Genus one mirror symmetry and the arithmetic RiemannRoch theorem - - PowerPoint PPT Presentation

Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Genus one mirror symmetry and the arithmetic Riemann–Roch theorem

Gerard Freixas i Montplet

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

Based on joint work with D. Eriksson and C. Mourougane

Inaugural France – Korea conference

November 2019

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Functorial BCOV conjecture

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The Grothendieck–Riemann–Roch theorem

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The Grothendieck–Riemann–Roch theorem

Let f : X → S be a smooth projective morphism of non-singular, connected algebraic varieties over C.

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The Grothendieck–Riemann–Roch theorem

Let f : X → S be a smooth projective morphism of non-singular, connected algebraic varieties over C. Let E be a vector bundle on X.

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The Grothendieck–Riemann–Roch theorem

Let f : X → S be a smooth projective morphism of non-singular, connected algebraic varieties over C. Let E be a vector bundle on X.

Theorem (Grothendieck–Riemann–Roch (GRR))

The following equality holds in CH•(S)Q : ch(Rf∗E) = f∗

• ch(E) td(TX/S)
• .

In particular, for the determinant of cohomology : c1(det Rf∗E) = f∗

• ch(E) td(TX/S)

(1) in CH1(S)Q.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Assume the fibers Xs are Calabi–Yau (CY), i.e. KXs ≃ OXs.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Assume the fibers Xs are Calabi–Yau (CY), i.e. KXs ≃ OXs. Define the virtual vector bundle DΩ•

X/S =

• p

(−1)ppΩp

X/S.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Assume the fibers Xs are Calabi–Yau (CY), i.e. KXs ≃ OXs. Define the virtual vector bundle DΩ•

X/S =

• p

(−1)ppΩp

X/S.

Then GRR simplifies to : c1(det Rf∗DΩ•

X/S) = χ

12c1(f∗KX/S) in CH1(S)Q, with χ = χ(Xs) the topological Euler characteristic of the fibers.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Definition (BCOV line bundle)

The BCOV line bundle on S is defined by λBCOV (f ) = det Rf∗DΩ•

X/S

=

• p

(det Rqf∗Ωp

X/S)(−1)p+qp.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Definition (BCOV line bundle)

The BCOV line bundle on S is defined by λBCOV (f ) = det Rf∗DΩ•

X/S

=

• p

(det Rqf∗Ωp

X/S)(−1)p+qp.

It commutes with arbitrary base change.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Definition (BCOV line bundle)

The BCOV line bundle on S is defined by λBCOV (f ) = det Rf∗DΩ•

X/S

=

• p

(det Rqf∗Ωp

X/S)(−1)p+qp.

It commutes with arbitrary base change.

Corollary

There exists an isomorphism of Q-line bundles on S λBCOV (f )⊗12

∼

− → (f∗KX/S)⊗χ.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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But : there are as many as H0(S, O×

S ) such isomorphisms.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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But : there are as many as H0(S, O×

S ) such isomorphisms.

Theorem (Eriksson, Franke)

There exists a canonical isomorphism of Q-line bundles GRR: λBCOV (f )⊗12 ∼ → (f∗KX/S)⊗χ, commuting with arbitrary base change. If f : X → S is defined over Q, GRR is defined over Q as well.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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But : there are as many as H0(S, O×

S ) such isomorphisms.

Theorem (Eriksson, Franke)

There exists a canonical isomorphism of Q-line bundles GRR: λBCOV (f )⊗12 ∼ → (f∗KX/S)⊗χ, commuting with arbitrary base change. If f : X → S is defined over Q, GRR is defined over Q as well. The arithmetic Riemann–Roch theorem of Gillet–Soul´ e provides a weak variant, enough for most purposes :

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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But : there are as many as H0(S, O×

S ) such isomorphisms.

Theorem (Eriksson, Franke)

There exists a canonical isomorphism of Q-line bundles GRR: λBCOV (f )⊗12 ∼ → (f∗KX/S)⊗χ, commuting with arbitrary base change. If f : X → S is defined over Q, GRR is defined over Q as well. The arithmetic Riemann–Roch theorem of Gillet–Soul´ e provides a weak variant, enough for most purposes :

◮ natural isomorphism up to a constant of norm one.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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But : there are as many as H0(S, O×

S ) such isomorphisms.

