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Algebraic non-hyperbolicity of hyperkahler manifolds with Picard rank greater than one
Kamenova L., Verbitsky M.
A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperk¨ahler manifolds are non algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperk¨ahler manifold is infinite then it is algebraically non-hyperbolic.
Keywords: hyperkahler manifoldsгиперкэлеровы многообразияalgebraic non-hyperbolicityалгебраичная негиперболичность
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Added: May 1, 2026
Ovcharenko M., / Series arXiv "math". 2026.
We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in n⩽4 variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, ...
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Added: April 18, 2026
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Added: April 3, 2026
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Added: April 2, 2026
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Added: February 13, 2026
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Added: December 25, 2025
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