Semicontinuity of Kodaira dimension

Kodaira Dimension and Symplectic Sums

Modulo trivial exceptions, we show that symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is d...

Kodaira Dimension and the Yamabe Problem

The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar curvature Riemannian metrics g on M . (To be absolutely precise, one only considers constant-scalar-curvature metrics which are Yamabe minimizers, but this does not affect the sign of the answer.) If M is the underlying smooth 4-manifold of a complex algebraic ...

The Kodaira Dimension of Certain Moduli Spaces of Abelian Surfaces

Moduli Spaces for polarized abelian surfaces are obtained as quotients of the Siegel upper half–plane H2 by arithmetic subgroups of the symplectic group Sp(4,Q). In this paper we will look at the Kodaira dimension of the moduli spaces At which parameterize abelian surfaces with a polarization of type (1, t). For many small values of t it is known that At is rational or unirational (e.g. see the...

The Kodaira Dimension of the Moduli Space of Prym Varieties

Prym varieties provide a correspondence between the moduli spaces of curves and abelian varieties Mg and Ag−1, via the Prym map Pg : Rg → Ag−1 from the moduli space Rg parameterizing pairs [C, η], where [C] ∈ Mg is a smooth curve and η ∈ Pic(C)[2] is a torsion point of order 2. When g ≤ 6 the Prym map is dominant and Rg can be used directly to determine the birational type ofAg−1. It is known t...

On the logarithmi Kodaira dimension of aÆne threefolds