Theorem (Eriksson, Franke)

There exists a canonical isomorphism of Q-line bundles GRR: λBCOV (f )⊗12 ∼ → (f∗KX/S)⊗χ, commuting with arbitrary base change. If f : X → S is defined over Q, GRR is defined over Q as well. The arithmetic Riemann–Roch theorem of Gillet–Soul´ e provides a weak variant, enough for most purposes :

◮ natural isomorphism up to a constant of norm one. ◮ isometry for auxiliary hermitian structures (Quillen metric).

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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But : there are as many as H0(S, O×

S ) such isomorphisms.

Theorem (Eriksson, Franke)

There exists a canonical isomorphism of Q-line bundles GRR: λBCOV (f )⊗12 ∼ → (f∗KX/S)⊗χ, commuting with arbitrary base change. If f : X → S is defined over Q, GRR is defined over Q as well. The arithmetic Riemann–Roch theorem of Gillet–Soul´ e provides a weak variant, enough for most purposes :

◮ natural isomorphism up to a constant of norm one. ◮ isometry for auxiliary hermitian structures (Quillen metric). ◮ over Q, the constant is necessarily ±1.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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But : there are as many as H0(S, O×

S ) such isomorphisms.

Theorem (Eriksson, Franke)

There exists a canonical isomorphism of Q-line bundles GRR: λBCOV (f )⊗12 ∼ → (f∗KX/S)⊗χ, commuting with arbitrary base change. If f : X → S is defined over Q, GRR is defined over Q as well. The arithmetic Riemann–Roch theorem of Gillet–Soul´ e provides a weak variant, enough for most purposes :

◮ natural isomorphism up to a constant of norm one. ◮ isometry for auxiliary hermitian structures (Quillen metric). ◮ over Q, the constant is necessarily ±1. ◮ compatible with Eriksson–Franke.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Mirror symmetry at genus one

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Mirror symmetry at genus one

Let X be a CY variety of dimension n.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Mirror symmetry at genus one

Let X be a CY variety of dimension n. Mirror symmetry predicts the existence of a mirror family of CY n-folds ϕ: X ∨ → D×, with :

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Mirror symmetry at genus one

Let X be a CY variety of dimension n. Mirror symmetry predicts the existence of a mirror family of CY n-folds ϕ: X ∨ → D×, with :

◮ D× = (D×)d is a punctured multi-disc, d = hn−1,1(X ∨ q ).

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Mirror symmetry at genus one

Let X be a CY variety of dimension n. Mirror symmetry predicts the existence of a mirror family of CY n-folds ϕ: X ∨ → D×, with :

◮ D× = (D×)d is a punctured multi-disc, d = hn−1,1(X ∨ q ). ◮ the monodromy on Rnϕ∗C is maximal unipotent :

if d = 1, (T − 1)n = 0 and (T − 1)n+1 = 0.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Mirror symmetry at genus one

Let X be a CY variety of dimension n. Mirror symmetry predicts the existence of a mirror family of CY n-folds ϕ: X ∨ → D×, with :

◮ D× = (D×)d is a punctured multi-disc, d = hn−1,1(X ∨ q ). ◮ the monodromy on Rnϕ∗C is maximal unipotent :

if d = 1, (T − 1)n = 0 and (T − 1)n+1 = 0.

◮ mirror Hodge numbers : hp,q(X) = hn−p,q(X ∨ q ).

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Mirror symmetry at genus one

Let X be a CY variety of dimension n. Mirror symmetry predicts the existence of a mirror family of CY n-folds ϕ: X ∨ → D×, with :

◮ D× = (D×)d is a punctured multi-disc, d = hn−1,1(X ∨ q ). ◮ the monodromy on Rnϕ∗C is maximal unipotent :

if d = 1, (T − 1)n = 0 and (T − 1)n+1 = 0.

◮ mirror Hodge numbers : hp,q(X) = hn−p,q(X ∨ q ).

