Threefolds of Kodaira dimension one

On the logarithmi Kodaira dimension of aÆne threefolds

On the Medvedev–Scanlon conjecture for minimal threefolds of nonnegative Kodaira dimension

Motivated by work of Zhang from the early ‘90s, Medvedev and Scanlon formulated the following conjecture. Let F be an algebraically closed field of characteristic 0 and let X be a quasiprojective variety defined over F endowed with a dominant rational self-map φ. Then there exists a point x ∈ X(F ) with Zariski dense orbit under φ if and only if φ preserves no nontrivial rational fibration, i.e...

Failure of the Integral Hodge Conjecture for Threefolds of Kodaira Dimension Zero

We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth project...

Kodaira dimension and zeros of holomorphic one-forms

We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge modules on abelian varieties.

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...