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

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? X t X 0 t 0

What if a family has singular fibers? X t X 0 t 0 Natural example : Deligne-Mumford compactification M g , 0 .

What if a family has singular fibers? X t X 0 t 0 Natural example : Deligne-Mumford compactification M g , 0 . Studying degeneration of �·� Q � g TX t , h ξ � = > 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 , X t = π − 1 ( t ) has at most double-point singularities (i.e. of the form { z 0 z 1 = 0 } )

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

Family of curves π : X → S proper holomorphic of relative dimension 1, t ∈ S , X t = π − 1 ( t ) has at most double-point singularities (i.e. of the form { z 0 z 1 = 0 } ) Σ X / S singular points of the fibers, ∆ = π ∗ (Σ X / S ) �·� ω X / S a Hermitian norm on ω X / S over X \ π − 1 ( | ∆ | ) g TX t - restriction on X t , t ∈ S \ | ∆ | ( ξ, h ξ ) a holomorphic Hermitian vector bundle over X

A theorem of Bismut-Bost L n = λ ( j ∗ ξ ) 12 ⊗ O S (∆) rk ( ξ )

A theorem of Bismut-Bost L n = λ ( j ∗ ξ ) 12 ⊗ O S (∆) rk ( ξ ) g TX t , h ξ �� 12 ⊗ ( �·� div �·� L n = �·� Q � ∆ ) rk ( ξ ) � �·� div ∆ is the canonical divisor norm on O S (∆) , i.e. over S \ | ∆ | , � s ∆ � div ∆ = 1 for the canonical holomoprhic section s ∆ .

A theorem of Bismut-Bost L n = λ ( j ∗ ξ ) 12 ⊗ O S (∆) rk ( ξ ) g TX t , h ξ �� 12 ⊗ ( �·� div �·� L n = �·� Q � ∆ ) rk ( ξ ) � �·� div ∆ is the canonical divisor norm on O S (∆) , 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 �·� L n on L n extends continuously to | ∆ |

A theorem of Bismut-Bost L n = λ ( j ∗ ξ ) 12 ⊗ O S (∆) rk ( ξ ) g TX t , h ξ �� 12 ⊗ ( �·� div �·� L n = �·� Q � ∆ ) rk ( ξ ) � �·� div ∆ is the canonical divisor norm on O S (∆) , 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 �·� L n on L n 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 L n = λ ( j ∗ ξ ) 12 ⊗ O S (∆) rk ( ξ ) g TX t , h ξ �� 12 ⊗ ( �·� div �·� L n = �·� Q � ∆ ) rk ( ξ ) � Continuity theorem (-, 2018) Under some hypothesizes on degeneration of metric near singularities (more general than in DM compactification), the norm L n on �·� L n extends continuously to | ∆ |

Continuity theorem (basic version) and the goal L n = λ ( j ∗ ξ ) 12 ⊗ O S (∆) rk ( ξ ) g TX t , h ξ �� 12 ⊗ ( �·� div �·� L n = �·� Q � ∆ ) rk ( ξ ) � Continuity theorem (-, 2018) Under some hypothesizes on degeneration of metric near singularities (more general than in DM compactification), the norm L n on �·� L n extends continuously to | ∆ | GOAL : study the geometric meaning of this extension.

What to expect? X t X 0 t 0

What to expect? X t X 0 Y 0 t 0

What to expect? X t X 0 Y 0 t 0 Belief : should be related to the Quillen metric of normalization.

What to expect? X t X 0 Y 0 t 0 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, obtained 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, obtained 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, obtained 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 D M = { P 1 , P 2 , . . . , P m } ⊂ M , M = M \ D M

What is a surface with cusps? M a compact Riemann surface D M = { P 1 , P 2 , . . . , P m } ⊂ M , M = M \ D M g TM is a Kähler metric on M z 1 , . . . , z m local holomorphic coordinates, z i ( 0 ) = { P i } Suppose g TM over {| z i | < ǫ } is induced by √ − 1 dz i dz i � 2 . � � � z i log | z i |

What is a surface with cusps? M a compact Riemann surface D M = { P 1 , P 2 , . . . , P m } ⊂ M , M = M \ D M g TM is a Kähler metric on M z 1 , . . . , z m local holomorphic coordinates, z i ( 0 ) = { P i } Suppose g TM over {| z i | < ǫ } is induced by √ − 1 dz i dz i � 2 . � � � z i log | z i | We call ( M , D M , g TM ) a surface with cusps

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

An important example Suppose 2 g ( M ) − 2 + # D M > 0, i.e. ( M , D M ) is stable By uniformization theorem, there is exactly one csc − 1 complete metric g TM hyp on M = M \ D M

