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In this paper we show some Lefschetz-type theorems for the effective cone ofHyperk\"ahler varieties. In particular we are able to show that the inclusionof any smooth ample divisor induces an isomorphism of effective cones. Moreoverwe deduce a similar statement for some effective exceptional divisors, whichyields the computation of the effective cone of e.g. projectivized cotangentbundles and some projectivized Lazarsfeld--Mukai bundles.

Lefschetz-type theorems for the effective cone on Hyperk\"ahler varieties