Kodaira dimension and zeros of holomorphic one-forms. (English) Zbl 1297.14011

Popa, Mihnea; Schnell, Christian

Ann. Math. (2) 179, No. 3, 1109-1120 (2014).

In this very interesting article, the authors use generic vanishing theorems for Hodge modules on abelian varieties to prove that if \(X\) is a smooth complex projective variety of general type and \(0\neq \omega \in H^0(\Omega ^1_X)\) is a global holomorphic 1-form, then the zero locus of \(\omega\) is non-empty.

Reviewer: Christopher Hacon (Salt Lake City)

MSC:

14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14K05	Algebraic theory of abelian varieties
32C38	Sheaves of differential operators and their modules, \(D\)-modules
32C05	Real-analytic manifolds, real-analytic spaces

Keywords:

abelian variety; generic vanishing theory; Hodge module; holomorphic one-form; Kodaira dimension; zero locus

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