Informally : ϕ: X ∨ → D× cusp in a moduli space of CY varieties.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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There should also exist a bihomolorphic map, called mirror map, τ : D× − → HX := H1,1

R (X)/H1,1 Z (X) + iKX,

where KX is the K¨ ahler cone of X, which relates the variation of complex structures of X ∨ → D× and the K¨ ahler “moduli” of X.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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There should also exist a bihomolorphic map, called mirror map, τ : D× − → HX := H1,1

R (X)/H1,1 Z (X) + iKX,

where KX is the K¨ ahler cone of X, which relates the variation of complex structures of X ∨ → D× and the K¨ ahler “moduli” of X. When d = 1, τ can be identified with a multi-valued function of the form τ(q) = 1 2πi

• γ1 η
• γ0 η,

where η is a local holomorphic frame for ϕ∗KX ∨/D× and γ0, γ1 are well-chosen homology n-cycles (Morrison).

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Definition

Define the formal generating series of genus one Gromov–Witten invariants of X by F1(τ) = − 1 24

• X

cn−1(X) ∩ 2πiτ +

• β

GW1(X, β)e2πiβ,τ, where

◮ τ ∈ HX. ◮ β ∈ H2(X, Z) runs over curve classes. ◮ GW1(X, β) = genus one Gromov–Witten invariant of class β.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Conjecture (Optimistic functorial BCOV)

Let X and ϕ: X ∨ → D× be mirrors as above.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Conjecture (Optimistic functorial BCOV)

Let X and ϕ: X ∨ → D× be mirrors as above. Assume that ϕ can be extended over an algebraic base.

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Conjecture (Optimistic functorial BCOV)

Let X and ϕ: X ∨ → D× be mirrors as above. Assume that ϕ can be extended over an algebraic base. Let τ : D× → HX be the mirror map.

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Conjecture (Optimistic functorial BCOV)

Let X and ϕ: X ∨ → D× be mirrors as above. Assume that ϕ can be extended over an algebraic base. Let τ : D× → HX be the mirror map. Then :

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Conjecture (Optimistic functorial BCOV)

Let X and ϕ: X ∨ → D× be mirrors as above. Assume that ϕ can be extended over an algebraic base. Let τ : D× → HX be the mirror map. Then :

◮ there exist canonical trivializations of λBCOV (ϕ) and

ϕ∗KX ∨/D×.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Conjecture (Optimistic functorial BCOV)

Let X and ϕ: X ∨ → D× be mirrors as above. Assume that ϕ can be extended over an algebraic base. Let τ : D× → HX be the mirror map. Then :

◮ there exist canonical trivializations of λBCOV (ϕ) and

ϕ∗KX ∨/D×.

◮ in these trivializations, GRR can be identified to a

holomorphic function in q ∈ D× of the form GRR(q) = exp

• (−1)nF1(τ(q))

24 .

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The BCOV invariant

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before. Recall the canonical GRR isomorphism of Q-line bundles GRR: λBCOV (f )⊗12

∼

− → (f∗KX/S)⊗χ.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before. Recall the canonical GRR isomorphism of Q-line bundles GRR: λBCOV (f )⊗12

∼

− → (f∗KX/S)⊗χ. To attack the functorial BCOV conjecture we need methods of explicitly computing GRR.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before. Recall the canonical GRR isomorphism of Q-line bundles GRR: λBCOV (f )⊗12

∼

− → (f∗KX/S)⊗χ. To attack the functorial BCOV conjecture we need methods of explicitly computing GRR. Idea :

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before. Recall the canonical GRR isomorphism of Q-line bundles GRR: λBCOV (f )⊗12

∼

− → (f∗KX/S)⊗χ. To attack the functorial BCOV conjecture we need methods of explicitly computing GRR. Idea :

◮ introduce natural hermitian metrics on λBCOV (f ) and f∗KX/S

(Hodge theory).

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before. Recall the canonical GRR isomorphism of Q-line bundles GRR: λBCOV (f )⊗12

∼

− → (f∗KX/S)⊗χ. To attack the functorial BCOV conjecture we need methods of explicitly computing GRR. Idea :

◮ introduce natural hermitian metrics on λBCOV (f ) and f∗KX/S

(Hodge theory).

◮ compute the norm of GRR with respect to these metrics.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before. Recall the canonical GRR isomorphism of Q-line bundles GRR: λBCOV (f )⊗12

∼

− → (f∗KX/S)⊗χ. To attack the functorial BCOV conjecture we need methods of explicitly computing GRR. Idea :

◮ introduce natural hermitian metrics on λBCOV (f ) and f∗KX/S

(Hodge theory).

