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Singularities of theta divisors and the birational geometry of irregular varieties. (English) Zbl 0901.14028
The 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.
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
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