Negative $K$-theory and Hodge theory
We study the negative $K$-groups of complex varieties from a mixed Hodge-theoretic perspective, making use of the theory of higher singularities, Chow groups, and the Minimal Model Program.
math.AG↗
arXiv subjects
Andrew Burke
Publications and source records attributed to Andrew Burke.
Local Cohomological Defect and a Conjecture of Mustata-Popa
We prove a general result on the depth of Du Bois complexes of a singular variety. We apply it to prove a conjecture of Mustata-Popa and to study the local cohomological defect, extending results of Ogus and Dao-Takagi over the complex numbers.
math.AG↗
Involution on the Graded Grothendieck Ring of Varieties and $\mathbb{D}$-Singularities
We realize a graded variant $K_0(Var_k^{dim})$ of the Grothendieck ring of varieties as a quadratic extension of the subring $K_0(Var_k^{sp})$ spanned by classes of smooth and proper varieties. As such, there exists a natural involution $\mathbb{D}$ on $K_0(Var_k^{dim})$. We show that $\mathbb{D}$ commutes with the symmetric power operations $Sym^m$ up to zero divisors. Moreover, we study varieties which are smooth up to cut-and-paste relations, which we call $\mathbb{D}$-singular varieties, and we give applications to compactifications of varieties and the irrationality of Kapranov zeta functions.
math.AG↗
Geometric Manin's Conjecture for Fano 3-Folds
We classify families of free rational curves on all smooth Fano threefolds over the complex numbers. In particular, we prove the family of very free rational curves representing any fixed numerical curve class is either irreducible or empty. This proves Geometric Manin's Conjecture in dimension three. For general Fano threefolds of each deformation type, our results allow us to explicitly count the number of components of the moduli space of irreducible, geometrically rational curves, which may not be free, representing any numerical class.
math.AG↗