\documentclass[12pt]{amsart} %\usepackage{amssymb} \def\thesisform{0} % \def\withfig{1} \def\withfig{0} \ifnum\withfig=1 \usepackage{psfig} \fi %\newtheorem{thm}{Theorem}[section] %\newtheorem{lemma}{Lemma}[section] %\newtheorem{cor}{Corollary}[section] %\newtheorem{conj}{Conjecture}[section] %\newtheorem{prop}{Proposition}[section] %\newtheorem{question}{Question}[section] %\theoremstyle{remark} %\newtheorem{rem}{Remark}[section] %\newtheorem{claim}{Claim}[section] %\numberwithin{equation}{section} \allowdisplaybreaks[4] %\ifnum\thesisform=1 \setlength{\oddsidemargin}{.0in} \setlength{\evensidemargin}{.0in} \setlength{\textwidth}{6.5in} \setlength{\topmargin}{0in} \setlength{\headsep}{0in} \setlength{\textheight}{8.6in} \pagestyle{plain} %\fi \begin{document} %\textwidth 7.5in %\setlength{\oddsidemargin}{.0in} %\setlength{\evensidemargin}{.0in} %\setlength{\textwidth}{8.5in} %\setlength{\topmargin}{0in} %\setlength{\headsep}{0in} %\setlength{\textheight}{8.6in} %\pagestyle{plain} %\title{Questions on the Fourier Mukai functor} %\title{Fourier transforms, generic vanishing theorems and %polarizations of abelian varieties} \def\nsubseteq{{\not\subset}} \def\vpp{{\vskip1cm}} \def\vpd{{\vskip.9cm}} \def\vnl{{\vskip.3cm}} \def\vvv{{\vskip.6cm}} \def\hhh{{\noindent \hskip1cm}} \def\proof{{{\it Proof.} ---\hskip.3cm}} \def\proofend{{ \hskip.5cm $\Box $}} \def\da{{\downarrow}} \def\OO{{\mathcal O}} \def\FF{{\mathcal F}} \def\EE{{\mathcal E}} \def\PP{{\mathcal P}} \def\PPP{{\mathbb P}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\CC{{\mathbb C}} \def\II{{\mathcal I}} \def\LL{{\mathcal L }} \def\lra{\longrightarrow} \def\ot{{\otimes}} \def\ss{{\mathcal S}} \def\sh{{\hat {\mathcal S}}} \def\Pic{{\rm {Pic}^0}} \def\dim{{\rm {dim}}} \def\codim{{\rm {codim}}} \def\alb{{\rm {alb}}} \def \Cal{\mathcal} \def \Hom{{\mathrm{Hom}}} \def \Supp{{\mathrm{Supp}}} \def \closure#1{\overline{#1}} \def \isom{\cong} \def \embed{\hookrightarrow} %\maketitle %\begin{center} {\bf by Christopher D. HACON }\end{center} %\vskip.5cm %\begin{center}{\bf -------- }\end{center} %\vskip.8cm \begin{center} {\bf \Large Fourier transforms, generic vanishing theorems \vskip.2cm and polarizations of abelian varieties} \vskip.4cm {By {\it Christopher D. Hacon} at {\it Utah}} \vskip.4cm ----------------------------- \end{center} \footnotetext{{\it Key words and phrases:} Abelian variety, polarization, singularity} \footnotetext{{\it 1991 Mathematics Subject Classification :} Primary 14K25; Secondary 14E05.} \vskip.4cm \noindent \hskip1cm {\bf Abstract.}\hskip.3cm 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 $\Theta $ a polarization of type $(1,...,1,2)$, then a general pair $(X,\Theta )$ is log canonical. \vvv \begin{center} {\bf Introduction} \end{center} \vvv \noindent \hskip1cm There is a well known connection between the geometry of principally polarized abelian varieties (PPAVs), and the singularities of their theta divisors. This was first discovered by Andreotti and Mayer, \cite {AM} in their work on the Schottky locus, and has since found applications in a variety of contexts. Subsequently Koll\'{a}r proved that the singularities of the theta divisors of a PPAV are mild in the sense that the pair $(A,\Theta )$ is log canonical. This implies for example that $\Sigma _k(\Theta ):=\{ x\in A|{\rm mult }_x ( \Theta )\geq k\}$ is of codimension at least $k$ in $A$. Ein and Lazarsfeld \cite{EL} prove that if $\Theta $ is irreducible then $\Theta $ is normal and has only rational singularities. In particular they show that if the codimension is exactly equal to $k$, then $(A, \Theta)$ splits as a $k$-fold product of PPAVs. In \cite{EL}, they also generalize the result of Koll{\'{a}}r to $\QQ$-divisors. They prove that if $D$ is a divisor in the linear series $|m\Theta |$ then the pair $(A,(1/m)D)$ is log canonical. In \cite{H}, the first result of Ein and Lazarsfeld was extended to the case of $\QQ$-divisors. Let $(A,\Theta )$ be a PPAV, and $D$ a divisor as above, such that $\lfloor (1/m)D\rfloor =0$. Then the pair $(A,(1/m)D )$ is log terminal. These results immediately generalize to arbitrary polarizations. Let $(X,L)$ be a polarization of type $(d_1,...,d_g)$, (as usual assume that $d_i|d_{i+1}$), then there exists an isogeny $X\lra X_1$ of degree $d_g$, such that $L$ is the pull back of a principal polarization $L_1$ on $X_1$. If $D$ is a divisor in the linear