44 problems

Let $k$ be the base field, let ${\mathcal {P}}$ be the class of schemes considered in the source, and let $X\in {\mathcal {P}}$. The monodromy operator is … The associated objects…

Let $({\rm {LM}}\flat)$ and $({\rm {LM}}\flat*)$ be the categories with the same objects and with morphisms defined using, respectively, the Chern class constructions from $K_0$ an…

Let $f:\mathcal{X}\rightarrow S$ be a toric degeneration, and write $X=\mathcal{X}_0$. Let $Z$ be the singular set of the degeneration, and let $\mathcal{M}_X$ denote the induced l…

Let $f\colon X\to Z$ be a fiber space and let $D$ be a $\mathbb{Q}$-divisor on $X$ such that $(X,D)$ is log canonical and $K_X+D$ is $\mathbb{Q}$-linearly trivial over $Z$. Let…

Let $(X,D)$ and $(X',D')$ be log smooth pairs, and let $E$ be a strong Fourier–Mukai kernel on $X\times^{\log}X'$. Let $\pi:X\times^{\log}X'\to X\times X'$ be the projection and re…

Let $(X,D)$ be a log smooth pair, and let $E_1\to E_2\to E_3\to E_1[1]$ be a distinguished triangle in $D(X\times^{\log}X)$ whose terms are strong log Fourier–Mukai kernels. Additi…

Let $A\to S$ be a family of log abelian varieties over a log smooth base $S$. Let $U\subseteq S$ be the dense open subset of $S$ on which the log structure is trivial. For a log sm…

Zero-mutable log resolution conjecture. The log scheme $X^\dagger$ admits a projective log crepant log resolution: there exist a log scheme $Y^\dagger$ and a projective morphism…

Let $\mathfrak{M}_{\mathrm{sat}}$ be the category of saturated log motives, let $E$ be a log Tate curve, and let $M$ be an object of $\mathfrak{M}_{\mathrm{sat}}$. Saturated log-mo…

Let $s$ be the standard log point over a finite field, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $d=\dim X$, and let $q$ be a section of the log struc…

Let $X$ be as above, let $d=\dim(X)$, and take integers $m,m',r,r'$ with $m,m'\geq 0$, $m+m'=2d$, and $m-2r=m'-2r'\geq 0$. The construction in the source gives a pairing … Perfect-…

Let $V$ be a variety with strictly semistable reduction $X$ over a local field with residue field $k$, assume that $k$ is finite, and define … where $\varphi$ is geometric Frobeniu…

Let $s$ be the standard log point over a finite field $k$, let $X$ be a projective vertical log smooth fs log scheme over $s$, and let $E$ be a log Tate curve. Define $K(X,m,r)$ as…

Let $s$ be the standard log point over a finite field $k$, let $X$ be a projective vertical log smooth fs log scheme over $s$, and define $K_0^{\log}(X)$ as the quotient of…

Let $s$ be the standard log point over a finite field $k$, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $m\geq 2r$, and let $E$ be a log Tate curve over…

Let $s=\operatorname{Spec}(k)$ be the standard log point, let $X$ be as above, let $\ell$ be invertible in $k$, and let $m\geq 2r$. Define $KH_{i,\lim}(X)=\varinjlim_{X'}KH_i(X')$…

Let $s=\operatorname{Spec}(k)$ be the standard log point, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $\ell$ be invertible in $k$, and define … where…

Let $s=\operatorname{Spec}(k)$ be the standard log point, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $\ell$ be invertible in $k$, and write … The fixed…

Semistable comparison conjecture. For all $i\in \mathbb{Z}$ and $n\in \mathbb{Z}_{\ge 0}$, there is a natural isomorphism

Let $(X,\Delta)$ be a klt pair, where $X$ is a normal projective variety and $\Delta$ is an effective $\mathbb{Q}$-divisor such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier. A divisor…

Let $X$ be a log scheme for which the constructions in the paper are defined, let $i:X\to B$ be the local complete intersection morphism used to define logarithmic Hochschild cohom…

Let $X$ be a projective log canonical variety, and suppose that $-K_X$ is a strictly nef divisor. Log version of Campana--Peternell's conjecture. The anti-canonical divisor $-K_X$…

Let $X$ be a projective log canonical variety with numerically trivial canonical divisor, so that $K_X\equiv 0$, and let $L$ be a strictly nef $\mathbb Q$-Cartier divisor on $X$. A…

Let $(X,\Delta)$ be a projective log canonical pair of dimension $n$, and let $L$ be a strictly nef Cartier divisor on $X$. Log version of Serrano's conjecture. The $\mathbb Q$-Car…

Log-smooth modification conjecture. There exists a finite extension $l/k$ and an admissible blowing up