Varieties with $P_3(X)=4$ and $q(X)=\dim (X)$
We classify varieties with $P_3(X)=4$ and $q(X)=\dim (X)$.
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arXiv subjects
Christopher D. Hacon
Publications and source records attributed to Christopher D. Hacon.
A derived category approach to generic vanishing
We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field $k$. When $k=\CC$ this implies a conjecture of Green and Lazarsfeld.
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Surfaces with $p_g=q=3$
We classify minimal complex surfaces of general type with $p_g=q=3$. More precisely, we show that such a surface is either the symmetric product of a curve of genus 3 or a free $\Z_2-$quotient of the product of a curve of genus 2 and a curve of genus 3. Our main tools are the generic vanishing theorems of Green and Lazarsfeld and Fourier--Mukai transforms. The same result has been obtained independently at the same time by G. Pirola using different methods.
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On algebraic fiber spaces over varieties of maximal Albanese dimension
We study algebraic fiber spaces $f:X \longrightarrow Y$ where $Y$ is of maximal Albanese dimension. In particular we give an effective version a theorem of Kawamata: If $P_m(X)=1$ for some $m \ge 2$, then the Albanese map of $X$ is surjective. Combining this with \cite{CH} it follows that $X$ is birational to an abelian variety if and only if $P_2(X)=1$ and $q(X)= \text{\rm dim} (X)$.
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Pluricanonical maps of varieties of maximal Albanese dimension
Let $X$ be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of $κ(X)$ in terms of the set $V^0(X,ω_{X})$ $:=\{P\in {\text{\rm Pic}}^0(X)|h^0(X, ω_X \otimes P) \ne 0\}$. An immediate consequence of this is that the Kodaira dimension $κ(X)$ is invariant under smooth deformations. We then study the pluricanonical maps $ϕ_m:X -> \Bbb{P} (H^0(X,mK_X))$. We prove that if $X$ is of general type, $ϕ_m$ is generically finite for $m\geq 5$ and birational for $m\geq 5 \text{\rm dim} (X) +1$. More generally, we show that for $m\geq 6$ the image of $ϕ_m$ is of dimension equal to $κ(X)$ and for $m\geq 6κ(X)+2$, $ϕ_m$ is the stable canonical map.
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Characterization of Abelian Varieties
We prove that any smooth complex projective variety $X$ with plurigenera $P_1(X)=P_2(X)=1$ and irregularity $q(X)=dim (X)$ is birational to an abelian variety.
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Fourier transforms, generic vanishing theorems and polarizations of abelian varieties
The purpose of this paper is to give two applications of Fourier transforms and generic vanishing theorems: - we give a cohomological characterization of principal polarizations - we prove that if $X$ an abelian variety and $Θ$ a polarization of type $(1,...,1,2)$, then a general pair $(X,Θ)$ is log canonical
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