series $|mL|$, then by pulling back by the corresponding isogeny $X_1\lra X$, one deduces for example that the pair $(A,(1/md_g)D)$ is log canonical. \vvv \hhh In this paper, exploiting the theory of Fourier transforms, we prove that a coherent sheaf which is generically of rank $1$ on an abelian variety and is "cohomologically" a principal polarization, must in fact be a line bundle and hence a principal polarization. A surprising consequence is that if $(X,L)$ is a general polarized abelian variety of type $(1,...,1,2)$, and $D$ is a divisor in the linear series $|mL|$, then the pair $(A, (1/m)D)$ is log canonical. When $(X,D)$ is an abelian surface of type $(1,2)$, and $D$ is irreducible, then the pair $(X,D)$ is log terminal. It is not clear whether the last statement generalizes to $\QQ$-divisors. Using the same technique, we are also able to give a cohomological criterion for a subvariety of an abelian variety to be a principal polarization. \vvv \noindent \hskip 1cm {\bf Acknowledgment.}\hskip.3cm I would like to thank R. Lazarsfeld, J. Koll\'{a}r, A. Bertram and A. Chen for the stimulating conversations. \vvv \begin{center}{\bf 0. Notation and conventions}\end{center} \vvv $f^*D$ pull-back %$D_{red}$ reduced divisor associated to $D$ % %$\Omega ^i _X$ sheaf of holomorphic $i$-forms with logarithmic poles along $D_{red}$ % $|D|$ linear series associated to the divisor $D$ $Bs|D|$ the base locus of the linear series associated to the divisor $D$ $\equiv $ numerical equivalence $\omega _X = \Omega ^n_X $ canonical sheaf of $X$ $\Omega ^i _X $ sheaf of holomorphic $i$-forms on $X$ $K_X$ linear equivalence class of a canonical divisor on $X$ $\omega _{X/Y}:=\omega _X \otimes (f^*\omega _Y)^*$ relative canonical sheaf of a morphism $f:X\lra Y$ $\kappa (X)$ Kodaira dimension of $X$ $k(x)$ the sheaf $\OO _X/m_{x}$ ${\rm ALB} (Z)$ albanese variety of $Z$ ${\rm alb}_Z: Z\lra {\rm ALB} (Z)$ albanese morphism $\bar {Z}:={\rm alb}_Z (Z)$ albanese image of $Z$ $T^0$ connected component containing the origin of the subgroup $T$ \vvv \noindent \hskip 1cm Unless otherwise stated $X$, $Y$ will denote smooth complex projective varieties. If $D$ is a $\QQ$-divisor we will denote by $\lfloor D \rfloor $ and by $\lceil D \rceil$ the round down and the round up of $D$ respectively. %If $Z$ is a normal variety, by abuse of notation we will denote by %$h^0(Z,\Omega ^i _Z)$, the integer $h^0(\tilde {Z},\Omega ^i _{\tilde {Z}})$ %for an appropriate smooth birational model $\tilde {Z}$ of $Z$. \vvv \begin{center}{\bf 1. Preliminaries}\end{center} \vvv \noindent \hskip 1cm Let $X$ be a smooth complex projective variety and let $D$ be a $\mathbb Q$-divisor on $X$. We will say that $f:Y\lra X$ is a log-resolution of the pair $(X,D)$, if $f$ is a proper birational morphism such that $f^{-1}D \cup \{ \ exceptional\ set\ of \ f\ \}$ is a divisor with normal crossing support. Given a log-resolution of the pair $(X,D)$, we can define the multiplier ideal sheaf associated to the divisor $D$, $$\II (D):=f_* (\OO _Y (K_{Y/X}-\lfloor f^* D \rfloor ))\ .$$ The definition is independent of the choice of log-resolution. Multiplier ideal sheaves may be defined in much grater generality. Their properties have been extensively studied eg. [N], [Sk], [De] and [E]. Under the above assumptions, we will say that the pair $(X,D)$ is log canonical (respectively log terminal) if the multiplier ideal sheaf associated to the divisor $(1-\epsilon )D$ is trivial, for $0<\epsilon < 1$ (respectively $0\leq \epsilon < 1$). \vvv \noindent \hskip 1cm One of the main tools that we will use is the theory of deformation of cohomology groups developed by Green and Lazarsfeld \cite{GL1}, \cite{GL2}. For the convenience of the reader, we include a brief summary of the main results. We wish to study the loci $$V^i:= \{ P \in \Pic (X)\ |\ h^i(\omega _X\otimes P )\neq 0\}\ .$$ The geometry of these loci is governed by the following theorem \cite{GL1}, \cite{GL2}, \cite{EL}: \vvv \noindent \hskip1cm {\bf Theorem 1.1 (Generic vanishing).} {\it \vskip.2cm \noindent (i) $\Pic (X)\supset V^0(\omega _X )\supset V^1(\omega _X ) \supset \ ...