An important example Suppose 2 g ( M ) − 2 + # D M > 0, i.e. ( M , D M ) is stable By uniformization theorem, there is exactly one csc − 1 complete metric g TM hyp on M = M \ D M The triple ( M , D M , g TM 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). X t X 0 Y 0 t 0

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). X t X 0 Y 0 t 0 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 ω g , m WP on the moduli space of m -pointed stable curves M g , 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 ω g , m WP on the moduli space of m -pointed stable curves M g , m of genus g satisfies ∗ ω g , m WP | M g 1 , m 1 × M g , m 2 = ω g 1 , m 1 ⊕ ω g 2 , m 2 , WP WP for g 1 + g 2 = g , m 1 + m 2 − 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 ω g , m WP on the moduli space of m -pointed stable curves M g , m of genus g satisfies ∗ ω g , m WP | M g 1 , m 1 × M g , m 2 = ω g 1 , m 1 ⊕ ω g 2 , m 2 , WP WP for g 1 + g 2 = g , m 1 + m 2 − 2 = m . Similarly, in Pic ( M g , m ) , λ 12 g , m ⊗ ψ − 1 � � g , m ⊗ O M g , m ( ∂ M g , m ) | M g 1 , m 1 × M g 2 , m 2 λ 12 g 1 , m 1 ⊗ ψ − 1 λ 12 g 2 , m 2 ⊗ ψ − 1 � � � � ≃ ⊠ . g 1 , m 1 g 2 , m 2

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 ω g , m WP on the moduli space of m -pointed stable curves M g , m of genus g satisfies ∗ ω g , m WP | M g 1 , m 1 × M g , m 2 = ω g 1 , m 1 ⊕ ω g 2 , m 2 , WP WP for g 1 + g 2 = g , m 1 + m 2 − 2 = m . Similarly, in Pic ( M g , m ) , λ 12 g , m ⊗ ψ − 1 � � g , m ⊗ O M g , m ( ∂ M g , m ) | M g 1 , m 1 × M g 2 , m 2 λ 12 g 1 , m 1 ⊗ ψ − 1 λ 12 g 2 , m 2 ⊗ ψ − 1 � � � � ≃ ⊠ . g 1 , m 1 g 2 , m 2 Finally, in H 2 ( M g , m ) , we have ∗ c 1 ( λ 12 g , m ) = [ ω g , m g , m ⊗ ψ − 1 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 ω g , m WP on the moduli space of m -pointed stable curves M g , m of genus g satisfies ∗ ω g , m WP | M g 1 , m 1 × M g , m 2 = ω g 1 , m 1 ⊕ ω g 2 , m 2 , WP WP for g 1 + g 2 = g , m 1 + m 2 − 2 = m . Similarly, in Pic ( M g , m ) , λ 12 g , m ⊗ ψ − 1 � � g , m ⊗ O M g , m ( ∂ M g , m ) | M g 1 , m 1 × M g 2 , m 2 λ 12 g 1 , m 1 ⊗ ψ − 1 λ 12 g 2 , m 2 ⊗ ψ − 1 � � � � ≃ ⊠ . g 1 , m 1 g 2 , m 2 Finally, in H 2 ( M g , m ) , we have ∗ c 1 ( λ 12 g , m ) = [ ω g , m g , m ⊗ ψ − 1 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 ω g , m WP on the moduli space of m -pointed stable curves M g , m of genus g satisfies ∗ ω g , m WP | M g 1 , m 1 × M g , m 2 = ω g 1 , m 1 ⊕ ω g 2 , m 2 , WP WP for g 1 + g 2 = g , m 1 + m 2 − 2 = m . Similarly, in Pic ( M g , m ) , λ 12 g , m ⊗ ψ − 1 � � g , m ⊗ O M g , m ( ∂ M g , m ) | M g 1 , m 1 × M g 2 , m 2 λ 12 g 1 , m 1 ⊗ ψ − 1 λ 12 g 2 , m 2 ⊗ ψ − 1 � � � � ≃ ⊠ . g 1 , m 1 g 2 , m 2 Finally, in H 2 ( M g , m ) , we have ∗ c 1 ( λ 12 g , m ) = [ ω g , m g , m ⊗ ψ − 1 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 X t X 0 Y 0 t 0

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

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

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

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

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

Restriction of the line bundle to singular locus We want to describe the restriction of the line bundle � 12 ⊗ O S (∆) rk ( ξ ) j ∗ ( ξ ⊗ ω n � L n = λ X / S ) to ∆ as some natural line bundle on the normalization.

Restriction of the divisor line bundle to the singular locus � 12 ⊗ O S (∆) rk ( ξ ) j ∗ ( ξ ⊗ ω n � L n = λ X / S )

Poincaré residue morphism We denote by k : = #Σ X / S , then ω k � � S ⊗ O S (∆) | | ∆ | → O | ∆ | .

