

extends continuously over r = 0.

Regularizing trace, IV Regularized heat trace Trr exp(−tEξ

n )

• = lim

r→0 Mr

Tr

• exp(−tEξ

n )(x, x)

• dvM(x)

− rk(ξ) · m · gn(r, t)

• .

Regularized zeta function ζEξ

n (s) =

1 Γ(s) +∞ Trr exp⊥(−tEξ

n )

• ts−1dt.

Regularized zeta function ζEξ

n (s) =

1 Γ(s) +∞ Trr exp⊥(−tEξ

n )

• ts−1dt.
• Theorem. (-, 2018)

ζEξ

n (s) is well-defined and extends meromorphically to C

0 ∈ C is a holomorphic point of ζEξ

n (s)

Finally, the determinant Definition of the determinant∗ det ′Eξ

n = exp

• − ζ′

Eξ

n (0)

• .

Finally, the determinant Definition of the determinant∗ det ′Eξ

n = exp

• − ζ′

Eξ

n (0)

• .

Quillen norm Hermitian norm on λ(Eξ

n), given by

·Q gTM, hEξ

n

=

• det ′Eξ

n 1/2 ··L2

• gTM, hEξ

n

Continuity theorem

A picture

Xt X0

t

What is a family of curves with cusps?

Family of curves with cusps π :X → S same family,

Family of curves with cusps π :X → S same family, σ1, . . . , σm :S → X \ ΣX/S hol. non intersect. sections DX/S = Im(σ1) + · · · + Im(σm)

Family of curves with cusps π :X → S same family, σ1, . . . , σm :S → X \ ΣX/S hol. non intersect. sections DX/S = Im(σ1) + · · · + Im(σm) ·ω

X/S Herm. norm on ωX/S over X \ (|DX/S| ∪ π−1(|∆|))

·ω

X/S |Xt induces metric gTXt on Xt \ |DX/S|, t ∈ S \ |∆|

So that (Xt, {σ1(t), . . . , σm(t)}, gTXt) is a surface with cusps

Family of curves with cusps π :X → S same family, σ1, . . . , σm :S → X \ ΣX/S hol. non intersect. sections DX/S = Im(σ1) + · · · + Im(σm) ·ω

X/S Herm. norm on ωX/S over X \ (|DX/S| ∪ π−1(|∆|))

·ω

X/S |Xt induces metric gTXt on Xt \ |DX/S|, t ∈ S \ |∆|

So that (Xt, {σ1(t), . . . , σm(t)}, gTXt) is a surface with cusps (π :X → S, DX/S,·ω

X/S) a family of curves with cusps

Quillen norm for families of surfaces with cusps Quillen norm We define the Quillen norm on λ(ξ ⊗ ωX/S(D)n) by ·Q gTXt, hξ ⊗·2n

X/S

• =
• det ′Eξ

n

t

1/2 ··L2

• gTXt, hξ ⊗·2n

X/S

• .

Wolpert norm for families Wolpert norm We define the Wolpert norm·W on ⊗iσ∗

i (ωX/S) over S by

gluing the Wolpert norms·W

t

• n ⊗iωX/S|σi(t) induced by gTXt.

Hypothesis on the metric We suppose that the metric ·X/S induced on ωX/S(D) is pre- log-log on X with singularities along π−1(|∆|) ∪ DX/S

Hypothesis on the metric We suppose that the metric ·X/S induced on ωX/S(D) is pre- log-log on X with singularities along π−1(|∆|) ∪ DX/S Notion defied by Burgos Gil-Kramer-Kühn, 2005 It is less restrictive than "good" condition of Mumford, 1977

Hypothesis on the metric We suppose that the metric ·X/S induced on ωX/S(D) is pre- log-log on X with singularities along π−1(|∆|) ∪ DX/S Notion defied by Burgos Gil-Kramer-Kühn, 2005 It is less restrictive than "good" condition of Mumford, 1977 If {z = 0} is a local equation for π−1(|∆|) ∪ DX/S around a smooth point log(υX/S) = O((log | log |z||)N) ∂ log(υX/S) = O