\ \supset V^n(\omega _X )=\{\OO _X \}$ \vskip.2cm \noindent (ii) Any irreducible component of $V_i$ is a translate of a torus and is of codimension at least $i-(\dim (X) -\dim\ \alb _X(X))$ in $\Pic (X)$ \vskip.2cm \noindent (iii) Given $P$ a general point of an irreducible component $T$ of $V^i$. Let $\phi \in H^0(X,\Omega ^1 _X)$. Suppose $\bar {\phi } \in H^1(X, \OO _X)\cong T_{P }\Pic (X)$ is not tangent to $T$. Then the sequence $$H^{0}(X,\Omega ^{n-i-1}_X\ot P) \stackrel{\wedge \phi} {\longrightarrow} H^{0}(X,\Omega ^{n-i}_X\ot P) \stackrel {\wedge \phi } {\longrightarrow } H^{0}(X,\Omega ^{n-i+1}_X\ot P) $$ is exact. If $\bar {\phi }$ is tangent to $T$, then the maps in the above sequence vanish.} \vvv \noindent \hskip1cm {\bf Remark 1.2.}\hskip.3cm It is possible to generalize the above definitions and results, to the more general case of a morphism $$\nu :X\lra A$$ from $X$ to an abelian variety $A$ see \cite {EL}. One must then consider the loci $$V^i(\omega _X,A):=\{P\in \Pic (A)|h^i(\omega _X\ot \nu ^*P)\neq 0\}.$$ The generic vanishing theorem still holds, with $\Pic (A) $ instead of $\Pic (X)$ and $\dim (\nu (X))$ instead of $\dim ( \alb _X(X))$. \vvv \begin{center}{\bf 2. A characterization of principal polarizations} \end{center} \vvv \hhh In this section we will assume the notation and results of \cite {M}. In particular $X$ will always be an abelian variety and $\hat {X}$ will denote the corresponding dual abelian variety. For any point $y\in \hat {X}$ let $P_y$ denote the associated topological trivial line bundle. Let $\PP$ be the normalized Poincar\'{e} bundle on $X\times \hat {X}$. One may define a functor $\sh $ of $\OO_X$-modules into the category of $\OO_{\hat {X}}$-modules by $$\sh (M)=\pi _{\hat {X},*}(\PP \ot \pi ^*_XM).$$ The derived functor $R\sh$ of $\sh$ then induces an equivalence of categories between the two derived categories $D(X) $ and $D(\hat {X})$ \cite {M} theorem 2.2: \vvv \noindent \hskip1cm {\bf Theorem 2.1 (Mukai).}{\it \hskip.3cm There are isomorphisms of functors: $$R\ss \circ R\sh \cong (-1_X)^*[-g]$$ and $$R\sh \circ R\ss \cong (-1_{\hat {X}})^*[-g],$$ where $[-g]$ denotes "shift the complex $g$ places to the right".} \vvv \hhh The index theorem (I.T.) is said to hold for a coherent sheaf $\FF$ on $X$ if there exists an integer $i(\FF )$ such that for all $j \not \hskip-.15cm {=}i(\FF )$, $H^j(X,\FF \ot P)=0$ for all $P\in \Pic (X)$. The weak index theorem (W.I.T.) holds for a coherent sheaf $\FF$ if there exists an integer which we again denote by $i(\FF )$ such that for all $j \not \hskip-.15cm {=}i(\FF )$, $R^j\sh (\FF )=0$. It is easily seen that the I.T. implies the W.I.T. We will denote the coherent sheaf $R^{i(\FF )}\sh (\FF )$ on $\hat {X} $ by $\hat {\FF }$. One of the main themes of \cite{M} is that information on the cohomology groups $H^i(X,\FF \ot P)$ may be interpreted as information on the coherent sheaves $\FF$, $R^i \sh (\FF )$ and $\hat {\FF }$ (if $\FF$ satisfies the W.I.T.). In this spirit we prove: \vvv \noindent \hskip1cm {\bf Proposition 2.2.}{\it \hskip.3cm Let $X$ be an abelian variety, $\FF$ a coherent sheaf generically of of rank $1$ (eg. $\FF \cong L\ot \II$ where $L$ is a line bundle and $\II \subset \OO _X$ is an ideal sheaf). For all $P\in \Pic (X)$, suppose that $h^0(X,\FF \ot P)=1$, and $h^i(X,\FF \ot P)=0$ for all $1\leq i\leq n=\rm {dim} (X)$. Then $\FF$ is a line bundle with $h^0(X,\FF )=1$, hence a principal polarization.} \vvv \noindent {\it Proof.} \hskip.3cm We will use the theory of Fourier functors developed by Mukai in \cite {M}. $\FF$ satisfies the I.T. and hence the W.I.T. and $i(\FF )=0$. We will need the following: \vvv \noindent \hskip1cm {\bf Claim 2.3.} {\it \hskip.3cm $\hat {\FF }$ is a line bundle and $h^{n}(\hat{X},\hat {\FF })=1$.} \vvv \noindent {\it Proof of claim 2.3.} \hskip.3cm We will use the isomorphism of \cite {M} proposition 2.7 $$Ext^i_{\OO _X}({k(x)},\FF )\cong H^i(\hat{X},\hat {\FF }\ot P_{-x})$$ to compute $h^i(\hat{X},\hat {\FF })$. Let $K^{p , q }$ be the bicomplex ${\Cal {C}}^p[X,{\Cal {H}}om(k(x),\II ^q(\FF ))]$ where $\II ^{q }(\FF )$ is an injective resolution of the sheaf $\FF $ (see \cite{G} II 7.3). Since $\FF$ is generically of rank $1$, a local computation shows that for generic $x\in X$, $$\EE xt ^{n}_{\OO _X}(k(x),\FF )\cong k(x),$$ and that for all $0\leq i \leq \dim (X)-1$ $$\EE xt ^{i}_{\OO _X}(k(x),\FF )\cong 0.$$ The corresponding spectral sequence therefore degenerates at the $E_2$ term and we have $'E^{p, q}_2=H^p(X, \EE xt ^{q}_{\OO _X}(k(x),\FF ))$ and hence $'E^{p, q}_2\cong \CC $ for $p=0$ and $q=\dim (X)$, and $'E^{p,q}_2\cong 0$ otherwise. Since the $E_{\infty }$ term gives a filtration of $Ext _{\OO _X}^{i }({k(x)},\FF )$, it follows that for $0\leq i\leq \dim (X)-1$ we have $Ext^i_{\OO (X)}({k(x)},\FF )\cong 0$. On the other hand, for $i=\dim (X)$ $$Ext^{n}_{\OO (X)}({k(x)},\FF )\cong H^0(X,{k(x)} ) \cong \CC .