## Dan Coman (Syracuse University): *Restricted spaces of holomorphic sections vanishing along subvarieties*

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.