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Hard Lefschetz Property for Isometric Flows
The hard Lefschetz property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian...
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Waring problems and the Lefschetz properties
We study three variations of the Waring problem for homogeneous polynomials, concerning the Waring rank, the border rank and the cactus rank of a...
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Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection...
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The Lefschetz standard conjectures for IHSMs of generalized Kummer deformation type in certain degrees
For a projective 2 n -dimensional irreducible holomorphic symplectic manifold Y of generalized Kummer deformation type and j the smallest prime number...
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Proof of the Hard Lefschetz Theorem
In this chapter, we outline the proof of Soergel’s conjecture via versions of the hard Lefschetz theorem and Hodge–Riemann bilinear relations.
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Remarks on Some Compact Symplectic Solvmanifolds
We study the hard Lefschetz property on compact symplectic solvmanifolds, i.e., compact quotients M = Γ G of a simply-connected solvable Lie group G ...
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A note on the Hard Lefschetz property of symplectic structures
In (Trans Am Math Soc 368(11), 8223–8248, 2016), Cho has constructed the first known example of a deformation equivalence between a Kähler form and a...
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Asymptotically holomorphic theory for symplectic orbifolds
We extend Donaldson’s asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a compact symplectic orbifold such that...
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Hodge Theory and Lefschetz Linear Algebra
The cohomology rings of smooth complex projective algebraic varieties satisfy a “package” of properties: Poincaré duality, weak Lefschetz, hard...
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The Lefschetz Theorem for Hyperplane Sections
In these notes we consider different theorems of the Lefschetz type. We start with the classical Lefschetz Theorem for hyperplane sections on a...
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A proof of the Donaldson–Thomas crepant resolution conjecture
We prove the crepant resolution conjecture for Donaldson–Thomas invariants of hard Lefschetz 3-Calabi–Yau (CY3) orbifolds, formulated by...
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Twisted basic Dolbeault cohomology on transverse Kähler foliations
In this paper, we study the twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem on a transverse Kähler foliation. And we give...
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Topological and geometric aspects of almost Kähler manifolds via harmonic theory
The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of...
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Known Cases of Global Langlands Functoriality
Over number fields the global Langlands functoriality conjecture is wide open. Despite this, several important cases have been established. We survey...