◮ compute the norm of GRR with respect to these metrics. ◮ how ?

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Construction of τBCOV

Let f : X → S be a family of Calabi–Yau manifolds as before. Recall the canonical GRR isomorphism of Q-line bundles GRR: λBCOV (f )⊗12

∼

− → (f∗KX/S)⊗χ. To attack the functorial BCOV conjecture we need methods of explicitly computing GRR. Idea :

◮ introduce natural hermitian metrics on λBCOV (f ) and f∗KX/S

(Hodge theory).

◮ compute the norm of GRR with respect to these metrics. ◮ how ? Arithmetic Riemann–Roch !

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Hermitian metrics via Hodge theory

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Hermitian metrics via Hodge theory

We endow the line bundle f∗KX/S with the L2 (or Hodge) metric : hL2,s(α, β) = in2 (2π)n

• Xs

α ∧ β, for α, β are local sections of f∗KX/S.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Hermitian metrics via Hodge theory

We endow the line bundle f∗KX/S with the L2 (or Hodge) metric : hL2,s(α, β) = in2 (2π)n

• Xs

α ∧ β, for α, β are local sections of f∗KX/S. The normalization (2π)n is standard in Arakelov geometry.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The line bundle λBCOV (f ) has a canonical metric :

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The line bundle λBCOV (f ) has a canonical metric :

◮ by the Hodge decomposition, there is a C∞ isomorphism

λBCOV (f )⊗2 ⊗ λBCOV (f )

⊗2 ∼

− →

• k

(det Rkf∗C)(−1)k2k.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The line bundle λBCOV (f ) has a canonical metric :

◮ by the Hodge decomposition, there is a C∞ isomorphism

λBCOV (f )⊗2 ⊗ λBCOV (f )

⊗2 ∼

− →

• k

(det Rkf∗C)(−1)k2k.

◮ using the lattice of integral cohomology :

• k

(det Rkf∗C)(−1)k2k =

• k

(det Rkf∗Z)(−1)k2k

nt

⊗ C ≃ C. The isomorphism is canonical, since (det Rkf∗Z)⊗2

nt ≃ Z

canonically.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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◮ All in all, we have a C∞ isomorphism

λBCOV (f )⊗2 ⊗ λBCOV (f )

⊗2 ∼

− → C, which actually defines a smooth hermitian metric on λBCOV (f )⊗2, and hence on λBCOV (f ).

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◮ All in all, we have a C∞ isomorphism

λBCOV (f )⊗2 ⊗ λBCOV (f )

⊗2 ∼

− → C, which actually defines a smooth hermitian metric on λBCOV (f )⊗2, and hence on λBCOV (f ).

Definition (L2-BCOV metric)

The above canonical hermitian metric on λBCOV (f ) is called the L2-BCOV metric, and denoted hL2,BCOV .

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Now the BCOV invariant of the family f : X → S is defined as the norm of GRR with respect to hL2 and hL2,BCOV :

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Now the BCOV invariant of the family f : X → S is defined as the norm of GRR with respect to hL2 and hL2,BCOV :

Definition (BCOV invariant)

We define the BCOV invariant of the family f : X → S as the C∞(S) function τBCOV (Xs) = GRR(θ)2

L2,s

θ2

L2,BCOV ,s

, where θ is any local trivialization of λBCOV (f ).

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Relation to holomorphic analytic torsion

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Relation to holomorphic analytic torsion

Assume for simplicity that X has a K¨ ahler form ω with :

◮ the restriction of ω on fibers is Ricci flat. ◮ the cohomology class of ω on fibers is rational.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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Relation to holomorphic analytic torsion

Assume for simplicity that X has a K¨ ahler form ω with :

◮ the restriction of ω on fibers is Ricci flat. ◮ the cohomology class of ω on fibers is rational.

With respect to ω, we form the ∂-Laplacian ∆p,q

∂,s on Ap,q(Xs).

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Relation to holomorphic analytic torsion

Assume for simplicity that X has a K¨ ahler form ω with :

◮ the restriction of ω on fibers is Ricci flat. ◮ the cohomology class of ω on fibers is rational.

