Abstract: The goal of this article is three-fold; first, we establish an interesting link between some sets related to cohomologies of complex local systems and the stratification of the Albanese morphism of smooth irregular complex projective varieties. This gives a different interpretation of a result due to Green and Lazarsfeld. This observation originated out of our interest in studying zeros of holomorphic 1-forms. Secondly, we show that the set of 1-forms that admit codimension 1 zeros are in fact a linear subvariety of $H^0(X,\Omega_X^1)$. We ask the same question for all codimensions, in particular for the set of all 1-forms that admit zeros. Thirdly, in a somewhat different direction, we deduce some geometric consequences. The key ingredient in the third part is a result of Kashiwara which relates the singular support produced by Albanese morphism to 1-forms that admit zeros. We show that when a smooth projective variety with simple Albanese admits a $\mathscr{C}^{\infty}$-fibre bundle structure over the circle, its Albanese morphism is cohomologically trivial, i.e., the decomposition theorem behaves like that of a smooth morphism. Such a $\mathscr{C}^{\infty}$-fibre bundle structure is conjecturally equivalent to the existence of holomorphic 1-forms without zeros.
Publication Year: 2021
Publication Date: 2021-04-14
Language: en
Type: preprint
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