Let $X$ be a smooth Fano variety and $H$ an ample line bundle with $-K_X=rH$ and $r\le\dim X-2$. We prove that the appropriately defined Kuznetsov component of $X$ with respect to $H$, together with its rotation functor, determines $(X,H)$ when $H$ is very ample, and more generally when $H$ is globally generated and th...
Let $X\subset\PP^{M-1}$ be a smooth hypersurface of degree $d<M$, and let $\Ku(X)$ be its Kuznetsov component, the orthogonal complement of the line bundles $\OO_X,\dots,\OO_X(M-d-1)$ in the derived category. We prove that $\Ku(X)$ determines $X$ for every smooth quartic in $\PP^{4k+1}$, $k\geq1$; together with earlier...
We prove categorical Torelli theorems for ordinary and weighted hypersurfaces using the singularity category of the completed affine cone. For Fano hypersurfaces we use the Kuznetsov component, whereas for hypersurfaces of general type we use the full category of graded matrix factorizations. When the degree is coprime...
Let $m\geq 2$ and $d\geq 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$. We prove the rational noncommutative Hodge conjecture for the dg category $\mathrm{MF}^{\mathrm{gr}...
Xun Lin, Shi-Zhuo Zhang· 0 citations
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