With respect to ω, we form the ∂-Laplacian ∆p,q

∂,s on Ap,q(Xs).

Theorem (Arithmetic Riemann–Roch + ε)

There exists a constant C such that τBCOV (Xs) = C

• p,q

(det ∆p,q

∂,s )(−1)p+qpq,

where det ∆p,q

∂,s is the ζ-regularized determinant of ∆p,q ∂,s .

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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◮ The arithmetic Riemann–Roch relationship

GRR 2 = C

• p,q

(det ∆p,q

∂,s )(−1)p+qpq

is a C∞, spectral evaluation of GRR.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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◮ The arithmetic Riemann–Roch relationship

GRR 2 = C

• p,q

(det ∆p,q

∂,s )(−1)p+qpq

is a C∞, spectral evaluation of GRR.

◮ The original BCOV conjecture was formulated in terms of the

function F1(s) := 1 2 log

• p,q

(det ∆p,q

∂,s )(−1)p+qpq.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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◮ The arithmetic Riemann–Roch relationship

GRR 2 = C

• p,q

(det ∆p,q

∂,s )(−1)p+qpq

is a C∞, spectral evaluation of GRR.

◮ The original BCOV conjecture was formulated in terms of the

function F1(s) := 1 2 log

• p,q

(det ∆p,q

∂,s )(−1)p+qpq. ◮ Determining the singularities of τBCOV for degenerations of

CY’s is central in our approach to the conjecture. This relies

• n the spectral interpretation.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The BCOV conjecture for hypersurfaces in Pn

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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In the remaining of the talk, we discuss the following statement :

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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In the remaining of the talk, we discuss the following statement :

Theorem (Eriksson–F.–Mourougane)

The functorial BCOV conjecture holds, up to a constant, for CY hypersurfaces in Pn

C and their mirror family.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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In the remaining of the talk, we discuss the following statement :

Theorem (Eriksson–F.–Mourougane)

The functorial BCOV conjecture holds, up to a constant, for CY hypersurfaces in Pn

C and their mirror family.

The 3-dimensional case was obtained by Fang–Lu–Yoshikawa.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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In the remaining of the talk, we discuss the following statement :

Theorem (Eriksson–F.–Mourougane)

The functorial BCOV conjecture holds, up to a constant, for CY hypersurfaces in Pn

C and their mirror family.

The 3-dimensional case was obtained by Fang–Lu–Yoshikawa. First instance of higher dimensional mirror symmetry of BCOV type at genus one.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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In the remaining of the talk, we discuss the following statement :

Theorem (Eriksson–F.–Mourougane)

The functorial BCOV conjecture holds, up to a constant, for CY hypersurfaces in Pn

C and their mirror family.

The 3-dimensional case was obtained by Fang–Lu–Yoshikawa. First instance of higher dimensional mirror symmetry of BCOV type at genus one. It builds on our previous work on the singularities of τBCOV for degenerations of CY’s, and some refinements of Schmid’s asymptotics of Hodge metrics.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The mirror family

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The mirror family

Let Xn+1 be a general degree n + 1 hypersurface in Pn

C.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The mirror family

Let Xn+1 be a general degree n + 1 hypersurface in Pn

C.

The construction of a mirror family for Xn+1 begins with the Dwork pencil : xn+1 + . . . + xn+1

n

− (n + 1)ψx0 · . . . · xn = 0, ψ ∈ U := C \ µn+1.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The mirror family

Let Xn+1 be a general degree n + 1 hypersurface in Pn

C.

The construction of a mirror family for Xn+1 begins with the Dwork pencil : xn+1 + . . . + xn+1

n

− (n + 1)ψx0 · . . . · xn = 0, ψ ∈ U := C \ µn+1. We mod out by a group of symmetries of fibers.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

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The mirror family

Let Xn+1 be a general degree n + 1 hypersurface in Pn

C.

The construction of a mirror family for Xn+1 begins with the Dwork pencil : xn+1 + . . . + xn+1

n

− (n + 1)ψx0 · . . . · xn = 0, ψ ∈ U := C \ µn+1. We mod out by a group of symmetries of fibers. Then we perform a crepant resolution.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

The mirror family

Let Xn+1 be a general degree n + 1 hypersurface in Pn

C.

