A Barth-Lefschetz theorem for toric varieties (Q1433422)

The author considers the situation of a Barth-Lefschetz theorem (the comparison of the cohomology of \(\mathbb C\) of a smooth variety \(X^n\) with that of its ambient space \(Y^r\) up to degree \(2n-r+1\)) if \(Y\) is toric rather than just \(\mathbb P^r\). He generally assumes that \(\mathcal N_{X|Y}\) is ample on \(X\) and, moreover, \(X\) intersects all toric strata transversally. The usage of \textit{M.-N. Ishida}'s representation [Tohoku Math. J., II. Ser. 32, 111--146 (1980; Zbl 0454.14021)] of the toric \(\Omega^p_Y\) by a complex involving the structure sheaf on the toric strata leads to a vanishing theorem for the symmetric powers of \(\mathcal N_{X|Y}\) as well as to a comparison of \(H^i(X, \Omega^j_Y|X)\) with \(H^i(X, \Omega^j_X)\). Then, the generalized Euler sequence of \textit{V. V. Batyrev} and \textit{D. A. Cox} [Duke Math. J. 75, 293--338 (1994; Zbl 0851.14021)] is used to determine the Hodge numbers \(h_X^{ij}\) for \(i+j\leq 2n-r\) in combinatorial terms -- only the \(h_X^{ij}\) on the diagonal (\(i=j\)) show up, and these depend on the number of connected components of the intersection of \(X\) with the toric strata of \(Y\), cf.\ Theorem 10. Finally, this leads to an exact characterization of when the Barth-Lefschetz theorem holds -- it is equivalent to a combinatorial condition called ``BL'' (Definition 13) which again involves the intersection behavior of \(X\) with the toric strata.