Collaborative Research: New Birational Invariants

Algebraic varieties are shapes defined by solution sets of systems of polynomial equations. They appear naturally in different fields of science and engineering, including physics, cryptography, control theory, robotics, computer vision, etc,. A fundamental problem in geometry is the classification of algebraic varieties, as it helps us gain a better understanding of the structures and relations between them. The first step in classification is called birational classification, i.e. two algebraic varieties are called birational if they are equal outside some lower-dimensional loci. In this proposal, the PIs will investigate new birational invariants. These invariants will shed new light on the birational classification problem. The Principal Investigators will bring new ideas from differential equations, category theory, mirror symmetry and conformal field theory for achieving this goal. This project will provide research training opportunities for graduate students and early-career researchers. More concretely, the Principal Investigators will develop an extended theory of variations of non-commutative Hodge structures. It will be based on a new singularity theory of Landau-Ginzburg models and a non-commutative refinement of the notion of spectrum of quantum multiplication operators. These new non-commutative spectra will provide natural obstructions to rationality and equivariant rationality of Fano varieties. Additionally the PIs will investigate the connection between non-commutative spectra and R-charges of conformal field theories. This will lead to even stronger birational invariants, as well as to new unexpected bridges between geometry and other branches of mathematics, including: a new connection between Steenbrink spectra and the spectra of conformal weights in vertex operator algebras; a connection between topological invariants of 3-manifolds and non-commutative spectra of complex surfaces; semicontinuity of non-commutative spectra of algebraic varieties and the RG-flows on sigma-models with targets on such varieties; and a relation between the Kaehler-Ricci flow and the RG-flow. 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.