TOPICS IN ALGEBRAIC GEOMETRY
TH 2:00-3:20 AM - LCB 222
Office Hours: By appointment
After reviewing vanishing theorems and singularities of pairs, the course will focus on properties and applications
of multiplier ideals and log canonical thresholds. The general aim is to eventually
explore the normalized volume function on the space of real valuation introduced in recent work of Chi Li
and, time permitting, its connection to K-stability.
Specific topics (subject to change) are:
Vanishing theorems: Kodaira vanishing, Kawamata-Viehweg vanishing, Grauert-Riemenschneider and Fujita vanishings,
Singularities of pairs: Pairs and log discrepancies, Shokurov-Kollar connectedness theorem.
Log canonical thresholds: Definition, examples, basic properties, multiplicity bounds, semicontinuity,
m-adic semicontinuity, ACC on smooth varieties.
Multiplier ideals: Definition, examples, first properties, Nadel vanishing theorem, asymptotic multiplier ideals,
adjoint ideals and the restriction theorem, subadditivity formula, uniform approximation of valuation ideals,
ampleness via asymptotic Serre vanishing.
Valuations: Definition and examples, the space of real valuations, log discrepancy function, multiplier ideals
revisited, normalized volume function, normalized multiplicity function, existence of minimizers on klt varieties
with isolated singularities.
K-stability: Motivations, test configurations, Donaldson-Futaki invariant, Ding invariant, bound of the volume of
K-stable Fano varieties, algebraic proof of K-stability of projective spaces, connection to the normalized volume function.
Prerequisites: Students are expected to be familiar with the basics of algebraic geometry
(Hartshorne + Chapter 1 of Lazarsfeld's Positivity book).