• (log | log |z||)N

dz z log |z|

• ∂∂ log(υX/S) = O
• (log | log |z||)N

dzdz |z log |z||2

Hypothesis on the metric We suppose that the metric ·X/S induced on ωX/S(D) is pre- log-log on X with singularities along π−1(|∆|) ∪ DX/S Notion defied by Burgos Gil-Kramer-Kühn, 2005 It is less restrictive than "good" condition of Mumford, 1977 If {z = 0} is a local equation for π−1(|∆|) ∪ DX/S around a smooth point log(υX/S) = O((log | log |z||)N) ∂ log(υX/S) = O

• (log | log |z||)N

dz z log |z|

• ∂∂ log(υX/S) = O
• (log | log |z||)N

dzdz |z log |z||2

• Wolpert, 1990, (compact case) and Freixas, 2007, (pointed

case) proved : the metric of csc −1 on the relative twisted canonical line bundle of universal curve is good

Continuity theorem, general case Ln = λ(j∗Eξ

n)12 ⊗ (⊗iσ∗ i ωX/S)−rk(ξ) ⊗ OS(∆)rk(ξ) ⊗ (⊗iσ∗ i det ξ)6

Continuity theorem, general case Ln = λ(j∗Eξ

n)12 ⊗ (⊗iσ∗ i ωX/S)−rk(ξ) ⊗ OS(∆)rk(ξ) ⊗ (⊗iσ∗ i det ξ)6

·Ln =

• ·Q

gTXt, hξ ⊗·2n

X/S

12 ⊗

• ·W −rk(ξ)

⊗ (·div

∆ )rk(ξ) ⊗ (⊗iσ∗ i hdet ξ)3

Continuity theorem, general case Ln = λ(j∗Eξ

n)12 ⊗ (⊗iσ∗ i ωX/S)−rk(ξ) ⊗ OS(∆)rk(ξ) ⊗ (⊗iσ∗ i det ξ)6

·Ln =

• ·Q

gTXt, hξ ⊗·2n

X/S

12 ⊗

• ·W −rk(ξ)

⊗ (·div

∆ )rk(ξ) ⊗ (⊗iσ∗ i hdet ξ)3

Continuity theorem. (-, 2018) Under assumption above∗,·Ln extends continuously over |∆|.

Restriction theorem

A picture

Xt X0 Y0

t

Normalization of singular fiber and metric over it For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0}

Normalization of singular fiber and metric over it For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0} Let ρ :Y → X0 be the normalisation of the singular fiber Let DY : = ρ−1(DX/S) + ρ−1(ΣX/S), here ΣX/S singular points

Normalization of singular fiber and metric over it For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0} Let ρ :Y → X0 be the normalisation of the singular fiber Let DY : = ρ−1(DX/S) + ρ−1(ΣX/S), here ΣX/S singular points ωY(D) : = ωY ⊗ OY(DY)

Normalization of singular fiber and metric over it For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0} Let ρ :Y → X0 be the normalisation of the singular fiber Let DY : = ρ−1(DX/S) + ρ−1(ΣX/S), here ΣX/S singular points ωY(D) : = ωY ⊗ OY(DY) ωY(D) ≃ ρ∗(ωX/S(D))

Normalization of singular fiber and metric over it For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0} Let ρ :Y → X0 be the normalisation of the singular fiber Let DY : = ρ−1(DX/S) + ρ−1(ΣX/S), here ΣX/S singular points ωY(D) : = ωY ⊗ OY(DY) ωY(D) ≃ ρ∗(ωX/S(D)) Suppose metric ·X/S on ωX/S(D) extends continuously over X \ (|DX/S| ∪ ΣX/S), and·Y = ρ∗(·X/S) has cusps at DY

Normalization of singular fiber and metric over it For the sake of simplicity, S = D(1) ⊂ C, ∆ = k{0} Let ρ :Y → X0 be the normalisation of the singular fiber Let DY : = ρ−1(DX/S) + ρ−1(ΣX/S), here ΣX/S singular points ωY(D) : = ωY ⊗ OY(DY) ωY(D) ≃ ρ∗(ωX/S(D)) Suppose metric ·X/S on ωX/S(D) extends continuously over X \ (|DX/S| ∪ ΣX/S), and·Y = ρ∗(·X/S) has cusps at DY Wolpert, 1990, (compact case) and Freixas, 2007, (pointed case) proved the pinching expansion, which implies that the metric of csc −1 on the relative twisted canonical line bundle of universal curve satisfies this assumption

