代数几何

This expository article elaborates upon my talk at the 2025 AMS Summer Institute on Algebraic Geometry. It gives an introduction to a conjecture from Tate's 1966 S\'eminaire Bourbaki report, predicting the existence of a symplectic form on…

In this paper, by using the generalized Borel theorems in $\mathbb{CP}^2$, we show the hyperbolicity of Fermat type curves and their complement in $\mathbb{CP}^2$. This improves Noguchi-Shirosaki's and Demailly-El Goul's degree bounds.

Let $X$ be a mildly singular Fano variety such that the tangent sheaf is a direct sum. We show that the direct factors are algebraically integrable, so the infinitesimal decomposition induces a product structure on a quasi-\'etale cover of…

We associate to every analytic surface singularity $(V,0)$ in $(\mathbb C^3,0)$, not necessarily isolated, an invariant $mult^* (V)$ and show that an analytic family of such singularities $(V_t,0)$, $t\in (\mathbb C^l,0)$, is generically…

We study moduli spaces of Higgs bundles with two poles on an elliptic curve. We describe all singular fibers of the Hitchin map, including the nilpotent cone. To achieve this, we consider a modular map that lifts Higgs bundles with five…

We study tensor network varieties associated with the triangular graph, with a focus on the case where one of the physical dimensions is 2. This allows us to interpret the tensors as pencils of matrices. We provide a complete…

We give an overview of the fundamental definitions and results concerning hypersurface singularities, defined by convergent power series over an arbitrary real valued field. This approach combines, on the one hand, the classical case of…

Let $\mathrm G$ be an isotrivial reductive group over a scheme $S$. We construct a smooth projective $S$-scheme containing $\mathrm G$ as a fiberwise-dense open subscheme equipped with left and right actions of $\mathrm G$ which extend the…

A main conjecture in the field of Positive Geometry states that amplituhedra, which are certain semi-algebraic sets in the Grassmannian, are positive geometries. It is motivated by examples showing that the canonical forms of certain…

Here we give an explicit construction of a globally defined strictly plurisubharmonic function on projectivized strata of strictly meromorphic differentials with prescribed orders of zeros and poles. In particular, this yields a…

The aim of this paper is to review how some approximation results in commutative algebra are being used to construct equisingular deformations of singularities. The first example of such an approximation result appeared for the first time…

For an endofunctor $F\colon\mathcal{C}\to\mathcal{C}$ on an ($\infty$-)category $\mathcal{C}$ we define the $\infty$-category $\operatorname{Cart}(\mathcal{C},F)$ of generalized Cartier modules as the lax equalizer of $F$ and the identity.…

Let $G$ be a generically reductive group over a smooth projective curve $X$ over a finite field. For any finite set $I$, we show that nearby cycles commute with proper direct image from stacks of shtukas to $X^I$. This generalizes some…

The signature of a membrane is a sequence of tensors whose entries are iterated integrals. We study algebraic properties of membrane signatures, with a focus on signature matrices of polynomial and piecewise bilinear membranes. Generalizing…

For connected reductive groups together with a Frobenius root $F$, we show that the cohomology of the structure sheaf and respectively the canonical sheaf for compactified Deligne--Lusztig varieties associated to an element in the free…

Let a reductive group $G$ act on a smooth variety $X$ such that a good quotient $X{/\!\!/}G$ exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of $X{/\!\!/} G$, obtained from a $G$-equivariant vector…

This paper studies a flat degeneration P_n of the classical coinvariant algebra R_n, a bigraded Artinian Gorenstein algebra that arises from the coordinate ring of the Segre embedding of the n-fold self-product of the projective line. The…

This paper undertakes the study of the topology of T-manifolds of arbitrary codimension obtained by combinatorial patchworking with real phase structure as described by Brugall\'e, L\'opez de Medrano and Rau (2024). We prove new bounds on…

We study sections of the relative Picard bundle of a family of curves of genus $g \geq 2$ through the rank of the associated normal function. Using Griffiths' formula for the infinitesimal invariant and higher Schiffer variations, we…

We introduce the notion of good pair of generalized nef partitions to describe Calabi-Yau complete intersections in Q-Fano toric varieties whose equations do not necessarily have maximal Newton polytopes. Moreover, we define a natural…