The construction of a mirror family for Xn+1 begins with the Dwork pencil : xn+1 + . . . + xn+1

n

− (n + 1)ψx0 · . . . · xn = 0, ψ ∈ U := C \ µn+1. We mod out by a group of symmetries of fibers. Then we perform a crepant resolution. We obtain a family f : Z → U of CY’s of dimension n − 1.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

The mirror family

Let Xn+1 be a general degree n + 1 hypersurface in Pn

C.

The construction of a mirror family for Xn+1 begins with the Dwork pencil : xn+1 + . . . + xn+1

n

− (n + 1)ψx0 · . . . · xn = 0, ψ ∈ U := C \ µn+1. We mod out by a group of symmetries of fibers. Then we perform a crepant resolution. We obtain a family f : Z → U of CY’s of dimension n − 1. In a neighborhood of ψ = ∞, this is indeed a mirror family. In particular, the monodromy on Rn−1f∗C at infinity is maximal unipotent.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

We restrict to a disc neighborhood of ψ = ∞, with parameter q f : Z → D×.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

We restrict to a disc neighborhood of ψ = ∞, with parameter q f : Z → D×. We fix a polarization, and thus have a primitivity notion on Hodge bundles.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

We restrict to a disc neighborhood of ψ = ∞, with parameter q f : Z → D×. We fix a polarization, and thus have a primitivity notion on Hodge bundles. Relevant Hodge bundles : (Rqf∗Ωp

Z/D×)prim with p + q = n − 1.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

We restrict to a disc neighborhood of ψ = ∞, with parameter q f : Z → D×. We fix a polarization, and thus have a primitivity notion on Hodge bundles. Relevant Hodge bundles : (Rqf∗Ωp

Z/D×)prim with p + q = n − 1.

The relevant Hodge bundles have rank one.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

We restrict to a disc neighborhood of ψ = ∞, with parameter q f : Z → D×. We fix a polarization, and thus have a primitivity notion on Hodge bundles. Relevant Hodge bundles : (Rqf∗Ωp

Z/D×)prim with p + q = n − 1.

The relevant Hodge bundles have rank one. To canonically trivialize them, we first need some information on Hn−1

lim . This is the limiting mixed Hodge structure on the

cohomology of a general fibre, associated to the degeneration of Hodge structures on (Rn−1f∗C)prim.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Let F •

∞ be the limiting Hodge filtration and W• the monodromy

weight filtration, on Hn−1

lim .

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Let F •

∞ be the limiting Hodge filtration and W• the monodromy

weight filtration, on Hn−1

lim .

For all k = 0, . . . , n − 1, GrW

2k Hn−1 lim

= Grk

F∞ GrW 2k Hn−1 lim

≃ Grk

F∞ Hn−1 lim

is 1-dimensional. All the other possible graded pieces are trivial.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Let F •

∞ be the limiting Hodge filtration and W• the monodromy

weight filtration, on Hn−1

lim .

For all k = 0, . . . , n − 1, GrW

2k Hn−1 lim

= Grk

F∞ GrW 2k Hn−1 lim

≃ Grk

F∞ Hn−1 lim

is 1-dimensional. All the other possible graded pieces are trivial. The flag W• hence looks like W0 = W1 ⊆ W2

• dim 1

= W3 ⊆ . . . ⊆ W2n−4 = W2n−3 ⊆ W2n−2

• dim 1

= Hn−1

lim .

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Let F •

∞ be the limiting Hodge filtration and W• the monodromy

weight filtration, on Hn−1

lim .

For all k = 0, . . . , n − 1, GrW

2k Hn−1 lim

= Grk

F∞ GrW 2k Hn−1 lim

≃ Grk

F∞ Hn−1 lim

is 1-dimensional. All the other possible graded pieces are trivial. The flag W• hence looks like W0 = W1 ⊆ W2

• dim 1

= W3 ⊆ . . . ⊆ W2n−4 = W2n−3 ⊆ W2n−2

• dim 1

= Hn−1

lim .