Restriction of the line bundle to the singular locus, a reminder Ln = λ

• j∗(ξ ⊗ ωX/S(D)n)

12 ⊗ (⊗m

i=1σ∗ i ωX/S)−rk(ξ)

⊗ OS(∆)rk(ξ) ⊗ (⊗m

i=1σ∗ i det ξ)6

L ′

n = λ

• ρ∗(ξ) ⊗ ωY(D)n12 ⊗ (det ωY|DY )−rk(ξ)

⊗

• det(ρ∗ξ)|DY

6

Restriction of the line bundle to the singular locus, a reminder Ln = λ

• j∗(ξ ⊗ ωX/S(D)n)

12 ⊗ (⊗m

i=1σ∗ i ωX/S)−rk(ξ)

⊗ OS(∆)rk(ξ) ⊗ (⊗m

i=1σ∗ i det ξ)6

L ′

n = λ

• ρ∗(ξ) ⊗ ωY(D)n12 ⊗ (det ωY|DY )−rk(ξ)

⊗

• det(ρ∗ξ)|DY

6 Canonical isomorphism : Ln||∆| → L ′

n

Restriction of the line bundle to the singular locus, a reminder Ln = λ

• j∗(ξ ⊗ ωX/S(D)n)

12 ⊗ (⊗m

i=1σ∗ i ωX/S)−rk(ξ)

⊗ OS(∆)rk(ξ) ⊗ (⊗m

i=1σ∗ i det ξ)6

L ′

n = λ

• ρ∗(ξ) ⊗ ωY(D)n12 ⊗ (det ωY|DY )−rk(ξ)

⊗

• det(ρ∗ξ)|DY

6 Canonical isomorphism : Ln||∆| → L ′

n

·Ln =

• ·Q (gTXt, hξ ⊗ ·2n

X/S)

12 ⊗

• ·W

X/S

−rk(ξ) ⊗

• ·div

∆

rk(ξ) ⊗ (⊗m

i=1σ∗ i hdet ξ)3

Restriction of the line bundle to the singular locus, a reminder Ln = λ

• j∗(ξ ⊗ ωX/S(D)n)

12 ⊗ (⊗m

i=1σ∗ i ωX/S)−rk(ξ)

⊗ OS(∆)rk(ξ) ⊗ (⊗m

i=1σ∗ i det ξ)6

L ′

n = λ

• ρ∗(ξ) ⊗ ωY(D)n12 ⊗ (det ωY|DY )−rk(ξ)

⊗

• det(ρ∗ξ)|DY

6 Canonical isomorphism : Ln||∆| → L ′

n

·Ln =

• ·Q (gTXt, hξ ⊗ ·2n

X/S)

12 ⊗

• ·W

X/S

−rk(ξ) ⊗

• ·div

∆

rk(ξ) ⊗ (⊗m

i=1σ∗ i hdet ξ)3

·L ′

n =

• ·Q (gTY, ρ∗(hξ) ⊗ ·2n

Y )

12 ⊗

• ·W

Y

−rk(ξ) ⊗

• det(ρ∗hξ)|DY

3

Restriction theorem Ln = λ

• j∗(ξ ⊗ ωX/S(D)n)

12 ⊗ (⊗m

i=1σ∗ i ωX/S)−rk(ξ)

⊗ OS(∆)rk(ξ) ⊗ (⊗m

i=1σ∗ i det ξ)6

L ′

n = λ

• ρ∗(ξ) ⊗ ωY(D)n12 ⊗ (det ωY|DY )−rk(ξ)

⊗

• det(ρ∗ξ)|DY

6

Restriction theorem Ln = λ

• j∗(ξ ⊗ ωX/S(D)n)

12 ⊗ (⊗m

i=1σ∗ i ωX/S)−rk(ξ)

