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 1/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Sommaire Introduction 1 Definition 2 Statement of the result 3 Tœplitz operators 4 Asymptotic heat kernel 5 2/21

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 of the real torsion. It is given as a weighted determinant of the Kodaira Laplacian of a holomorphic Hermitian bundle ( E , h E ) on a compact complex Riemannian manifold ( M , g TM ) . � � dim M � det ζ ( � E | Ω 0 , k ) ( − 1 ) k k . τ ( g TM , h E ) exp := k = 0 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. 3/21

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 L p = 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 L p ← → S p ( E ) , the p th symmetric power of a positive bundle E . 4/21

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 { ( X b , g TX b ) } b ∈ B of Riemannian compact complex manifolds, more precisely: π : M → B holo. proper fibration with fiber X and B compact. ω M (1,1)-form on M inducing metrics on the fibers. Assume ω M is closed. Consider also ( E , h E ) be a holo. Herm. vector bundle on M . � T ( ω M , h E ) ∈ Ω 2 • ( B ) . 5/21

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: � � R • π ∗ E , h R • π ∗ E � Td ( TX , h TX ) ch ( E , h E ) ch − X ¯ ∂∂ 2 i π T ( ω M , h E ) . = Also plays a role in the theory of direct image in Arakelov geometry: π ! : � K 0 ( M ) → � K 0 ( B ) given by � ( − 1 ) j ( R j π ∗ E , h R j π ∗ E , 0 ) π ! ( E , h E , α ) = � � � 0 , 0 , −T ( ω M , h E ) + Td ( TX , g TX ) α + . X Thus it appears in the arithmetic RRG theorem. Here we want to extend the results of Bismut-Vasserot for T . 6/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Sommaire Introduction 1 Definition 2 Statement of the result 3 Tœplitz operators 4 Asymptotic heat kernel 5 7/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Finite dim. model : transgression of the Chern character ( B , g TB ) Riemannian compact complex manifold. E • = 0 δ δ → E m → 0 complex of holomorphic δ → E 1 → . . . vector bundles. h E • 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 ) . 8/21

