arXiv subjects

Christopher Hacon

Publications and source records attributed to Christopher Hacon.

29 records · Page 2Linked to original sources

arXiv2017

Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun

We present a simplified proof for a recent theorem by Junyan Cao and Mihai Paun, confirming a special case of Iitaka's conjecture: if $f \colon X\to Y$ is an algebraic fiber space, and if the Albanese mapping of $Y$ is generically finite over its image, then we have the inequality of Kodaira dimensions $κ(X)\geq κ(Y)+κ(F)$, where $F$ denotes a general fiber of $f$. We include a detailed survey of the main algebraic and analytic techniques, especially the construction of singular hermitian metrics on pushforwards of adjoint bundles (due to Berndtsson, Paun, and Takayama).

math.AG↗

arXiv2017

On the characterization of abelian varieties in characteristic $p>0$

This article is superseded by 1703.06631. We keep this version here since some of the arguments for the special cases treated here are different than those of 1703.06631.

math.AG↗

arXiv2013

On the numerical dimension of pseudo-effective divisors in positive characteristic

Let X be a smooth projective variety over an algebraically closed field of positive characteristic. We prove that if D is a pseudo-effective R-divisor on X which is not numerically equivalent to the negative part in its divisorial Zariski decomposition, then the numerical dimension of D is positive. In characteristic zero, this was proved by Nakayama using vanishing theorems.

math.AG↗

arXiv2012

On the birational automorphisms of varieties of general type

We show that the number of birational automorphisms of a variety of general type X is bounded by c \cdot \vol(X,K_X), where c is a constant which only depends on the dimension of X.

math.AG↗

arXiv2012

ACC for log canonical thresholds

We show that log canonical thresholds satisfy the ACC

math.AG↗

arXiv2012

On Finiteness of B-representation and Semi-log Canonical Abundance

We give a new proof of the finiteness of B-representations. As a consequence of the finiteness of B-representations and Kollár's gluing theory on lc centers, we prove that the (relative) abundance conjecture for slc pairs is implied by the abundance conjecture for log canonical pairs.

math.AG↗

arXiv2009

Rigidity properties of Fano varieties

We overview some recent results on Fano varieties giving evidence of their rigid nature under small deformations.

math.AG↗

arXiv2006

Singularities of divisors of low degree on abelian varieties

Building on previous work of Kollar, Ein, Lazarsfeld, and Hacon, we show that ample divisors of low degree on an abelian variety have mild singularities in case the abelian variety is simple or the degree of the polarization is two.

math.AG↗

arXiv2006

Termination of (many) 4-dimensional log flips

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(K+D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and termination of flips in a relative birational setting. We also prove termination of directed flips with big K+D. As a consequence, we prove existence of minimal models of 4-dimensional dlt pairs of general type, existence of 5-dimensional log flips, and rationality of Kodaira energy in dimension 4.

math.AG↗

arXiv2005

On the existence of flips

We prove the existence of flips in dimension n, contingent on the termination of real flips in dimension n-1.

math.AG↗

arXiv2000

Deformations of the trivial line bundle and vanishing theorems

This paper reproves a general form of the Green-Lazarsfeld 'generic vanishing' theorem and more recent strengthenings, as well as giving some new applications.

math.AG↗