Hodge Filtration, Singularities, and Complex Birational Geometry

    Project: Research project

    Project Details

    Description

    Overview. The PI proposes to continue applying the theory of mixed Hodge modules to
    concrete problems in complex and birational geometry. Continuing his long-term project with
    M. Mustata, he intends to pursue the development of the theory of Hodge ideals, as well as
    of the Hodge fi�ltration on the local cohomology, both associated to arbitrary subschemes of
    smooth varieties. This requires signi�ficant new ideas compared to the study of Hodge ideals
    associated to Q-divisors, which has recently been completed. Once this program is achieved,
    the Hodge fi�ltration on local cohomology will provide an enhancement of the theory of multiplier
    ideals in its full generality. Consequently, he intends to provide applications that reflect this. In
    their work the PI and Mustata have already obtained applications regarding the singularities of
    theta divisors, hypersurfaces in projective space, or minimal exponents. In addition to further
    consequences along these lines, they are planning to use the proposed extensions in order to
    study, for instance, effective bounds for linear series, or roots of the Bernstein-Sato polynomial.
    The PI has also been involved in applying the theory of Hodge modules towards the study of the
    variation of families smooth projective varieties of varieties, e.g. Brody hyperbolicity or Viehweg-
    type questions for parameter spaces. He would like to extend this study to families of singular
    varieties, especially those that appear in the theory of moduli of higher dimensional varieties
    according to Kollar and others, perhaps using those Hodge modules that extend variations of
    mixed Hodge structure. The PI would also like to continue working towards the classi�fication
    of subvarieties with minimal cohomology class on principally polarized abelian varieties, and its
    link with generic vanishing subschemes and with the singularities of theta divisors.

    Intellectual merit. Some of the problems the PI proposes to attack, like the study of roots of
    Bernstein-Sato polynomials, effective bounds in birational geometry, hyperbolicity of parameter
    spaces of varieties, subvarieties of minimal class in abelian varieties, or singularities of theta
    divisors, are among the most prominent problems in their respective areas and will have a high
    impact as proved statements. Others, especially the development of the theory of Hodge ideals
    or of the Hodge �filtration on local cohomology are part of newly emerging directions of research,
    where a deeper understanding is guaranteed to lead to more applications. All parts of the project
    will have a broad range of applications, further our knowledge in the �field, lead to interaction
    with experts in other areas, and produce problems suitable for students. The work of my Ph.D.
    students already contains results directly related to the problems and techniques proposed here.

    Broader impacts. The PI will continue to contribute towards improving the environment
    for women in mathematics through his mentorship at both the post-graduate and graduate
    level (and hopefully undergraduate as well), as a follow-up to his participation in the GROW
    program at Northwestern. In the international mathematical community, he has been and will
    continue to be involved in organizing conferences and workshops, editing expository volumes, and
    continuing work on editorial (like the Journal of Algebraic Geometry, and Algebraic Geometry
    until this year) and scientifi�c (Superior Normal School, Bucharest) boards. The PI will continue
    to deliver lectures at summer schools and conferences in the U
    StatusActive
    Effective start/end date7/1/206/30/23

    Funding

    • National Science Foundation (DMS-2000610)

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