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 ) := Tr s [ e −∇ 2 u ] = Tr | E even [ e −∇ 2 u ] − Tr | E odd [ e −∇ 2 u ] . Let N s.t. N | E k = k . Id E k . Then � 1 � � � ∂ Ne −∇ 2 ∂ u ch ( ∇ u ) = − ¯ u Tr s ∂∂ . u Let � + ∞ � � ζ ( s ) = − 1 u s − 1 Tr s Ne −∇ 2 du . u Γ( s ) 0 � � � + ∞ Ne −∇ 2 Then ζ ′ ( 0 ) = du Tr s u . u 0 Conclusion: ch ( E • , ∇ E • ) − ch ( H ( E • , δ ) , ∇ H ) = − ¯ ∂∂ζ ′ ( 0 ) . 9/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Infinite dimensional bundle π : M → B a proper holomorphic fibration with fiber X . ω M a (1,1)-form on M . Definition ( π, ω M ) is a Kähler fibration if ω is closed, 1 � U , V � TX = ω ( U , JV ) for U , V ∈ TX define a fiberwise metric. 2 ( E , h E ) holo. Herm. vector bundle on M . E infinite dimensional bundle on B : E • b = Ω 0 , • ( X b , E | X b ) ( C -antilinear forms ) . Endowed with the L 2 -product. Chern connections on E , TX � ∇ E • connection on E • . 10/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Bismut’s superconnection X b = ¯ X b + ¯ ∂ E , ∗ D E ∂ E X b Dirac operator of the fiber X b , acting on E • b . Definition The Bismut’s superconnection is an operator acting on Ω • ( B , E • ) : B u = ∇ E • + √ uD E X + . . . Theorem (Bismut) u = uD E , 2 B 2 + N E ∈ Ω • ( B , End ( E • )) , X u where N E u is a fiberwise nilpotent operator of order 1. In particular, B 2 u fiberwise elliptic operator of order 2 and its fiberwise heat kernel exp ( − B 2 u ) is well-defined. 11/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Holomorphic torsion form N number operator on E • . ω H = ω M | T H M × T H M and N u = N − i ω H u . Let � + ∞ � � ζ ( s ) = − 1 u s − 1 Tr s N u e − B 2 du . u Γ( s ) 0 Then ζ admits a holomorphic extension near 0. Definition The holomorphic analytic torsion form is T ( ω M , h E ) = ζ ′ ( 0 ) . 12/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Sommaire Introduction 1 Definition 2 Statement of the result 3 Tœplitz operators 4 Asymptotic heat kernel 5 13/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel π 1 : N → M and π 2 : M → B two proper holomorphic fibrations with fibers Y and X . π 3 := π 2 ◦ π 1 : N → B , with fiber Z . ( π 2 , ω M ) a structure of Kähler fibration on M . ( L , h L ) holo. Herm. line bundle on N , with Chern curvature R L . Assume that L | Z is a positive bundle, i.e., R L | Z positive (1,1)-form. For p ≫ 1, F p := H 0 � � Y , L p | Y is a holo. Herm. vector bundle on M . � T ( ω M , h F p ) associated torsion form. 14/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel ⇒ we can identify R L | TX to a Herm. matrix ˙ R X , L �· , ·� TX = n Z = dim C Z . Let ψ p ∈ End (Λ even ( T ∗ B )) such that for α ∈ Λ 2 k ( T ∗ R B ) , ψ p α = p − k α . Theorem (P.) As p → + ∞ , � �� � � �� � � p ˙ R X , L T ( ω M , h F p ) − 1 e p . c 1 ( L , h L ) ψ p log det 2 2 π Z = o ( p n Z ) , for the C ∞ topology on B . 15/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Steps of the proof: � � � + ∞ u s − 1 Tr s N u e − B 2 1 ζ p ( s ) = − 1 du implies p , u Γ( s ) 0 � � du � 1 � � − C p , − 1 T ( ω M , h F p ) = − N u exp ( − B 2 − C p , 0 Tr s p , u ) u u 0 � + ∞ � � du N u exp ( − B 2 u + C p , − 1 + Γ ′ ( 1 ) C p , 0 . − Tr s u ) 1 2 Get the asymptotics of the heat kernel + asymptotics of the C p , i + dominations and use dominated convergence theorem ⇒ “ ζ ′ p ( 0 ) → ζ ′ = ∞ ( 0 ) ”. 3 Compute ζ ′ ∞ ( 0 ) . Main difficulty: e − B p , u ( · , · ) ∈ Λ • ( T ∗ B ) ⊗ End (Λ 0 , • ( T ∗ X ) ⊗ F p ) , acts on changing bundles, and dim F p → ∞ . Sense of convergence? 16/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Sommaire Introduction 1 Definition 2 Statement of the result 3 Tœplitz operators 4 Asymptotic heat kernel 5 17/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel For m ∈ M , we denote Y m simply by Y . P p orthogonal proj. L 2 ( Y , L p ) → H 0 ( Y , L p ) = F p . Définition A Tœplitz operator on Y is a family of operators T p ∈ End ( L 2 ( Y , L p )) satisfying: 1 ∀ p ∈ N , we have T p = P p T p P p , thus T p ∈ End ( F p ) . 2 ∃ ϕ r ∈ C ∞ ( Y ) s.t. for k ∈ N , ∃ C k > 0 s.t. � � � � k � 1 1 � � � T p − � ≤ C k � p r P p ϕ r P p � p k + 1 , r = 0 where � · � is the operator norm on End ( L 2 ( Y , L p )) . “Elementary” Tœplitz operators of the form P p ϕ P p play the role of the limits: u p → ϕ ← → � u p − P p ϕ P p � → 0. 18/21

Introduction Definition Statement of the result Tœplitz operators Asymptotic heat kernel Sommaire Introduction 1 Definition 2 Statement of the result 3 Tœplitz operators 4 Asymptotic heat kernel 5 19/21