Let $X$ be a smooth, irreducible, projective algebraic surface, and let $α\in \mathbb{Q}[m]_{>0}$ be a polynomial. In this paper, we determine topological and geometric properties of the...

We generalize to dimensions $n\ge3$ the compactified moduli stack of elliptic curves $\overline{M}_{1,1}=\mathbb{P}(4,6)$ and Brieskorn's family of $U\oplus E_8$-polarized K3 surfaces over a...

Donaldson showed that the constant scalar curvature Kähler metrics can be quantized by the balanced Hermitian norms on the spaces of global sections. We explore an analogous problem in the...

We prove the multiplicative version of the dimensional reduction theorem in cohomological Donaldson--Thomas theory. More precisely, we show that the BPS cohomology associated with the loop stack...

This paper studies the Cohomological Donaldson-Thomas theory of loop stacks of $0$-shifted symplectic stacks. In particular, we compare $(-1)$-shifted tangent stacks of these moduli problems,...

We show that the optimal Jordan constant for the volume-preserving plane Cremona group $\mathrm{Bir}(\mathbb P^2, Δ)$ is 12. We provide a Jordan constant of $60$ for the three-dimensional...

We study real linear spaces in projective space that avoid the real points of a non-degenerate projective variety. For a variety $X \subset \mathbb{P}^{n-1}$ with a real smooth point, we define...

It is known that the normalization of a quasi-ordinary complex singularity is a Hirzebruch-Jung, see [Gon00; Pop04; AS05]. We extend this result to Puiseux hypersurfaces. Moreover, we prove that...

The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus $g \geq 2$ over a discretely valued field admits a unique stable model after a finite...

Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of...

We use derived methods to study the Gauss-Manin connection in Hochschild homology, infinitesimal cohomology, and derived de Rham cohomology. As applications, we give new approaches to...

Let $\mathbf{B}$ be the ring of analytic functions on the Fargues-Fontaine curve $Y_{\rm FF}$. We show that adding $p$-adic analogs of $\log p$ and $\log 2πi$ kills its Galois cohomology in...

We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth...

We study the quasi-Albanese morphisms for log canonical Calabi-Yau pairs.