The asymptotics of the holomorphic torsion form Martin Puchol - - PowerPoint PPT Presentation

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

The asymptotics of the holomorphic torsion form

Martin Puchol Institut Camille Jordan – Université Lyon 1

Index Theory and Singular Structures – Toulouse

May 30, 2017

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Sommaire

1

Introduction

2

Definition

3

Statement of the result

4

Tœplitz operators

5

Asymptotic heat kernel

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

The holomorphic analytic torsion is a spectral invariant introduced by Ray and Singer in 1973 as the complex analogue

• f the real torsion.

It is given as a weighted determinant of the Kodaira Laplacian

• f a holomorphic Hermitian bundle (E, hE) on a compact

complex Riemannian manifold (M, gTM). exp

• τ(gTM, hE)
• :=

dim M

• k=0

det ζ(E|Ω0,k)(−1)kk. It plays a role in the study of the determinant of the direct image of a holomorphic vector bundle under a holomorphic proper fibration (Bismut, Gillet et Soulé). used in Arakelov geometry.

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

In 1989, Bismut and Vasserot computed the asymptotics when p → +∞ of the torsion associated with Lp = L⊗p, L a positive line bundle.

• Ex. of application: arithmetic version of the Hilbert-Samuel

theorem by Gillet-Soulé in Arakelov geometry (estimation of the number of arithmetic sections of small norm). In 1990, Bismut and Vasserot extended their result with Lp ← → Sp(E), the pth symmetric power of a positive bundle E.

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Extension for families: the holomorphic analytic torsion form. Defined first by Bismut-Gillet-Soulé and in greater generality by Bismut-Köhler (1992). Consider a family {(Xb, gTXb)}b∈B of Riemannian compact complex manifolds, more precisely:

π: M → B holo. proper fibration with fiber X and B compact. ωM (1,1)-form on M inducing metrics on the fibers. Assume ωM is closed.

Consider also (E, hE) be a holo. Herm. vector bundle on M. T (ωM, hE) ∈ Ω2•(B).

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

T (0) = Ray-Singer torsion along the fiber, and it appears in a refinement of the Riemann-Roch-Grothendieck theorem: ch

• R•π∗E, hR•π∗E

−

• X

Td(TX, hTX) ch(E, hE) = ¯ ∂∂ 2iπT (ωM, hE). Also plays a role in the theory of direct image in Arakelov geometry: π! : K0(M) → K0(B) given by π!(E, hE, α) =

• (−1)j(Rjπ∗E, hRjπ∗E, 0)

+

• 0, 0, −T (ωM, hE) +
• X

Td(TX, gTX)α

• .

Thus it appears in the arithmetic RRG theorem. Here we want to extend the results of Bismut-Vasserot for T .

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Sommaire

1

Introduction

2

Definition

3

Statement of the result

4

Tœplitz operators

5

Asymptotic heat kernel

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Finite dim. model : transgression of the Chern character

(B, gTB) Riemannian compact complex manifold. E • = 0

δ

→ E 1

δ

→ . . .

δ

→ E m → 0 complex of holomorphic vector bundles. hE • metric on E •, ∇E • associated Chern connection (preserves Herm. and holo. structures). H(E •, δ) cohomology. Assume it is a smooth bundle. Let ∇H be the connection induced by ∇E •. We know: ch(E •) = ch(H(E •, δ)) ∈ H(B).

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Finite dim. model : transgression of the Chern character

D = δ + δ∗. Then ∇u = ∇E • + √uD connection on E • and ch(∇u) := Trs[e−∇2

u] = Tr |E even[e−∇2 u] − Tr |E odd[e−∇2 u].

Let N s.t. N|E k = k. IdE k. Then ∂ ∂u ch(∇u) = −¯ ∂∂ 1 u Trs

• Ne−∇2

u

. Let ζ(s) = − 1 Γ(s) +∞ us−1 Trs

• Ne−∇2

u

• du.

Then ζ′(0) = +∞ Trs

• Ne−∇2

u

• du

u .

Conclusion: ch(E •, ∇E •) − ch(H(E •, δ), ∇H) = −¯ ∂∂ζ′(0).

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Infinite dimensional bundle

π: M → B a proper holomorphic fibration with fiber X. ωM a (1,1)-form on M. Definition (π, ωM) is a Kähler fibration if

1

ω is closed,

2

U, V TX = ω(U, JV ) for U, V ∈ TX define a fiberwise metric.

(E, hE) holo. Herm. vector bundle on M. E infinite dimensional bundle on B: E•

b = Ω0,•(Xb, E|Xb)

(C-antilinear forms). Endowed with the L2-product. Chern connections on E, TX ∇E• connection on E•.

