Tannakian categories of perverse sheaves on abelian varieties (Q2844799)

Perverse sheaves on a variety \(X/\mathbb{C}\) are bounded constructible sheaf complexes \(K\) on \(X\) such that its cohomology sheaves \(\mathcal{H}^{-i}(K)\) and those of its Verdier dual \(D(K)\) satisfy \(\dim(\text{supp}(\mathcal{H}^{-i}(K))) \leq i\) and \(\dim(\text{supp}(\mathcal{H}^{-i}(D(K)))) \leq i\) for all integers \(i\) see [\textit{A. A. Beilinson} et al., Astérisque 100, 172 p. (1982; Zbl 0536.14011)]. They form a full abelian subcategory of the derived category of bounded constructible sheaf complexes \(D_c^b(X, \mathbb{C})\) (or over some other base ring instead of \(\mathbb{C}\)). If the variety \(X\) is an abelian or semiabelian variety, the group law \(a:X \times X \to X\) induces a convolution product on \(D_c^b(X, \mathbb{C})\) which turns \(D_c^b(X, \mathbb{C})\) into a rigid symmetric monoidal category (see [\textit{R. Weissauer}, in: Modular forms on Schiermonnikoog. Based on the conference on modular forms, Schiermonnikoog, Netherlands, October 2006. Cambridge: Cambridge University Press. 267--274 (2008; Zbl 1160.14034)]). By the intrinsic characterisation of Tannakian categories by \textit{P. Deligne} [Prog. Math. 87, 111--195 (1990; Zbl 0727.14010)], the lack of being a Tannakian category relies only on the fact that the ranks of the objects (in this case the Euler characteristics) are not always natural numbers but only integers. For perverse sheaves the ranks are non-negative but the category of perverse sheaves is not stable under the convolution product. However, there is a quotient (so called André-Kahn quotient) of the tensor category generated by all degree shifts of semisimple perverse sheaves such that the (images of) semisimple perverse sheaves are stable under the convolution product. Hence these form a Tannakian category. In particular, to every semisimple perverse sheaf \(P\) there is attached a linear algebraic group, the Tannaka group of the Tannakian subcategory generated by \(P\).NEWLINENEWLINEAll this is explained in the first two chapters of the thesis under review in detail accessible also to non-experts. The author continues with using this Tannakian framework to give a new proof for a vanishing theorem for perverse sheaves on abelian varieties which he developed in joint work with \textit{R. Weissauer} [J. Algebr. Geom. 24, No. 3, 531--568 (2015; Zbl 1338.14023)]. This is generalised to semiabelian varieties in the third chapter. In the following chapters, the author studies in detail the case where the perverse sheaf is the intersection cohomology sheaf \(\delta_{\theta}\) associated to the theta divisor \(\theta\) of a principally polarized abelian variety. In particular, the Tannaka group of \(\delta_{\theta}\) is computed.NEWLINENEWLINEMost parts of the thesis are given in broader generality, and even in positive characteristic. For example, the vanishing theorem is also valid in positive characteristic if the category mentioned above is also Tannakian in that case. As the intrinsic characterisation of P. Deligne only works in characteristic zero, this is only conjectured in the thesis.