Singularities of theta divisors and the birational geometry of irregular varieties.

*(English)*Zbl 0901.14028The paper applies the generic vanishing theorems of Green and Lazarsfeld to settle some questions and conjectures raised by J. Kollár [in “Shafarevich maps and automorphic forms” (Princeton 1995; Zbl 0871.14015)] concerning the geometry of irregular complex projective varieties. There are essentially 3 types of results:

(1): Let \((A,\Theta)\) denote a principally polarized abelian variety. The first theorem is that if \(\Theta\) is an irreducible theta divisor, then \(\Theta\) is normal and has only rational singularities. This generalizes a theorem of Kempf, who proved this for Jacobian varieties. Kollar showed (loc. cit.) that every component of \(\Sigma_k(\Theta):= \{x\in A: \text{mult}_x(\Theta)\geq k\}\) has codimension \(\geq k\) in \(A\). A consequence of the first theorem is that if \(k\geq 2\), then \(\Sigma_k (\Theta)\) contains an irreducible component of codimension \(k\) in \(A\) if and only if \((A, \Theta)\) splits as a \(k\)-fold product of p.p.a.v’s. This generalizes a theorem of Smith and Varley who proved this for \(k=g\).

(2): Kollár conjectured that if \(X\) is of general type, then the holomorphic Euler characteristic \(\chi(X,\omega_X)\) is positive. The third theorem of the paper is that this is true if \(X\) is birationally a subvariety of \(\text{Alb} (X)\). The picture is completed by showing that the conjecture fails in general. The counterexample is a threefold whose Albanese mapping is a branched covering with a rather degenerate branching divisor.

(3): The fourth theorem is that if \(X\) is a smooth projective variety of dimension \(n\) with plurigenera \(P_1(X)\) and \(P_2(X)\) equal to 1, then the Albanese mapping of \(X\) is surjective. This implies a result of Kawamata saying that the Albanese mapping is surjective if the Kodaira dimension of \(X\) is zero. The proof of the theorem is surprisingly simple.

(1): Let \((A,\Theta)\) denote a principally polarized abelian variety. The first theorem is that if \(\Theta\) is an irreducible theta divisor, then \(\Theta\) is normal and has only rational singularities. This generalizes a theorem of Kempf, who proved this for Jacobian varieties. Kollar showed (loc. cit.) that every component of \(\Sigma_k(\Theta):= \{x\in A: \text{mult}_x(\Theta)\geq k\}\) has codimension \(\geq k\) in \(A\). A consequence of the first theorem is that if \(k\geq 2\), then \(\Sigma_k (\Theta)\) contains an irreducible component of codimension \(k\) in \(A\) if and only if \((A, \Theta)\) splits as a \(k\)-fold product of p.p.a.v’s. This generalizes a theorem of Smith and Varley who proved this for \(k=g\).

(2): Kollár conjectured that if \(X\) is of general type, then the holomorphic Euler characteristic \(\chi(X,\omega_X)\) is positive. The third theorem of the paper is that this is true if \(X\) is birationally a subvariety of \(\text{Alb} (X)\). The picture is completed by showing that the conjecture fails in general. The counterexample is a threefold whose Albanese mapping is a branched covering with a rather degenerate branching divisor.

(3): The fourth theorem is that if \(X\) is a smooth projective variety of dimension \(n\) with plurigenera \(P_1(X)\) and \(P_2(X)\) equal to 1, then the Albanese mapping of \(X\) is surjective. This implies a result of Kawamata saying that the Albanese mapping is surjective if the Kodaira dimension of \(X\) is zero. The proof of the theorem is surprisingly simple.

Reviewer: H.Lange (Erlangen)

##### MSC:

14K25 | Theta functions and abelian varieties |

14J99 | Surfaces and higher-dimensional varieties |

32J27 | Compact Kähler manifolds: generalizations, classification |

14E99 | Birational geometry |

##### Keywords:

irregular varieties; theta divisors; generic vanishing theorems; principally polarized abelian variety; holomorphic Euler characteristic; Albanese mapping
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\textit{L. Ein} and \textit{R. Lazarsfeld}, J. Am. Math. Soc. 10, No. 1, 243--258 (1997; Zbl 0901.14028)

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