On Thursday, March 14, at 4 p.m., we invite you to the following guest lecture presented within the Geometry group:

Restricted spaces with holomorphic sections that vanish along subvarieties,

supported by Dan Coman of Syracuse University.

The lecture will be held live using Microsoft Teams (access link).

Abstract: Let L be a holomorphic line bundle on a compact normal complex space X of dimension n , let Σ = ( Σ ₁ ,…, Σ l ) be an l -tuple of distinct irreducible proper analytic subsets of X , and τ = ( τ ₁ ,…, τ l ) be an l -tuple of positive real numbers. We consider the space H ₀ ⁰ ( X , L p ) of global holomorphic sections of L p := L ^⊗ p that vanish to order at least τ j p along Σ j , 1 ≤ j ≤ l , and give necessary and sufficient conditions to ensure that dim H ₀ ⁰ ( X , L p ) ∼ p n . If Y ⊂ X is an irreducible analytic subset of dimension m , we also consider the space H ₀ ⁰ ( X | Y , L p ) of holomorphic sections of L p | Y that extend to global holomorphic sections in H ₀ ⁰ ( X , L p ) , and we give a general condition on Y to ensure that dim H ₀ ⁰ ( X | Y , L p ) ∼ p m . When L is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces H ₀ ⁰ ( X | Y , L p ) converge to a certain equilibrium current on Y , and we apply this to the study of the equidistribution of zeros in Y of random holomorphic sections in H ₀ ⁰ ( X | Y , L p ) as p → ∞. This is joint work with George Marinescu and Viêt-Anh Nguyên.