Poincaré residue morphism We denote by k : = #Σ X / S , then ω k � � S ⊗ O S (∆) | | ∆ | → O | ∆ | . Double-point singularities give a natural isomorphism ω k S | | ∆ | → ⊗ P ∈ ρ − 1 (Σ X / S ) ω Y 0 | P

Poincaré residue morphism We denote by k : = #Σ X / S , then ω k � � S ⊗ O S (∆) | | ∆ | → O | ∆ | . Double-point singularities give a natural isomorphism ω k S | | ∆ | → ⊗ P ∈ ρ − 1 (Σ X / S ) ω Y 0 | P By combining, we have a natural isomorphism � − 1 � O S (∆) | | ∆ | → ⊗ P ∈ ρ − 1 (Σ X / S ) ω Y 0 | P

Restriction of the determinant to the singular locus � 12 ⊗ O S (∆) rk ( ξ ) j ∗ ( ξ ⊗ ω n � L n = λ X / S )

Restriction of the determinant to the singular locus Short exact sequence ξ ⊗ ω n � � 0 → O X 0 X / S ρ ∗ ( ξ ) ⊗ ω Y ( D ) n � � → ρ ∗ O Y � � → O Σ X / S ξ | Σ X / S → 0 ,

Restriction of the determinant to the singular locus Short exact sequence ξ ⊗ ω n � � 0 → O X 0 X / S ρ ∗ ( ξ ) ⊗ ω Y ( D ) n � � → ρ ∗ O Y � � → O Σ X / S ξ | Σ X / S → 0 , Induces isomorphism j ∗ ( ξ ⊗ ω n � � λ X / S ) | | ∆ | ρ ∗ ( ξ ) ⊗ ω Y ( D ) n � � � � ≃ λ ⊗ det π ∗ ( ξ | Σ X / S ) .

A combination is the answer For the line bundle L n , defined by � 12 ⊗ O S (∆) rk ( ξ ) , j ∗ ( ξ ⊗ ω n � L n = λ X / S ) we have the following isomorphism of line bundles on | ∆ | ρ ∗ ( ξ ) ⊗ ω Y ( D ) n � 12 � L n | | ∆ | ≃ λ � − rk ( ξ ) � 12 ⊗ � � ⊗ det π ∗ ( ξ | Σ X / S ) ⊗ P ∈ ρ − 1 (Σ X / S ) ω Y 0 | P .

Wolpert norm for surfaces with cusps

The Wolpert norm ( M , D M , g TM ) , D M = { P 1 , . . . , P m } surface with cusps

The Wolpert norm ( M , D M , g TM ) , D M = { P 1 , . . . , P m } surface with cusps z 1 , . . . , z m local holomorphic coordinates, z i ( 0 ) = { P i } g TM over {| z i | < ǫ } is induced by √ − 1 dz i dz i � 2 � � � z i log | z i |

The Wolpert norm ( M , D M , g TM ) , D M = { P 1 , . . . , P m } surface with cusps z 1 , . . . , z m local holomorphic coordinates, z i ( 0 ) = { P i } g TM over {| z i | < ǫ } is induced by √ − 1 dz i dz i � 2 � � � z i log | z i | Wolpert norm �·� W on ⊗ m i = 1 ω M | P i is defined by � W = 1 . � ⊗ i dz i | P i � �

The Wolpert norm ( M , D M , g TM ) , D M = { P 1 , . . . , P m } surface with cusps z 1 , . . . , z m local holomorphic coordinates, z i ( 0 ) = { P i } g TM over {| z i | < ǫ } is induced by √ − 1 dz i dz i � 2 � � � z i log | z i | Wolpert norm �·� W on ⊗ m i = 1 ω M | P i is defined by � W = 1 . � ⊗ i dz i | P i � � √ − 1 dzdz � W = 1 D ∗ � � � dz | 0 on � � 2 � � � z log | z |

The Wolpert norm ( M , D M , g TM ) , D M = { P 1 , . . . , P m } surface with cusps z 1 , . . . , z m local holomorphic coordinates, z i ( 0 ) = { P i } g TM over {| z i | < ǫ } is induced by √ − 1 dz i dz i � 2 � � � z i log | z i | Wolpert norm �·� W on ⊗ m i = 1 ω M | P i is defined by � W = 1 . � ⊗ i dz i | P i � � √ − 1 dzdz � W = 1 D ∗ � � � dz | 0 on � � 2 2 � � � z log | 2 z |

Quillen metric for surfaces with cusps

The L 2 -norm � 1 / 2 ·�·� L 2 �·� Q = det ′ � �

The L 2 -norm Let ( M , D M , g TM ) be a surface with cusps �·� ω M the induced Hermitian norm on ω M over M