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

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant 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 1 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Functorial BCOV conjecture 2 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C The Grothendieck–Riemann–Roch theorem 3 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C The Grothendieck–Riemann–Roch theorem Let f : X → S be a smooth projective morphism of non-singular, connected algebraic varieties over C . 3 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C 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 . 3 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C 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( T X / S ) . In particular, for the determinant of cohomology : � � (1) CH 1 ( S ) Q . c 1 (det Rf ∗ E ) = f ∗ ch( E ) td( T X / S ) in 3 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Assume the fibers X s are Calabi–Yau (CY), i.e. K X s ≃ O X s . 4 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Assume the fibers X s are Calabi–Yau (CY), i.e. K X s ≃ O X s . Define the virtual vector bundle � D Ω • ( − 1) p p Ω p X / S = X / S . p 4 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Assume the fibers X s are Calabi–Yau (CY), i.e. K X s ≃ O X s . Define the virtual vector bundle � D Ω • ( − 1) p p Ω p X / S = X / S . p Then GRR simplifies to : X / S ) = χ c 1 (det Rf ∗ D Ω • CH 1 ( S ) Q , 12 c 1 ( f ∗ K X / S ) in with χ = χ ( X s ) the topological Euler characteristic of the fibers. 4 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Definition (BCOV line bundle) The BCOV line bundle on S is defined by λ BCOV ( f ) = det Rf ∗ D Ω • X / S � X / S ) ( − 1) p + q p . (det R q f ∗ Ω p = p 5 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Definition (BCOV line bundle) The BCOV line bundle on S is defined by λ BCOV ( f ) = det Rf ∗ D Ω • X / S � X / S ) ( − 1) p + q p . (det R q f ∗ Ω p = p It commutes with arbitrary base change. 5 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Definition (BCOV line bundle) The BCOV line bundle on S is defined by λ BCOV ( f ) = det Rf ∗ D Ω • X / S � X / S ) ( − 1) p + q p . (det R q f ∗ Ω p = p It commutes with arbitrary base change. Corollary There exists an isomorphism of Q -line bundles on S ∼ λ BCOV ( f ) ⊗ 12 → ( f ∗ K X / S ) ⊗ χ . − 5 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C But : there are as many as H 0 ( S , O × S ) such isomorphisms. 6 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C But : there are as many as H 0 ( S , O × S ) such isomorphisms. Theorem (Eriksson, Franke) There exists a canonical isomorphism of Q -line bundles GRR: λ BCOV ( f ) ⊗ 12 ∼ → ( f ∗ K X / S ) ⊗ χ , commuting with arbitrary base change. If f : X → S is defined over Q , GRR is defined over Q as well. 6 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C But : there are as many as H 0 ( S , O × S ) such isomorphisms. Theorem (Eriksson, Franke) There exists a canonical isomorphism of Q -line bundles GRR: λ BCOV ( f ) ⊗ 12 ∼ → ( f ∗ K X / 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 : 6 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C But : there are as many as H 0 ( S , O × S ) such isomorphisms. Theorem (Eriksson, Franke) There exists a canonical isomorphism of Q -line bundles GRR: λ BCOV ( f ) ⊗ 12 ∼ → ( f ∗ K X / 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. 6 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C But : there are as many as H 0 ( S , O × S ) such isomorphisms. Theorem (Eriksson, Franke) There exists a canonical isomorphism of Q -line bundles GRR: λ BCOV ( f ) ⊗ 12 ∼ → ( f ∗ K X / 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). 6 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C But : there are as many as H 0 ( S , O × S ) such isomorphisms. Theorem (Eriksson, Franke) There exists a canonical isomorphism of Q -line bundles GRR: λ BCOV ( f ) ⊗ 12 ∼ → ( f ∗ K X / 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. 6 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C But : there are as many as H 0 ( S , O × S ) such isomorphisms. Theorem (Eriksson, Franke) There exists a canonical isomorphism of Q -line bundles GRR: λ BCOV ( f ) ⊗ 12 ∼ → ( f ∗ K X / 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. 6 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Mirror symmetry at genus one 7 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C Mirror symmetry at genus one Let X be a CY variety of dimension n . 7 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C 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 : 7 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C 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 = h n − 1 , 1 ( X ∨ q ). 7 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C 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 = h n − 1 , 1 ( X ∨ q ). ◮ the monodromy on R n ϕ ∗ C is maximal unipotent : if d = 1, ( T − 1) n � = 0 and ( T − 1) n +1 = 0. 7 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C 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 = h n − 1 , 1 ( X ∨ q ). ◮ the monodromy on R n ϕ ∗ C is maximal unipotent : if d = 1, ( T − 1) n � = 0 and ( T − 1) n +1 = 0. ◮ mirror Hodge numbers : h p , q ( X ) = h n − p , q ( X ∨ q ). 7 / 28

CY hypersurfaces in P n Functorial BCOV conjecture The BCOV invariant C 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 = h n − 1 , 1 ( X ∨ q ). ◮ the monodromy on R n ϕ ∗ C is maximal unipotent : if d = 1, ( T − 1) n � = 0 and ( T − 1) n +1 = 0. ◮ mirror Hodge numbers : h p , q ( X ) = h n − p , q ( X ∨ q ). Informally : ϕ : X ∨ → D × � cusp in a moduli space of CY varieties . 7 / 28