# On the generalized lower bound conjecture for polytopes and spheres

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The Hard Lefschetz Theorem (HLT) and the Hodgeâ€“Riemann bilinear relations (HRR) hold in various contexts: they impose restrictions on the cohomology algebra of a smooth compact KÃ¤hler manifold; they restrict the local monodromy of a polarized variation of Hodge structure; they impose...

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We extend our previous work [8] on coherentâ€“constructible correspondence for toric varieties to toric Deligneâ€“Mumford (DM) stacks. Following Borisov et al. [3], a toric DM stack is described by a â€œstacky fanâ€ Î£=(N,Î£,Î²), where N is a finitely generated abelian...

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We show that every 4-representative graph embedding in the double torus contains a noncontractible cycle that separates the surface into two pieces. As a special case, every triangulation of the double torus in which every noncontractible cycle has length at least 4 has a noncontractible cycle...