Related papers: Categorical Torelli theorem for hypersurfaces

We use the "closed point sieve" to prove a variant of a Bertini theorem over finite fields. Specifically, given a smooth quasi-projective subscheme X of P^n of dimension m over F_q, and a closed subscheme Z in P^n such that Z intersect X is…

We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$ is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate…

We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of…

We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective toric DM stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton…

We prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from $2$ are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof…

Let $X$ be a projective variety with an isolated $A_2$ singularity. We study its bounded derived category and prove that there exists a crepant categorical resolution $\pi_*\colon \widetilde{\mathcal{D}} \to D^b(X)$, which is a Verdier…

We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $\alpha\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a…

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism…

Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We the propose the definition of a categorical resolution of singularities. Our main example is the derived category $D(X)$ of quasi-coherent…

General hyperplane sections of a Fano threefold $Y$ of index 2 and Picard rank 1 are del Pezzo surfaces, and their Picard group is related to a root system. To the corresponding roots, we associate objects in the Kuznetsov component of $Y$…

We show that the general Enriques surface can be recovered from the Kuznetsov component of its bounded derived category of coherent sheaves.

We provide descriptions of the derived categories of degree $d$ hypersurface fibrations which generalize a result of Kuznetsov for quadric fibrations and give a relative version of a well-known theorem of Orlov. Using a local generator and…

We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary.…

In this paper, we show the fundamental theorems for rotationally symmetric hypersurfaces, and thus, together with the earlier results in [3] and [4], provide a complete classification of umbilic hypersurfaces in the Heisenberg groups…

Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear…

Many quantum invariants of knots and 3-manifolds (e.g. Jones polynomials) are special cases of the Witten-Reshetikhin-Turaev 3D TQFT. The latter is in turn a part of a larger theory - the Crane-Yetter 4D TQFT. In this work, we compute the…

We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties.…

We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics revealing its relation to projective duality. As an application we describe the derived categories of…

Let $X\subset \P^5$ be a smooth cubic fourfold. A well known conjecture asserts that $X$ is rational if and only if there an Hodge theoretically associated K3 surface $S$. The surface $S$ can be associated to $X$ in two other different…

In this short note we prove a version of Bertini's theorem for unipotent rigid fundamental groups, stating that for every smooth, projective, geometrically connected variety $X$ over an infinite perfect field $k$ of characteristic $p>0$,…