54 problems

Log Iitaka conjecture. One has

Let $\mathcal{V}$ be a complete discrete valuation ring of mixed characteristics $(0,p)$ with perfect residue field $k$. Let $\mathfrak{X}$ be a separated smooth formal…

Let $Y$ be a smooth variety over $k$ (that is, a separated smooth $k$-scheme of finite type), and let $E$ be an overconvergent $F$-isocrystal on $Y$. A morphism…

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…

Strict cdh descent conjecture. The complex $\mathbb{Q}_p^\mathrm{syn}(i)$ is a strict cdh sheaf on the category of quasi-compact, quasi-separated fs log schemes over $\mathbb{F}_p$…

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

Unobstructedness conjecture. For every $b\in H_2$ and every $a\in H_1$, the deformations of both restricted log maps are unobstructed; equivalently, $H_1$ and $H_2$ are log smooth…

Let $(X,D)$ be a smooth projective simple normal crossing pair and set $U=X\backslash\operatorname{Supp}D$. A quasi-projective variety is $\mathbb{A}^1$-uniruled if it is covered b…

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…