$$ The claim now follows since for a generic point $x\in X$ we have $h^{n}(\hat{X},\hat {\FF }\ot P_{-x})=1$. \hskip.5cm $\Box $ \vvv \noindent \hskip1cm Consequently $\hat {\FF }\cong \OO _{\hat {X}} (-\Theta )$ for an appropriate theta divisor $\Theta$. $\hat {\FF }$ also satisfies the I.T., and in fact $h^i(\hat{X},\hat {\FF }\ot P_{-x})=0$ for all $0\leq i \leq \dim (X)-1$ and hence $\FF \cong (-1_X)^*\hat {\hskip -.07cm \hat {\FF}}$ is also a line bundle with only one section (see also \cite {M} proposition 3.11). $\Box $ \vvv \noindent \hskip1cm {\bf Remark 2.4.} \hskip.3cm The same proof also shows that if $\FF$ is a coherent sheaf generically of rank $r$ such that for all $P\in \Pic (X)$, $h^0(X,\FF \ot P)=1$ and $h^i(X,\FF \ot P)=0$ for all $1\leq i\leq n=\dim (X)$. Then $\hat {\FF }$ is dual to an ample line bundle with $r$ sections, and $\FF$ is a locally free sheaf. We will say that a sheaf satisfying the above properties is a cohomological principal polarization. Our proposition states that a cohomological principal polarization of generic rank $1$ is a principal polarization. \vvv \noindent \hskip1cm {\bf Remark 2.5.} \hskip.3cm The hypothesis on the cohomological groups may not be weakened. Consider an irreducible theta divisor $\Theta \subset X$. Let $f:\tilde {\Theta }\lra \Theta$ be an appropriate smooth birational model Since $\Theta $ has only log-terminal singularities, $h^0(f_*(\omega _{\tilde {\Theta }}\ot P))$ is $1$ if $P\not \hskip-.15cm {=}\OO_X$ and is $n$ if $P=\OO _X$. Similarly for $1\leq i\leq n-1$, $h^i(f_*(\omega _{\tilde {\Theta }}\ot P))$ is $0$ if $P\not \hskip-.15cm {=}\OO_X$ and is $\binom {n}{i+1}$ if $P=\OO _X$. It follows that the sheaf $f_*(\omega _{\tilde {\Theta }})\oplus \OO_X$ is coherent, generically of rank $1$ and satisfies the conditions for being a cohomological principal polarization, for all $P \not \hskip-.15cm {=}\OO_X$. Analogously one could also consider the sheaf $k(0)\oplus \OO _X$. It is not clear however whether one can find a similar example for a sheaf of the form $L\ot \II$ where $L$ is a line bundle and $\II \subset \OO _X$ is an ideal sheaf. \vvv \noindent \hskip1cm {\bf Remark 2.6.} \hskip.3cm One might expect that an analogue of proposition 2.2 might hold for more general polarizations. This is however not the case. Consider an abelian surface with polarization $L$ of type $(1,3)$. It is easy to show that for any point $x$, and any topologically trivial line bundle $P$, $h^i(L\ot \II _x \ot P)=0$ for all $i\geq 1$. Of course it follows that the sheaf $L\ot \II _x$ satisfies the W.I.T., and in general has all the cohomological properties of a polarization of type $(1,2)$. However the sheaf $L\ot \II _x$ is not a line bundle. \vvv \begin{center}{\bf 3. Sub-varieties of abelian varieties}\end{center} \vvv \noindent \hskip1cm Let X be an abelian variety of dimension $n$. We will say that a reduced irreducible divisor $Z$ is a cohomological theta divisor if for any desingularization $\nu :\tilde {Z}\lra Z$ i) $h^i(\omega _{\tilde {Z}})=\binom {n}{i+1}$ for all $0\leq i\leq n-1$ ii) $h^i(\omega _{\tilde {Z}}\ot \nu ^*P)=0$ for all $1\leq i\leq n-1$ and all $\OO_X \not \hskip-.15cm {=}P\in \Pic (X)$. \vvv \noindent \hskip1cm By the Hodge and Serre dualities, condition i) is equivalent to $h^0(\Omega ^j_{\tilde {Z}})=\binom {n}{j}$ for all $0\leq j\leq n-1$, and condition ii) is equivalent to $h^0(\Omega ^j_{\tilde {Z}}\ot \nu ^* P)=0$ for all $0\leq j\leq n-2$ and all $\OO_X \not \hskip-.15cm {=}P\in \Pic (X)$. The above definition, immediately generalizes to the case of a map $\nu : Z\lra X$ from a smooth $n-1$ dimensional variety $Z$ to a $n$ dimensional abelian variety $X$. The following lemma will be useful: \vvv \noindent \hskip1cm {\bf Lemma 3.1.}{\it \hskip.3cm Let $Z$ be a reduced irreducible divisor of a $n$ dimensional abelian variety $X$. Let ${\tilde {Z}}$ be a smooth birational model of $Z$. Then $Z$ is of general type if and only if $|\chi (\OO_{\tilde{Z}})|>0$.} \vvv \noindent {\it Proof.