代数几何

We establish a stronger Bogomolov--Gieseker type inequality for slope-semistable sheaves on the smooth quintic threefold. Our approach combines a refined restriction theorem for tilt-stable objects with explicit Clifford-type bounds for…

We use categorification of monoid actions to study algebraic geometry over symmetric monoidal categories. This brings together the relative algebraic geometry over symmetric monoidal categories developed by To\"{e}n and Vaqui\'{e}, along…

We define a derived enhancement of the hyperquot scheme (also known as nested Quot scheme), which classically parametrises flags of quotients of a perfect coherent sheaf on a projective scheme. We prove it is representable by a derived…

We develop the theory of "virtual morphisms" in logarithmic algebraic geometry, introduced by Howell. It allows one to give algebro-geometric meaning to various useful maps of topological spaces that do not correspond to morphisms of (log)…

We show that the minimal volume of surfaces of log general type, with non-empty non-klt locus on the ample model, is $\frac{1}{825}$. Furthermore, the ample model $V$ achieving the minimal volume is determined uniquely up to isomorphism.…

We partially resolve conjectures of Deligne and Simpson concerning $\mathbb{Z}$-local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with $\mathbb{Q}$-anisotropic monodromy,…

We introduce the concept of non-Archimedean metrics attached to a transcendental pseudoeffective cohomology class on a compact K\"ahler manifold. This is obtained via extending the Ross-Witt Nystr\"om correspondence to the relative case,…

We present an approach to detecting Zariski pairs in conic line arrangements. Our method introduces a combinatorial condition that reformulates the tubular neighborhood homeomorphism criterion arising in the definition of Zariski pairs.…

Let $k$ be an algebraic field extension of $\mathbb{Q}$ and let $X$ be a smooth projective variety over $k$ of dimension $d \geq 2$. We study the pro-representability of the Chow group $CH^{p}(X)$ with $2 \leq p \leq d$. When certain Hodge…

We study moduli space of higher rank marginally stable pairs (E,s:= (s_1,..., s_r)) consisting of torsion free coherent sheaf E of rank r and r sections (s_1,..., s_r) on a smooth projective surface. Having fixed the Chern character of E,…

Saito theory associates to a quasihomogeneous isolated singularity the structure of a Dubrovin--Frobenius manifold. This structure is not unique, depending on the special choice of a primitive form or, equivalently, a good basis. We study…

We study the K-stability of $\mathbb{Q}$-Fano spherical varieties using compatible divisors. More precisely, if the $\mathbb{Q}$-Fano variety, with a reductive group action, has an open Borel subgroup orbit, then there is a unique…

Miyaoka proved that a smooth surface of degree $d$ in ${\mathbf{P}}^3({\mathbb{C}})$ contains at most $2d(d-2)$ pairwise disjoint lines. In this note, we verify that the Maschke octic contains $96$ pairwise disjoint lines, thereby proving…

We use the cohomological interpretation of anti-holomorphic derivatives of the isomonodromic deformation of a Higgs bundle, as established in our previous work \cite{HSZ}, to provide a short new proof of the non-existence of holomorphic…

Algebraic geometry codes on the Hermitian curve have been the subject of several papers, since they happen to have good performances and large automorphism groups. Here, those arising from the Singer cycle of the Hermitian curve are…

The results of A.Chiodo, Y.Ruan and M.Krawitz associate the mirror partner Calabi-Yau variety $X$ to a Landau--Ginzburg orbifold $(f,G)$ if $f$ is an invertible polynomial satisfying Calabi-Yau condition and the group $G$ is a diagonal…

We study semi-stable degenerations of quasi-Fano varieties to unions of two pieces. We conjecture that the higher rank Landau-Ginzburg models mirror to these two pieces can be glued together to lower rank Landau-Ginzburg models which are…

We prove an orbifold type decomposition theorem for the Hochschild homology of the symmetric powers of a small DG category $\mathcal{A}$. In noncommutative geometry, these can be viewed as the noncommutative symmetric quotient stacks of…

In this article, we introduce a new class of smooth partially proper rigid analytic varieties over a $p$-adic field that satisfy Poincar\'e duality for \'etale cohomology with mod $p$-coefficients : the varieties satisfying "primitive…

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a K\"ahler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form…