Some Applications of Perverse Sheaves and Hodge Theory to Birational Geometry

Olano, Sebastian

doi:https://doi.org/10.21985/n2-8zgs-1s14

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Some Applications of Perverse Sheaves and Hodge Theory to Birational Geometry

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In this thesis, we study applications of the theory of perverse sheaves and their enhancements to problems in birational geometry. In the first application, we give positive results towards a conjecture of Batyrev about the nonnegativity of stringy Hodge numbers. In particular, we prove the nonnegativity of $(p,1)$-stringy Hodge numbers using methods of mixed Hodge theory and a result about the extension of holomorphic forms along divisors from a resolution of Gorenstein canonical singularities. Based on Kawamata's result on the existence of $\QQ$-factorializations of threefolds with isolated singularities, we obtain new results about the stringy Hodge diamond of threefolds. We study the nonnegativity of stringy Hodge numbers for fourfolds using the Decomposition Theorem and several results about its interaction with the Hodge filtration due to de Cataldo and Migliorini. Using this, we prove the nonnegativity of stringy Hodge numbers for a class of fourfolds. In the second application, we extend some of the work of Musta\c{t}\u{a} and Popa on the theory of Hodge ideals. This theory is based on an extension of the category of perverse sheaves called mixed Hodge modules, introduced by Saito. The sheaf of functions with poles along a divisor underlies a mixed Hodge module. In particular, this sheaf is endowed with two filtrations, called the Hodge and weight filtrations. Hodge ideals arise from the study of the Hodge filtration of this sheaf. Studying together the Hodge and weight filtrations, it is natural to extend the theory to introduce weighted Hodge ideals. We study the local and global properties of the first sequence of these ideals called weighted multiplier ideals. Using this, we give a Hodge theoretic interpretation of the adjoint ideal of a divisor. Moreover, using this combination of local and global results, we prove results about the geometry of certain isolated singularities in projective spaces.

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