}\hskip.3cm If $Z$ is of general type, then $|\chi (\OO_{\tilde{Z}})|>0$ by \cite {EL} theorem 3 and theorem 3.3. If $Z$ is not of general type, then one may consider the abelian subvariety $B:=\{ x\in X|x+Z\subset Z\}^{0}$. By \cite{U}, we have i) $Z\lra Z/B$ is an \'{e}tale fiber bundle with fiber $B$, ii) $Z\lra Z/B$ is birational to the Iitaka fibering of $Z$, and iii) ${Z/B}$ is of general type. \noindent We may assume that $\tilde {Z}\lra \widetilde {Z/B}$ is the Iitaka fibration. Restricting to a general fiber, since $(\omega _{\tilde{Z}} \ot P)_{|B}\cong P_{|B}$, it follows that $|\omega _{\tilde{Z}} \ot P|$ is empty unless $P_{|B}\cong \OO _B$. So $H^0(\tilde{Z}, \omega _{\tilde{Z}} \ot P)=0$ for general $P\in \Pic (X)$, and hence $|\chi (\OO_{\tilde{Z}})|=0$.\proofend \vvv \noindent \hskip1cm {\bf Lemma 3.2.}{\it \hskip.3cm Let $\nu :Z\lra X$ be a morphism from a smooth variety to an $n$ dimensional abelian variety. Assume that $\nu$ restricted to $Z$ is generically finite onto a divisor $\bar {Z}=\nu (Z)$. Then i) If $\bar {Z}$ is of general type, then $|\chi (\OO _Z)| >0$. ii) If $|\chi (\OO _Z)|>0$, then $Z$ is of general type. iii) If $h^1(\omega _Z\ot \nu ^*P)=0$ for all but finitely many $P\in \Pic (X)$, then $\bar {Z}$ is of general type if and only if $Z$ is of general type.} \vvv \noindent {\it Proof.}\hskip.3cm i) If $\chi (\omega _Z)=0$, then $h^0(\omega _Z\ot \nu ^*P)=0$ for general $P\in \Pic (X)$, so $h^0(\omega _{\bar {Z}}\ot P)$ also vanishes for generic $P\in \Pic (X)$. Hence $\chi (\omega _{\bar {Z}})=0$, i.e. ${\bar {Z}}$ is not of general type. \vvv \hhh ii) If $Z$ is not of general type, then $\bar {Z}$ is not of general type. Let $\beta$ be the generic fiber of the Iitaka fibration $Z\lra Y$, then $\beta$ is \'{e}tale onto its image $B\subset X$. In fact $\kappa (\beta )=\kappa (B)=0$, and hence $\beta$ and $B$ are abelian varieties of positive dimension. Now $(\omega _Z\ot \nu ^*P) _{|{\beta }}= \nu ^*P_{|\beta }$, and since $\beta \lra B$ is an \'{e}tale map of abelian varieties, we see that $h^0(\nu ^*P_{|\beta })=0$ for generic $P\in \Pic (X)$. So $h^0(\omega _Z\ot \nu ^* P)$ also vanishes for generic $P\in \Pic (X)$ and by the generic vanishing theorem $\chi (\omega _Z)=0$. \vvv \hhh iii) Assume now that $Z$ is of general type, and $h^1(\omega _Z\ot \nu ^*P)=0$ for all but finitely many $P\in \Pic (X)$. If $\bar {Z}$ is not of general type, then $\chi (\OO _{\bar {Z}})=0$. Let $\tilde {Z}$ be a desingularization of $\bar {Z}$. Since $\bar {Z}$ is a divisor of $X$, by \cite {EL} proposition 2.2, there exists a positive dimensional subgroup $T$ of $\Pic (X)$ such that for all $P\in T$, $h^0(\omega _{\tilde {Z}}\ot P)\geq 1$. Since $\chi (\OO _{\tilde {Z}})=0$, then also $h^0(\Omega ^{n-2} _{\tilde {Z}}\ot P )=h^1(\omega _{\tilde {Z}}\ot P)\geq 1$ for $P\in T$. It follows that also $h^0(\Omega ^{n-2} _{ {Z}}\ot \nu ^*P )\geq 1$ for $P\in T$ which is a contradiction. \proofend \vvv \noindent \hskip1cm {\bf Theorem 3.3.}{\it \hskip.3cm Let $\Theta $ be a reduced irreducible divisor on an abelian variety $X$. Then $\Theta $ is a principal polarization if and only if ${\Theta}$ is a cohomological theta divisor.} \vvv \noindent {\it Proof.}\hskip.3cm We may assume that $(\tilde {X}, \tilde {\Theta})$ is a log resolution of the pair $(X, \Theta )$. One has the exact sequence of sheaves: $$0\lra \omega _{\tilde {X}}\lra \omega _{\tilde {X}} \ot \OO _{\tilde {X}}(\tilde {\Theta}) \lra \omega _{\tilde {\Theta}}\lra 0.$$ Pushing forward this sequence yields \cite {EL} the exact sequence of sheaves on $X$: $$0\lra \OO _{ {X}}\lra \OO _{{X}}({\Theta})\ot \II (\Theta ) \lra f_*\omega _{\tilde {\Theta}}\lra 0.$$ If $\Theta$ is an irreducible principal polarization, then by \cite{EL} the pair $(X, \Theta )$ is log terminal, so the multiplier ideal sheaf $\II (\Theta )$ is trivial. The cohomology groups of $\omega _{\tilde {\Theta}}$ may now be easily computed from the second exact sequence, and we conclude that ${\Theta}$ is a cohomological theta divisor. \vvv \hhh Assume now that $\Theta $ is a cohomological theta divisor. The goal here is to show that the sheaf $\OO _{{X}}({\Theta})\ot \II (\Theta )$ is a cohomological principal polarization and then to apply proposition 2.2. Of course this will follow from the equivalent statement for the sheaf $\omega _{\tilde {X}}\ot \OO _{\tilde {X}}(\tilde {\Theta})$. For all $\OO_X \not \hskip-.15cm {=}P\in \Pic (X)$ and $0\leq i\leq n$, we have $h^i(\OO _X \ot P)=0$, so $H^i(X, \OO _{{X}}({\Theta})\ot \II (\Theta )\ot P) \cong H^i(X,f_*\omega _{\tilde {\Theta}}\ot P)$. We therefore need only consider the case $P=\OO _X$. There is an injection $H^0(X,\Omega ^i_X)\lra H^0({\tilde {\Theta}}, \Omega ^i _{{\tilde {\Theta}}})$. Using the Hodge and Serre dualities, one may translate this into the isomorphisms $H^{n-i-1}({\tilde {\Theta}},\omega _{\tilde {\Theta}})\cong H^{n-i}( \tilde {X},\omega _{\tilde {X}})$. Consequently $h^0(\omega _{\tilde {X}}\ot \OO _{\tilde {X}}(\tilde {\Theta}))=1$ and $h^i(\omega _{\tilde {X}}\ot \OO _{\tilde {X}}(\tilde {\Theta}))=0$ for all $1\leq i \leq n$.