Collapsing of abelian fibred Calabi-Yau manifolds and hyperkähler

125 Pages·2011·1 MB·English

by  Valentino Tosatti

Collapsing of abelian fibred Calabi-Yau manifolds and
hyperk¨ahler mirror symmetry
Valentino Tosatti
ColumbiaUniversity
Geometric Structures in Mathematical Physics
Golden Sands, September 23, 2011
joint work with Mark Gross and Yuguang Zhang
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Plan
1 Introduction
2 Some Results
3 Abelian fibred Calabi-Yau manifolds
4 Hyperk¨ahler manifolds and mirror symmetry
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Calabi-Yau manifolds
A compact K¨ahler manifold Xn is called Calabi-Yau if c (X) =0 in
1
H2(X,R).
n = dim X
C
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Calabi-Yau manifolds
A compact K¨ahler manifold Xn is called Calabi-Yau if c (X) =0 in
1
H2(X,R).
n = dim X
C
This is equivalent to requiring that the canonical bundle K be torsion,
X
ℓK = , ℓ > 1.
X ∼ X
O
ValentinoTosatti (Columbia) CollapsingofCalabi-Yaus September23,2011 3/36
Calabi-Yau manifolds
A compact K¨ahler manifold Xn is called Calabi-Yau if c (X) =0 in
1
H2(X,R).
n = dim X
C
This is equivalent to requiring that the canonical bundle K be torsion,
X
ℓK = , ℓ > 1.
X ∼ X
O
Often more restrictive definitions are considered (X projective, K trivial,
X
H1(X, ) = 0,Hp(X, ) = 0 for 0< p < n,...).
X X
O O
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Examples of Calabi-Yau manifolds
X = Cn/Λ, a complex torus
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Examples of Calabi-Yau manifolds
X = Cn/Λ, a complex torus
X a simply connected Calabi-Yau surface n = 2 is called a K3 surface.
ValentinoTosatti (Columbia) CollapsingofCalabi-Yaus September23,2011 4/36
Examples of Calabi-Yau manifolds
X = Cn/Λ, a complex torus
X a simply connected Calabi-Yau surface n = 2 is called a K3 surface.
Every Calabi-Yau surface is either a torus, K3, or a finite unramified
quotient of these (bielliptic surfaces, Enriques surfaces)
ValentinoTosatti (Columbia) CollapsingofCalabi-Yaus September23,2011 4/36
Examples of Calabi-Yau manifolds
X = Cn/Λ, a complex torus
X a simply connected Calabi-Yau surface n = 2 is called a K3 surface.
Every Calabi-Yau surface is either a torus, K3, or a finite unramified
quotient of these (bielliptic surfaces, Enriques surfaces)
X a smooth complex hypersurface in CPn+1 of degree n+2 is
Calabi-Yau. n = 1 : torus, n = 2 : K3
ValentinoTosatti (Columbia) CollapsingofCalabi-Yaus September23,2011 4/36
Examples of Calabi-Yau manifolds
X = Cn/Λ, a complex torus
X a simply connected Calabi-Yau surface n = 2 is called a K3 surface.
Every Calabi-Yau surface is either a torus, K3, or a finite unramified
quotient of these (bielliptic surfaces, Enriques surfaces)
X a smooth complex hypersurface in CPn+1 of degree n+2 is
Calabi-Yau. n = 1 : torus, n = 2 : K3
Complete intersections in products of (weighted) projective spaces
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