Mirror Symmetry, Birational Geometry, and Moduli.

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

String theory posits that interactions of fundamental particles at small scales are explained by hidden dimensions of our universe which are wrapped up to form a tiny geometric space called a Calabi--Yau manifold at each point. The mirror symmetry phenomenon asserts that Calabi--Yau manifolds come in mirror pairs X and Y which determine the same physics, implying surprising relations between geometric properties of X and Y. Birational geometry is the study of surgeries of spaces obtained by cutting out a subspace of lower dimension and gluing another in its place. The moduli space of a space Y parametrizes all possible spaces obtained by deforming Y. If X and Y are a mirror pair, then the birational geometry of X determines the structure of the moduli space of Y near a certain limit point. Based on this heuristic, Morrison conjectured that a Calabi--Yau manifold X admits only finitely many possible surgeries up to symmetries of X. The PI aims to show the conjecture is false in general, but a weaker statement sufficient for applications to moduli holds, as suggested by recent work of the PI with graduate students on unbounded Calabi--Yau manifolds. The PI will also study positively curved spaces called Fano manifolds and singularities that arise at limit points of the moduli space of Calabi--Yau manifolds via mirror symmetry. These projects will be pursued together with graduate students supported by the grant. The PI will also organize seminars and a conference focused on training of graduate students. The PI will study Morrison's cone conjecture and applications to birational geometry and moduli, joint with a collaborator. Morrison conjectured that the automorphism group of a Calabi--Yau manifold acts on its nef cone with rational polyhedral fundamental domain, so that a neighborhood of a cusp of the moduli space of the mirror manifold admits a compactification determined by this data via a construction of Looijenga. Recent work of the PI with graduate students on log Calabi--Yau manifolds and mirror deformations of singularities suggests that the conjecture does not hold in general, but a weaker version sufficient for applications to moduli should hold. The PI will study mirror symmetry for Q-Fano 3-folds and applications to classification and non-arithmetic curves on moduli of K3 surfaces, joint with a graduate student. Q-Fano 3-folds arise as end products of the minimal model program and so are basic to our understanding of 3-folds. Mirror symmetry heuristics suggest that the mirror of a Q-Fano 3-fold is a K3 fibration over the affine line with monodromy at infinity that is maximally unipotent after a finite base change. The mirror corresponds to a rigid rational curve on a moduli space of polarized K3 surfaces. Computations suggest that these curves are not Shimura curves but are uniformized by non-arithmetic groups. The PI will study mirror symmetry for Milnor fibers of surface singularities and applications to symplectomorphism groups and moduli of surfaces, a project that joint with others. 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.