K-Trivial Varieties - Degenerations, Automorphisms, and Periods

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Algebraic geometry is concerned with the study of algebraic varieties, that is geometric objects defined by polynomial equations. Such objects are ubiquitous in mathematics, and are relevant to a variety of real world applications ranging from cryptography, to computational biology, to models of the universe in physics. Indeed, the Calabi-Yau threefolds, a special class of algebraic varieties, are abstract representations of the shape of the universe in string theory. A very consequential, wide-open question regarding the Calabi-Yau threefolds is the existence of finitely many types of such objects. This proposal is concerned with the study of Calabi-Yau threefolds and of a related, wider class of algebraic varieties, the so-called K-trivial varieties. A number of questions ranging from the above-mentioned finiteness question to more tangible questions will be investigated. This study will involve a number of the PI’s graduate students and postdocs. Some additional research and outreach activities related to the subject are also planned. The study of K-trivial varieties, that is algebraic varieties with trivial canonical class, is a central subject in algebraic geometry. The proposed projects will focus on two main classes of K-trivial varieties: hyper-Kaehler manifolds and Calabi-Yau threefolds. The motivational goals driving this study are the finiteness of deformation types for such objects, and the complementary question of constructing new deformation classes (especially in the hyper-Kaehler case). Intermediate steps towards these challenging objectives include questions regarding the automorphism groups, that is the symmetries of such objects; the deformations and degenerations, that is breaking up the K-trivial varieties into simpler, more manageable pieces; and the fibrations of K-trivial varieties (especially Lagrangian fibrations for hyper-Kaehler manifolds), that is constructing such varieties from lower dimensional objects. A main tool in this investigation is Hodge theory, and the associated period maps.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.