\proofend \vvv \noindent \hskip1cm {\bf Corollary 3.4.} {\it Let $Z$ be a smooth variety mapping finitely onto a divisor $\bar {Z}$ of an $n$ dimensional abelian variety $X$. Assume that $\bar {Z}$ generates $X$, and that $Z$ satisfies cohomological properties analogous to those of a cohomological theta divisor. Then $\bar {Z}$ is a theta divisor.} \vvv \noindent \proof Let $\bar {Z}$ be the image of $Z$ by the map $\nu :Z\lra X$. By assumption $h^i(\omega _Z)=\binom {n}{i+1}$ for all $0\leq i\leq n-1$ and $h^i(\omega _Z \ot \nu ^* P)=0$ for all $1\leq i\leq n-1$ and $\OO_X \not \hskip-.15cm {=}P\in \Pic (X)$. The induced map $H^0(X,\Omega ^1_X)\lra H^0(Z, \Omega ^1_Z)$ is an isomorphism. Let $\tilde {Z}$ be a smooth birational model of $\bar {Z}$. We may assume that $Z\lra \bar {Z}$ factors through $\tilde {Z}$. By lemma 3.2, $\bar {Z}$ is of general type. By \cite{KV} theorem 1, $h^i(\omega _{\tilde {Z}})\geq \binom {n}{i+1}$ for all $0\leq i\leq n-1$. Clearly $$h^i(\omega _Z\ot \nu ^*P)=h^0(\Omega ^{n-i-1}_Z\ot \nu ^*P)\geq h^0(\Omega ^{n-i-1}_{\tilde {Z}}\ot P)=h^i(\omega _{\tilde {Z}}\ot P)$$ for all $0\leq i\leq n-1$ and $P\in \Pic (X)$. It follows that ${ {Z}}$ is also a cohomological theta divisor. Theorem 3.3 now proves the corollary. \proofend \vvv \noindent \hskip1cm If $Z$ is a reduced and irreducible divisor of an abelian variety, then the cohomological theta divisor condition may be substantially weakened. \vvv \noindent \hskip1cm {\bf Corollary 3.5.} {\it An irreducible reduced divisor $Z$ of an $n$ dimensional abelian variety $X$ is a principal polarization if and only if one of the following equivalent conditions holds i) $Z$ is a cohomological theta divisor ii) $Z$ is of general type, $h^0(\omega _Z )=n$, $h^1(\omega _Z \ot P)=0$ for all $\OO_X \not \hskip-.15cm {=}P\in \Pic (X)$ iii) $h^0(\omega _Z )=n$, $h^0(\omega _Z \ot P)=1$ for all $\OO_X \not \hskip-.15cm {=}P\in \Pic (X)$} \vvv \noindent \proof The equivalence of condition $i)$ is proposition 2. Condition $i)$, clearly implies the other two conditions (see lemma 3.1). \vvv \hhh Assume that condition $ii)$ is satisfied. Let ${\tilde {Z}}$ be a smooth birational model of $Z$. By \cite {KV} theorem 1, it follows that $h^0(\Omega ^i _{\tilde {Z}})= \binom {n}{i}$ for all $0\leq i\leq n-1$, and $|\chi (\OO _{\tilde {Z}})|=1$. By \cite {EL} lemma 1.8, for any $P\in \Pic (X)$, the condition $h^1(\omega _{\tilde{Z}} \ot P)=0$ implies $h^i(\omega _{\tilde {Z}} \ot P)=0$ for all $1\leq i\leq n-1$. Since $|\chi (\OO _{\tilde {Z}})|=1$, we conclude that $h^0(\omega _{\tilde {Z}} \ot P)=1$ for all $\OO_X \not \hskip-.15cm {=}P\in \Pic (X)$. Thus condition $ii)$ implies that $Z$ is a cohomological theta divisor. \vvv \hhh Assume now that condition $iii)$ is satisfied. Since $h^0(\omega _{\tilde {Z}}\ot P) >0$ for generic $P\in \Pic (X)$, then $|\chi (\OO_{\tilde {Z}})|>0$ and by lemma 3.1, $Z$ is of general type. We will now argue that condition ii) must also hold. Suppose therefore that there exists some irreducible component $T$ of $V^1:=\{ P\in \Pic (X)\ s.t.\ H^1({\tilde{Z}},\omega _{\tilde{Z}}\ot P)\not \hskip-.15cm {=}0\}$. By \cite {EL} or \cite {GL1}, \cite {GL2}, $T$ is a subtourus of $\Pic (X)$ of codimension at least $1$. Let $P\not \hskip-.15cm {=}\OO _Z$ be a general point in $T$, let $\phi \in H^0(X, \Omega ^1 _X)$ be any holomorphic $1$-form which is not the pullback of a $1$-form on $S=T^*$. Consider the following complex of vector spaces: $$H^0(X,\Omega ^{n-i-2}_{\tilde {Z}}\ot P)\stackrel {\wedge \phi }{\lra }H^0(X,\Omega ^{n-1-i}_{\tilde {Z}}\ot P) \stackrel {\wedge \phi }{\lra }H^0(X,\Omega ^{n-i}_{\tilde {Z}}\ot P).