⊗ OS(∆)rk(ξ) ⊗ (⊗m

i=1σ∗ i det ξ)6

L ′

n = λ

• ρ∗(ξ) ⊗ ωY(D)n12 ⊗ (det ωY|DY )−rk(ξ)

⊗

• det(ρ∗ξ)|DY

6

• Theorem. (-, 2019)

Isomorphism Ln||∆| → L ′

n is isometry up to universal constant

Restriction theorem Ln = λ

• j∗(ξ ⊗ ωX/S(D)n)

12 ⊗ (⊗m

i=1σ∗ i ωX/S)−rk(ξ)

⊗ OS(∆)rk(ξ) ⊗ (⊗m

i=1σ∗ i det ξ)6

L ′

n = λ

• ρ∗(ξ) ⊗ ωY(D)n12 ⊗ (det ωY|DY )−rk(ξ)

⊗

• det(ρ∗ξ)|DY

6

• Theorem. (-, 2019)

Isomorphism Ln||∆| → L ′

n is isometry up to universal constant

·Ln ||∆| = exp(k · rk(ξ) · C−n) ··L ′

n .

k : = #ΣX/S, C0 = −6 log(π), Ck = −6(1 + k) log(2) − 6(1 + 2k) log(π) − 6 log((2k)!).

Sketch of the proof in the case m = 0

• 1. Endow the family with smooth metric and apply Bismut, 1997

·B

Ln ||∆| = exp(k · rk(ξ) · D−n) ··B L ′

n .

Sketch of the proof in the case m = 0

• 1. Endow the family with smooth metric and apply Bismut, 1997

·B

Ln ||∆| = exp(k · rk(ξ) · D−n) ··B L ′

n .

• 2. Relate·B

Ln ||∆| and·Ln ||∆| by anomaly formula,

Bismut-Gillet-Soulé, 1987

Sketch of the proof in the case m = 0

• 1. Endow the family with smooth metric and apply Bismut, 1997

·B

Ln ||∆| = exp(k · rk(ξ) · D−n) ··B L ′

n .

• 2. Relate·B

Ln ||∆| and·Ln ||∆| by anomaly formula,

Bismut-Gillet-Soulé, 1987, if m > 0, then by its generalization to surfaces with cusps (-, 2018) ·B

Ln = exp(k · rk(ξ) · E−n) ··Ln .

Sketch of the proof in the case m = 0

• 1. Endow the family with smooth metric and apply Bismut, 1997

·B

Ln ||∆| = exp(k · rk(ξ) · D−n) ··B L ′

n .

• 2. Relate·B

Ln ||∆| and·Ln ||∆| by anomaly formula,

Bismut-Gillet-Soulé, 1987, if m > 0, then by its generalization to surfaces with cusps (-, 2018) ·B

Ln = exp(k · rk(ξ) · E−n) ··Ln .

• 3. Relate·B

L ′

n ||∆| and·L ′ n ||∆| by relative compact

perturbation theorem (-, 2018)

Sketch of the proof in the case m = 0

• 1. Endow the family with smooth metric and apply Bismut, 1997

·B

Ln ||∆| = exp(k · rk(ξ) · D−n) ··B L ′

n .

• 2. Relate·B

Ln ||∆| and·Ln ||∆| by anomaly formula,

Bismut-Gillet-Soulé, 1987, if m > 0, then by its generalization to surfaces with cusps (-, 2018) ·B

Ln = exp(k · rk(ξ) · E−n) ··Ln .

• 3. Relate·B

L ′

n ||∆| and·L ′ n ||∆| by relative compact

perturbation theorem (-, 2018)

Sketch of the proof in the case m = 0

• 1. Endow the family with smooth metric and apply Bismut, 1997

·B

Ln ||∆| = exp(k · rk(ξ) · D−n) ··B L ′

n .

• 2. Relate·B

Ln ||∆| and·Ln ||∆| by anomaly formula,

Bismut-Gillet-Soulé, 1987, if m > 0, then by its generalization to surfaces with cusps (-, 2018) ·B

Ln = exp(k · rk(ξ) · E−n) ··Ln .