The monodromy weight filtration on (Hn−1)lim ≃ Hn−1

lim

has the same structure.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Theorem

Fix a basis γ• = {γk}k of (Hn−1)lim adapted to the weight filtration : γk ∈ W ′

2k \ W ′ 2k−2. Then :

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Theorem

Fix a basis γ• = {γk}k of (Hn−1)lim adapted to the weight filtration : γk ∈ W ′

2k \ W ′ 2k−2. Then : ◮ there exists a unique holomorphic trivialization ϑ• of

(Rn−1f∗Ω•

Z/D×)prim, adapted to the Hodge filtration, with

• γj

ϑk =

• if j < k

1 if j = k. Here, the convention is ϑk ∈ Fn−1−k(Rn−1f∗Ω•

Z/D×)prim.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Theorem

Fix a basis γ• = {γk}k of (Hn−1)lim adapted to the weight filtration : γk ∈ W ′

2k \ W ′ 2k−2. Then : ◮ there exists a unique holomorphic trivialization ϑ• of

(Rn−1f∗Ω•

Z/D×)prim, adapted to the Hodge filtration, with

• γj

ϑk =

• if j < k

1 if j = k. Here, the convention is ϑk ∈ Fn−1−k(Rn−1f∗Ω•

Z/D×)prim. ◮ the basis ϑ• extends to a trivialization of the Deligne

extension of (Rn−1f∗Ω•

Z/D×)prim to D.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Notice that, up to constants, ϑ• only depends on the weight filtration W• on Hn−1

lim .

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Notice that, up to constants, ϑ• only depends on the weight filtration W• on Hn−1

lim .

Definition

We denote by ηk the trivializing section of (Rkf∗Ωn−1−k

Z/D× )prim

• btained by projecting ϑk.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Notice that, up to constants, ϑ• only depends on the weight filtration W• on Hn−1

lim .

Definition

We denote by ηk the trivializing section of (Rkf∗Ωn−1−k

Z/D× )prim

• btained by projecting ϑk.

By the theorem and the observation above :

◮ the ηk trivialize the respective Deligne extensions. ◮ the ηk depend only on the limiting mixed Hodge structure,

up to constants.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Let · L2 be the L2 norm on (Rkf∗Ωn−1−k

Z/D× )prim. It does not

depend on the polarization canonical up to normalization.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Let · L2 be the L2 norm on (Rkf∗Ωn−1−k

Z/D× )prim. It does not

depend on the polarization canonical up to normalization. Let q → τ(q) be the mirror map and F1(τ) the generating series of genus one Gromov–Witten invariants of Xn+1.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

Let · L2 be the L2 norm on (Rkf∗Ωn−1−k

Z/D× )prim. It does not

depend on the polarization canonical up to normalization. Let q → τ(q) be the mirror map and F1(τ) the generating series of genus one Gromov–Witten invariants of Xn+1.

Theorem (Eriksson–F.–Mourougane)

The BCOV invariant of the mirror family f : Z → D× of Xn+1 has the form τBCOV (Zq) = C

• exp
• (−1)n−1F1(τ(q))
• 4 Θ2,

where Θ :=

• η0χ(Xn+1)/12

L2

n−1

k=0 ηk(n−1−k) L2

(−1)n−1 and C is some constant.

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

From Schmid’s asymptotics for L2 metrics, one derives :

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

From Schmid’s asymptotics for L2 metrics, one derives :

Corollary

As τ → i∞, there is an asymptotic expansion 1 2∂τ log τBCOV (Zq(τ)) = (−1)n−1∂τF1(τ)

• holomorphic part

+ ρ∞ Im τ

• 1 + O

1 Im τ

• real analytic in (Im τ)−1

, where ρ∞ is explicit and depends only on Hn−1

lim .

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Functorial BCOV conjecture The BCOV invariant CY hypersurfaces in Pn

C

From Schmid’s asymptotics for L2 metrics, one derives :

Corollary

As τ → i∞, there is an asymptotic expansion 1 2∂τ log τBCOV (Zq(τ)) = (−1)n−1∂τF1(τ)

• holomorphic part

+ ρ∞ Im τ

• 1 + O

1 Im τ

• real analytic in (Im τ)−1

, where ρ∞ is explicit and depends only on Hn−1

lim .

Taking ∂ log removes all the indeterminacies, and produces a canonical expression.

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