Some Effective Problems in Algebraic Geometry and Their Applications

Dutta, Yajnaseni

doi:https://doi.org/10.21985/n2-zhvm-n157

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Some Effective Problems in Algebraic Geometry and Their Applications

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In this thesis, we study pushforwards of canonical and log-pluricanonical bundles on projective log canonical pairs over the complex numbers. We partially answer a Fujita-type conjecture proposed by Popa and Schnell in the log canonical setting. Built on Kawamata’s result for morphisms that are smooth outside a simple normal crossing divisor, we show a global generation result for morphisms that are log-smooth with respect to a reduced snc pair outside such divisors. Furthermore, we partially generalize this result to arbitrary log canonical pairs and obtain generic effective global generation. In the pluricanonical setting, we show two different effective statements. First, when the morphism surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when the morphism is fibred over a smooth projective variety, we give a linear bound for twists by ample line bundles. In each context we give descriptions of the loci over which these global generations hold. These results in particular give effective nonvanishing statements. As an application, we prove an effective weak positivity statement for log-pluricanonical bundles in this setting, with a description of the loci where this positivity is valid. We discuss its most remarkable application by presenting a proof of a case of the Iitaka conjecture for subadditivity of log Kodaira dimensions. Finally, using the description of the positivity loci, we show an effective vanishing theorem for pushforwards of certain log-pluricanonical bundles under smooth morphisms.

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