• 3. Relate·B

L ′

n ||∆| and·L ′ n ||∆| by relative compact

perturbation theorem (-, 2018)

Sketch of the proof in the case m = 0

• 1. Endow the family with smooth metric and apply Bismut, 1997

·B

Ln ||∆| = exp(k · rk(ξ) · D−n) ··B L ′

n .

• 2. Relate·B

Ln ||∆| and·Ln ||∆| by anomaly formula,

Bismut-Gillet-Soulé, 1987, if m > 0, then by its generalization to surfaces with cusps (-, 2018) ·B

Ln = exp(k · rk(ξ) · E−n) ··Ln .

• 3. Relate·B

L ′

n ||∆| and·L ′ n ||∆| by relative compact

perturbation theorem (-, 2018) ·B

L ′

n = exp(k · rk(ξ) · G−n) ··L ′ n .

Sketch of the proof in the case m = 0

• 1. Endow the family with smooth metric and apply Bismut, 1997

·B

Ln ||∆| = exp(k · rk(ξ) · D−n) ··B L ′

n .

• 2. Relate·B

Ln ||∆| and·Ln ||∆| by anomaly formula,

Bismut-Gillet-Soulé, 1987, if m > 0, then by its generalization to surfaces with cusps (-, 2018) ·B

Ln = exp(k · rk(ξ) · E−n) ··Ln .

• 3. Relate·B

L ′

n ||∆| and·L ′ n ||∆| by relative compact

perturbation theorem (-, 2018) ·B

L ′

n = exp(k · rk(ξ) · G−n) ··L ′ n .

• 4. Use the proof of Freixas, 2007, of the arithmetic

Riemann-Roch theorem for stable pointed curves and our normalization of the analytic torsion to pin down the constant.

Applications

Compatibility theorem

• Theorem. (-, 2019)

Suppose (M, DM, gTM

hyp) is a hyperbolic surface, (ξ, hξ) trivial. For

any m ≥ 0, n ≤ 0, we have det ′Eξ

n =∗ det ′

TZEξ

n .

=∗ means up to some computed universal constant

Compatibility theorem

• Theorem. (-, 2019)

Suppose (M, DM, gTM

hyp) is a hyperbolic surface, (ξ, hξ) trivial. For

any m ≥ 0, n ≤ 0, we have det ′Eξ

n =∗ det ′

TZEξ

n .

=∗ means up to some computed universal constant m = 0, Phong-D’Hoker, 1986

Restriction theorem on M g,m, I Now, on the Deligne-Mumford compactification M g,m : λH,n

g,m : = λ(j∗(ωg,m(D)n))12 ⊗ (⊗σ∗ i ωg,m)−1 ⊗ OM g,m(∂Mg,m)

·H,n

g,m : = (·Q,n g,m)12 ⊗ (·W g,m)−1 ⊗ ·div ∂Mg,m

Restriction theorem on M g,m, I Now, on the Deligne-Mumford compactification M g,m : λH,n

g,m : = λ(j∗(ωg,m(D)n))12 ⊗ (⊗σ∗ i ωg,m)−1 ⊗ OM g,m(∂Mg,m)

·H,n

g,m : = (·Q,n g,m)12 ⊗ (·W g,m)−1 ⊗ ·div ∂Mg,m

We have clutching morphisms α :M g−1,m+2 → M g,m β :M g1,m1 × M g2,m2 → M g,m for g1 + g2 = g and m1 + m2 − 2 = m.

Restriction theorem on M g,m, II As in the restriction theorem, we have α∗λH,n

g,m ≃ λH,n g−1,m+1

β∗λH,n

g,m ≃ λH,n g1,m1 ⊠ λH,n g2,m2

Restriction theorem on M g,m, II As in the restriction theorem, we have α∗λH,n

g,m ≃ λH,n g−1,m+1

β∗λH,n

g,m ≃ λH,n g1,m1 ⊠ λH,n g2,m2

• Theorem. (-, 2019)

The isomorphisms above are isometries up to exp(C−n). C0 = −6 log(π), Ck = −6(1 + k) log(2) − 6(1 + 2k) log(π) − 6 log((2k)!).

Thank you!