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Bismut’s superconnection

DE

Xb = ¯

∂E

Xb + ¯

∂E,∗

Xb Dirac operator of the fiber Xb, acting on E• b.

Definition The Bismut’s superconnection is an operator acting on Ω•(B, E•): Bu = ∇E• + √uDE

X + . . .

Theorem (Bismut) B2

u = uDE,2 X

+ N E

u

∈ Ω•(B, End(E•)), where N E

u is a fiberwise nilpotent operator of order 1.

In particular, B2

u fiberwise elliptic operator of order 2 and its

fiberwise heat kernel exp(−B2

u) is well-defined.

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Holomorphic torsion form

N number operator on E•. ωH = ωM|T HM×T HM and Nu = N − i ωH

u .

Let ζ(s) = − 1 Γ(s) +∞ us−1 Trs

• Nue−B2

u

• du.

Then ζ admits a holomorphic extension near 0. Definition The holomorphic analytic torsion form is T (ωM, hE) = ζ′(0).

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Sommaire

1

Introduction

2

Definition

3

Statement of the result

4

Tœplitz operators

5

Asymptotic heat kernel

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

π1 : N → M and π2 : M → B two proper holomorphic fibrations with fibers Y and X. π3 := π2 ◦ π1 : N → B, with fiber Z. (π2, ωM) a structure of Kähler fibration on M. (L, hL) holo. Herm. line bundle on N, with Chern curvature RL. Assume that L|Z is a positive bundle, i.e., RL|Z positive (1,1)-form. For p ≫ 1, Fp := H0 Y , Lp|Y

• is a holo. Herm. vector bundle
• n M.

T (ωM, hFp) associated torsion form.

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

· , ·TX = ⇒ we can identify RL|TX to a Herm. matrix ˙ RX,L nZ = dimC Z. Let ψp ∈ End(Λeven(T ∗B)) such that for α ∈ Λ2k(T ∗

RB),

ψpα = p−kα. Theorem (P.) As p → +∞, ψp

• T (ωM, hFp) − 1

2

• Z

log

• det
• p ˙

RX,L 2π

• ep.c1(L,hL)

= o(pnZ ), for the C ∞ topology on B.

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Steps of the proof:

1 ζp(s) = −

1 Γ(s)

+∞ us−1 Trs

• Nue−B2

p,u

• du implies

T (ωM, hFp) = − 1

• Trs
• Nu exp(−B2

p,u)

• − Cp,−1

u − Cp,0 du u − +∞

1

Trs

• Nu exp(−B2

u)

du u + Cp,−1 + Γ′(1)Cp,0.

2 Get the asymptotics of the heat kernel

+ asymptotics of the Cp,i + dominations and use dominated convergence theorem = ⇒ “ζ′

p(0) → ζ′ ∞(0)”.

3 Compute ζ′

∞(0).

Main difficulty: e−Bp,u(·, ·) ∈ Λ•(T ∗B) ⊗ End(Λ0,•(T ∗X) ⊗ Fp), acts on changing bundles, and dim Fp → ∞. Sense of convergence?

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Sommaire

1

Introduction

2

Definition

3

Statement of the result

4

Tœplitz operators

5

Asymptotic heat kernel

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

For m ∈ M, we denote Ym simply by Y . Pp orthogonal proj. L2(Y , Lp) → H0(Y , Lp) = Fp. Définition A Tœplitz operator on Y is a family of operators Tp ∈ End(L2(Y , Lp)) satisfying:

1 ∀p ∈ N, we have Tp = PpTpPp, thus Tp ∈ End(Fp). 2 ∃ ϕr ∈ C ∞(Y ) s.t. for k ∈ N, ∃ Ck > 0 s.t.

• Tp −

k

• r=0

1 pr PpϕrPp

• ≤ Ck

1 pk+1 , where · is the operator norm on End(L2(Y , Lp)). “Elementary” Tœplitz operators of the form PpϕPp play the role of the limits: up → ϕ ← → up − PpϕPp → 0.

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Sommaire

1

Introduction

2

Definition

3

Statement of the result

4

Tœplitz operators

5

Asymptotic heat kernel

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

nX = dimC X. Theorem (P.) For p → +∞, we have for the C ∞ topology associated with the

• perator norm, uniformly u in a compact of R∗

+,

ψp exp(−B2

p,u/p)(m, m)

= pnX (2π)nX Pp,me−Ωu,(m,·) det( ˙ RX,L

(m,·))

det

• 1 − exp(−u ˙

RX,L

(m,·))

Pp,m + o(pnX ). Here, (·) stands for the variable in Ym and Ωu is an explicit element

• f Λ•(T ∗B) ⊗ End(Λ0,•(T ∗TX)).

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Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel

Thank you for your attention!

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