代数几何

In this paper, we categorize all isomorphism classes of quasi-elliptic surfaces over a field $k$ of characteristic 2 or 3. For every quasi-elliptic surface $X$, we classify all possible sequences of blow-downs from $X$ to the projective…

Peter Scholze has raised the question whether some variant of the $q$-de Rham complex is already defined over the Habiro ring $\mathcal H = \lim_{m\in\mathbb N}\mathbb Z[q]_{(q^m-1)}^\wedge$. We show that such a variant exists whenever the…

We develop a calculus based on graph enumeration for $S_n$-equivariant motivic invariants of graphically stratified moduli spaces. We apply our theory to the Deligne--Mumford moduli space $\overline{\mathcal{M}}_{g, n}$ and to the space of…

We study the problem of determining when the blowup $X \to \mathbb{P}^3$ along a smooth space curve $C$ is a Mori Dream Space. We obtain sufficient conditions, as well obstructions to the Mori dreamness of $X$ based on the external geometry…

We construct some version of the trace morphism between the Du Bois complexes, with applications towards the behavior of the local cohomological dimension and some Hodge theoretic aspects of singularities under finite morphisms.

The Chen-Ng\^o Conjecture predicts that the Hitchin morphism from the moduli stack of $G$-Higgs bundles on a smooth projective variety surjects onto the space of spectral data. The conjecture is known to hold for the group $GL_n$ and any…

We observe what the canonical bundle formula gives towards a conjecture of Schnell on algebraic fiber spaces, a question concerning the equivalence between the non-vanishing conjecture and the Campana--Peternell conjecture. As a result, we…

We prove that every simple flop of type $D_5$, i.e., resolved by blowups with exceptional divisor isomorphic to a generalized Grassmann bundle with fiber $OG(4, 10)$, induces a derived equivalence. This provides new evidence for the DK…

We verify part of a conjecture of Campana predicting that rational points on the weakly-special non-special simply-connected smooth projective threefolds constructed by Bogomolov-Tschinkel are not dense. To prove our result, we establish…

We study the interaction between Fourier-Mukai transforms and perverse filtrations for a certain class of dualizable abelian fibrations. Multiplicativity of the perverse filtration and the "Perverse $\supset$ Chern" phenomenon for these…

We give an analytic proof of the Saito vanishing theorem using $L^{2}$-methods, by going back to the original idea for the proof of the Kodaira vanishing theorem.

This work presents a generalization of derived blow-ups and of the derived deformation to the normal bundle from derived algebraic geometry to any geometric context. The latter is our proposed globalization of a derived algebraic context,…

Given a semistable non-isotrivial fibered surface $f:X\to \mathbb{P}^1$ it was conjectured by Tan and Tu that if $X$ is of general type, then $f$ admits at least $7$ singular fibers. In this paper we prove this conjecture in several…

We define the relative stability threshold of a family of Fano varieties over a DVR and show that it is computed by a divisorial valuation. In the case when the special fiber is K-unstable, but the generic fiber is K-semistable, we use the…

The local simple $9$-fold flop of Grassmannian type is a birational transformation between total spaces of vector bundles on the Grassmannians $\mathrm{Gr}(2, 5)$ and $\mathrm{Gr}(3, 5)$. We produce four different derived equivalences which…

We introduce and develop the notion of "unipotent spectra." This is defined to be the stabilization of To\"en's category of affine stacks, and is related to recent work of Mondal--Reinecke. Unipotent spectra give rise to unipotent stable…

We build a purely inseparable Galois theory using non-derived commutative algebra. Our theory works on fields and on normal varieties. It says that a purely inseparable morphism corresponds to a finite (saturated) subalgebra of differential…

We extend the computation of the invariant $\eta(\omega,C,a)$ defined in arXiv:2409.01751 to special points on the line at infinity and show that, as in the affine case, its value is determined purely by the geometry of the integral curve…

In this article, we extend the nonabelian Hodge correspondence in positive characteristic to the nonlinear setting.

This work investigates the vertical quantum cohomology and quantum spectra of flag bundles, uncovering new links between the Gromov-Witten theory of homogeneous fibrations and analytic number theory. Building on previous constructions by…