Hodge type subvarieties of compact hermitian symmetric spaces (Q1910198)

Let \(X\) be an irreducible compact hermitian symmetric space embedded minimally and equivariantly into \(\mathbb{P}^n\). Consider a subvariety \(T \subset X\) defined by \(f_1, \dots, f_s\) where \(f_i \in H^0 (X,O (d_i))\), \(d_1 \geq \cdots \geq d_s\). The Hodge type of \(U : = X \backslash T\) is defined as the largest \(a\) for which the Hodge-Deligne filtration \(F^\bullet\) on the de Rham cohomology of \(U\) with compact supports satisfies \(F^a H^i_c (U) = H^i_c (U)\) \(\forall i\). For \(X = \mathbb{P}^n\) and the equations defined over \(\mathbb{F}_q\). \textit{H. Esnault}, \textit{M. V. Nori} and \textit{V. Srinivas} [Math. Ann. 293, No. 1, 1-6 (1992; Zbl 0784.14003)] estimated the Hodge type of \(U\) from below. The author gets a similar result for a general \(X = G/P\) (where \(G\) is a simple group and \(P\) a maximal parabolic proper subgroup) depending on the type of \(G\) and on the form of \(f_i\). He uses the ideas of the previous authors.