K-Stability, Moduli Spaces, and Singularities

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Varieties are geometric objects as common solutions of multivariable polynomial equations. Varieties can vary by changing the coefficients of polynomials, and it is an important question on when their geometric shapes will remain in a nice way. When the varieties are positively curved called Fano varieties, a natural approach is to look at those with Einstein structures, a notion originated from general relativity. The Einstein structures are deeply related to K-stability, a stability theory in algebraic geometry. The PI will study Fano varieties with Einstein structures, and how they degenerate and develop singularities. The problems are foundational and have deep connections among different areas of mathematics, such as algebraic geometry, differential geometry, commutative algebra, and mathematical physics.The PI will investigate K-stability of Fano varieties, their moduli spaces, and geometry of singularities. The PI aims to show that moduli spaces of K-polystable Fano varieties are compact, which is related to finding a Harder-Narasimhan filtration on K-unstable Fano varieties. The PI will study explicit moduli spaces of log Fano pairs and their wall crossings, and estabilish connections to moduli spaces of curves and K3 surfaces. The PI will study distribution of local volumes of Kawamata log terminal singularities, where he aims to show that 0 is the only accumulation point. The proposed projects above will use classical techniques from the Minimal Model Program and Geometric Invariant Theory, as well as new tools from algebraic theory of K-stability, K-moduli spaces, and normalized volumes developed by the PI and collaborators.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.