We prove a formula, which, given a principally polarized abelian variety $(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda )$.

We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the canonical ribbon conjecture of Bayer and Eisenbud holds over a field of characteristic $0$ or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics the results of Voisin asserting that Green's conjecture holds for generic curves of each gonality.

Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then the locus in $X$ over which $f$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $X$, the same property holds for other local properties of morphisms, even if $f$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $F$-rationality, and the ‘Cohen–Macaulay and $F$-injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero.

We prove the generic Torelli theorem for hypersurfaces in $\mathbb {P}^{n}$ of degree $d$ dividing $n+1$, for $d$ sufficiently large. Our proof involves the higher-order study of the variation of Hodge structure along particular one-parameter families of hypersurfaces that we call ‘Schiffer variations.’ We also analyze the case of degree $4$. Combined with Donagi's generic Torelli theorem and results of Cox and Green, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.

For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat, Brav, Bussi and Joyce introduced certain perverse sheaves on them which can be regarded as sheaf-theoretic categorifications of the Donaldson–Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel–Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson–Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of the Thom isomorphism theorem for dual obstruction cones, we propose a sheaf-theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks.

Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$, and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$-modules. We write $\mathscr {C}_{\mathfrak {g}}^0$ for the monoidal subcategory of $\mathscr {C}_{\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\mathfrak {g}})$ in a natural way and show that the block decompositions of $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal {W}$ (respectively $\mathcal {W} _0$) arising from simple modules of $\mathscr {C}_{\mathfrak {g}}$ (respectively $\mathscr {C}_{\mathfrak {g}}^0$) by using the invariant $\Lambda ^\infty$ introduced in previous work by the authors. The groups $\mathcal {W}$ and $\mathcal {W} _0$ have subsets $\Delta$ and $\Delta _0$ determined by the fundamental representations in $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$, respectively. We prove that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta )$ is isomorphic to the direct sum of infinite copies of $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ as a root system.

Let $Q$ be an acyclic quiver and $w \geqslant 1$ be an integer. Let $\mathsf {C}_{-w}({\mathbf {k}} Q)$ be the $(-w)$-cluster category of ${\mathbf {k}} Q$. We show that there is a bijection between simple-minded collections in $\mathsf {D}^b({\mathbf {k}} Q)$ lying in a fundamental domain of $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$. This generalises the same result of Iyama–Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf {H}$ of a bounded t-structure in a Hom-finite, Krull–Schmidt, ${\mathbf {k}}$-linear saturated triangulated category $\mathsf {D}$ is functorially finite in $\mathsf {D}$ if and only if $\mathsf {H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.