$$ By \cite {EL}, this is exact for all $i\geq 1$ (notice that any component of $V^i$, $i\geq 2$ through the general point $P$ is also contained in $T$ and hence may be assumed to be either equal to $T$ or empty. We may of course also assume that $P$ is a general point of $V^i$, $i\geq 2$). Since $\chi$ is zero on exact sequences, and $|\chi (\OO _X)|=1$ it follows that $H^0 (X,\Omega ^{n-2}_{\tilde {Z}} \ot P) \stackrel {\wedge \phi }{\lra } 0$. We may choose local coordinates $\{ z_1,...,z_{n-1}\}$ on $Z$, centered at a general point $z\in Z$. Choose $\phi _i\in H^0 (X,\Omega ^{1}_X)$ such that $dz_1,...,dz_{n-1}$ correspond to the restriction of $\phi _i$ to $T_z(Z)^*$. Let $\eta \in H^0(X,\Omega ^1 _X)$ be a non zero holomorphic 1 form such that $\eta _{|T_z(Z)^*}=0$. Since $Z$ is of general type, $Z$ is not vertical with respect to $\pi :X\lra S$, and for general $z\in Z$, $\eta$ is not in $\pi ^* (H^0(S,\Omega ^1 _S))$. We may further assume that the forms $\phi _i$ are not pulled back from $S$. (If this is not the case, then consider instead $\phi _i+\epsilon _i \eta$.) By a local computation, it follows that given any non zero $\omega \in H^0 (X,\Omega ^{n-2}_{\tilde {Z}})$, there exists a $\phi _i$ such that $\omega \wedge \phi_i$ is not zero. Since this is a contradiction, $h^1(Z,\omega _{\tilde {Z}}\ot P)=0$ for all $\OO_X \not \hskip-.15cm {=}P\in \Pic (X)$.\hskip.5cm \proofend \vvv \begin{center}{\bf 4. Polarizations of type $(1,...,1,2)$}\end{center} \vvv \noindent \hskip1cm {\bf Theorem 4.1.} {\it Let $(A,Z )$ be an $n$ dimensional abelian variety with a polarization of type $(1,...,1,2)$ (i.e. $Z$ is the class of an ample line bundle with two sections), and let $D$ be any divisor in the linear series $|mZ |$. Then either i) $(X,L)$ splits as the product of a PPAV and an elliptic curve, or ii) $(A, ( {1}/{m})D)$ is log canonical.} \vvv \noindent {\it Proof.}\hskip.3cm Consider the exact sequence associated to the multiplier ideal sheaf $\II := \II (( 1-(\epsilon / m) )D)$ for an appropriate rational number $0<\epsilon <<1$: $$0\lra L\ot \II \lra L \lra L\ot \OO _X/\II \lra 0.$$ Since $h^i(L\ot \II \ot P)=0$ for all $1\leq i\leq n$ and all $P\in \Pic (X)$, it follows that the above sequence is exact on global sections, and that $h^0(L\ot \II \ot P)=\chi (L\ot \II )$ for all $P\in \Pic (X)$. Since $L$ is of type $(1,...,1,2)$, we must have $0\leq \chi (L\ot \II ) \leq 2$. If $\chi (L\ot \II )=0$ then $h^i(L\ot \II \ot P)=0$ for all $0\leq i\leq n$ and all $P\in \Pic (X)$, and hence by a result of Mukai $L\ot \II =0$ (as sheaves) which is impossible. If $\chi (L\ot \II )=1$, then $L\ot \II$ is a cohomological principal polarization and hence $L\ot \II$ is a line bundle with only one section. This is again a impossible, unless $\II \cong \OO_X (-Z ')$ where $Z'$ is a divisor of type $(0,...,0,1)$. In this case there is a map $X\lra E$ of $X$ onto some elliptic curve $E$ such that $Z'$ is the pull back of a principal polarization on $E$. Let $L':=L\ot \OO_X (-Z')$. Then $(X,L')$ is a PPAV that splits as the product of $(E,\OO _E (p))$ and the PPAV $(\bar {X},\bar {L})$. Finally if $\chi (L\ot \II )=2$, all sections of $L$ vanish along any translate of the cosupport of $\II$. This is only possible if the cosupport of $\II$ is empty, i.e. if $\II =\OO _X$.\proofend \vvv \noindent \hskip1cm {\bf Remark 4.2.}\hskip.3cm This statement is optimal in the sense that one may consider $X=E_1\times E_2$, the product of two elliptic curves, endowed with the polarization of type $(1,2)$ given by $Z =p_1^*(p)+p_2^*(q_1+q_2)$ where $p$ and $q_i$ are points of $E_1$ and of $E_2$ respectively. If the $q_i$ are distinct points, then $(X,Z )$ is log canonical but not log terminal. If $q_1=q_2$, then $(X,Z)$ is not log canonical. One may ask whether for $D$ a reduced and irreducible divisor of type $(1,...,1,2)$, the pair $(X, D)$ is log terminal. While this is not clear in higher dimensions, it does hold if $X$ is an abelian surface. \vvv \noindent \hskip1cm {\bf Theorem 4.3.}{\it \hskip.3cm Let $(X,D )$ be a polarized abelian surface with $D$ an ample reduced irreducible divisor such that $h^0(\OO_A(D))=2$. Then the pair $(X, D)$ is log terminal.} \vvv \noindent \proof As in the proof of proposition 2, one may consider an appropriate smooth birational model $f:\tilde {D}\lra D$ and the induced exact sequence of sheaves $$0\lra \OO _{ {X}}\lra \OO _{{X}}(D)\ot \II (D ) \lra f_*\omega _{\tilde {D}}\lra 0.$$ \noindent Assume that $(X,D)$ is not log terminal, then the cosupport of $\II :=\II (D )$ is not empty and has codimension two in $X$. Pick a general divisor $D'\in |D|$ which is distinct from $D$. These two divisors will intersect along a finite set of points. It follows that $h^0(\OO _X(D)\ot \II \ot P)=2$ only if the appropriate translate of the cosupport of $\II$ is contained in this finite set of points. (By \cite{LB} 10.1.2, $|D|$ has exactly four base points, corresponding to $D'\cap D''$ the intersection of two general members of $|D|$.) This means that $h^0(\OO _X(D)\ot \II \ot P)\leq 1$ for all but finitely many $P\in \Pic (X)$. For all topologically trivial line bundles $P \not \hskip -.2cm {=} \OO_X$ there is an isomorphism $H^0(\OO _{{X}}(D)\ot \II (D )\ot P)\cong H^0(f_*\omega _{\tilde {D}}\ot P)$. Since $D$ is ample and hence of general type, $h^0(f_*\omega _{\tilde {D}}\ot P)>0$ for all $P\in \Pic (X)$. So $h^0(f_*\omega _{\tilde {D}}\ot P)=h^0(\OO _{{X}}(D)\ot \II (D )\ot P) =1$ for all but finitely many $P\in \Pic (X)$. Hence $g({\tilde {D}})=2$ and $h^0(f_*\omega _{\tilde {D}}\ot P)=h^0(\OO _{{X}}(D)\ot \II (D )\ot P) =1$ for all $\OO _X$ $\not \hskip -.15 cm {=}P\in \Pic (X)$. By corollary 3.5 the proposition now follows. %Alternatively, for a direct proof, one may proceed as follows. %The ideal sheaf $\II $ %is cosupported in dimension $0$. If the cosupport $\II $ consists of at least %two points, then $2g(\tilde {D})-2=D^2-\sum m_i (m_i-1)\leq 0$ and then %$D$ is either a rational or an elliptic curve. In either case this is a %contradiction. So $\II $ is cosupported at a unique point. %Furthermore $D\cap D'$ must also consist of a single point, %otherwise $h^0(\OO _{{X}}(D)\ot \II (D )\ot P)$ would be equal to $2$ %for some nontrivial $P$. %In which case $\OO _X(D)$ would have a unique base point at which %every divisor in the linear series $|\OO _X(D)|$ vanishes with multiplicity %exactly $2$, and any two divisors intersect exclusively along this point. %This is a contradiction, eg. \cite{LB} 10.1.3. \hskip.5cm $\Box$ \vvv \noindent \hskip1cm {\bf Question 4.4.}{\hskip.3cm Let $(X,D)$ be a general polarized abelian of type $(1,...,1,2)$. Assume that $D$ is irreducible. Is $(X,D)$ a log terminal pair? Similarly let $(X,D)$ be a general polarized abelian of type $(1,...,1,d)$, with $d=3,4$. Assume that $D$ is irreducible. Is $(X,D)$ a log canonical pair?} \vvv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % References %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{99} \bibitem {AM} {\it A. Andreotti, A. Mayer}, {{On period relations for abelian integrals on algebraic curves}}, Ann. Scuola Normale Superiore Pisa {\bf 21} (1967) 189-238 \bibitem {EL} {\it L. Ein, R. Lazarsfeld}, {{Singularities of theta divisors, and the birational geometry of irregular varieties.}}, Jour. Amer. Math. Soc. {\bf 10} (1997), 243-258 \bibitem{G} {\it R. Godement}, {{Topologie alg\'{e}brique et théorie des faisceaux}}, Publications de l'Institut de Math\'{e}matique de l'Universit\'{e} de Strasbourg, XIII. Actualit\'{e}s Scientifiques et Industrielles, No. 1252. Hermann, Paris, 1973 \bibitem{GL1} {\it M. Green, R. Lazarsfeld}, {{Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville}}, Inventiones Math. {\bf 90} (1987), 389-407 \bibitem{GL2}{\it M. Green, R. Lazarsfeld}, {{Higher obstructions to deforming cohomology groups of line bundles}}, J. Amer. Math. Soc. {\bf 4} (1991), 87-103 \bibitem {H} {\it C. D. Hacon}, {{Divisors on principally polarized abelian varieties}}, Compositio Mathematica (to appear) \bibitem {Ke} {\it G. Kempf}, {{On the geometry of a theorem of Riemann}}, Ann. of Math. {\bf 98} (1973), 178-185 \bibitem{KV} {\it Y. Kawamata, E. Viehweg}, {{On a characterization of abelian varieties in the classification theory of algebraic varieties}}, Compositio Math. {\bf 41} (1980), 355-360 \bibitem {LB} {\it H. Lange C. Birkenhake}, {{Complex Abelian Varieties}}, Grundlehren der mathematischen Wissenschaften, Springer Verlag \bibitem{M} {\it S. Mukai}, {{Duality between $D(X)$ and $D(\hat {X})$, with application to Picard sheaves }}, Nagoya math. J. {\bf 81} (1981), 153-175 \bibitem{U}{\it K. Ueno}, {{Classification Theory of Algebraic Varieties and Compact Complex Spaces}}, Springer Verlag {\bf LNM 439 } (1975) \end{thebibliography} \begin{center} -------------------------------- \vskip.4cm Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090, USA. \vskip.3cm email: hacon@math.utah.edu\end{center} \end{document} \end