We study the rationality properties of the moduli space ${\mathcal{A}}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular, we show that any principally polarised abelian 3-fold over ${\mathbb{F}}_p$ may be lifted to an abelian variety over $\mathbb{Q}$. This is a phenomenon of low dimension: assuming the Bombieri–Lang conjecture, we also show that this is not the case for abelian varieties of dimension at least 7. Concerning moduli spaces, we show that ${\mathcal{A}}_g$ is unirational over $\mathbb{Q}$ for $g\le 5$ and stably rational for $g=3$. This also allows us to make unconditional one of the results of Masser and Zannier about the existence of abelian varieties over $\mathbb{Q}$ that are not isogenous to Jacobians.

We prove that the only Bott manifolds such that the Futaki invariant vanishes for any Kähler class are isomorphic to the products of the projective lines.

Dale Peterson has discovered a surprising result that the quantum cohomology ring of the flag variety $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is isomorphic to the coordinate ring of the intersection of the Peterson variety $\operatorname {\mathrm {Pet}}_n$ and the opposite Schubert cell associated with the identity element $\Omega _e^\circ $ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$. This is an unpublished result, so papers of Kostant and Rietsch are referred for this result. An explicit presentation of the quantum cohomology ring of $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is given by Ciocan–Fontanine and Givental–Kim. In this paper, we introduce further quantizations of their presentation so that they reflect the coordinate rings of the intersections of regular nilpotent Hessenberg varieties $\operatorname {\mathrm {Hess}}(N,h)$ and $\Omega _e^\circ $ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$. In other words, we generalize the Peterson’s statement to regular nilpotent Hessenberg varieties via the presentation given by Ciocan–Fontanine and Givental–Kim. As an application of our theorem, we show that the singular locus of the intersection of some regular nilpotent Hessenberg variety $\operatorname {\mathrm {Hess}}(N,h_m)$ and $\Omega _e^\circ $ is the intersection of certain Schubert variety and $\Omega _e^\circ $, where $h_m=(m,n,\ldots ,n)$ for $1<m<n$. We also see that $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $ is related with the cyclic quotient singularity.

We geometrize the mod p Satake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake transform of an arbitrary parahoric mod p Hecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric mod p Hecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step toward a geometrization of a mod p Local Langlands Correspondence.

In our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum K-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant K-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb {C})$.

Let $\mathfrak{g}$ be the queer superalgebra $\operatorname {\mathfrak{q}}(n)$ over the field of complex numbers $\mathbb C$. For any associative, commutative, and finitely generated $\mathbb C$-algebra A with unity, we consider the loop Lie superalgebra $\mathfrak{g} \otimes A$. We define a class of central operators for $\mathfrak{g} \otimes A$, which generalizes the classical Gelfand invariants. We show that they generate the algebra $U(\mathfrak{g} \otimes A)^{\mathfrak{g}}$. We also show that there are no non-trivial $\mathfrak{g}$-invariants of $U(\mathfrak{g} \otimes A)$ where $\mathfrak{g}=\mathfrak{p}(n)$, the periplectic Lie superalgebra.

We consider faithful actions of simple algebraic groups on self-dual irreducible modules and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the dimension of the variety. We prove that in all but a finite list of cases, there is a dense open subset where the stabilizer of any point is conjugate to a fixed subgroup, called the generic stabilizer. We use these results to determine whether there exists a dense orbit. This in turn lets us complete the answer to the problem of determining all pairs of maximal connected subgroups of a classical group with a dense double coset.

We construct families of non-toric ${\mathbb {Q}}$-factorial terminal Fano (${\mathbb {Q}}$-Fano) threefolds of codimension $\geq 20$ corresponding to 54 mutation classes of rigid maximally mutable Laurent polynomials. From the point of view of mirror symmetry, they are the highest codimension (non-toric) ${\mathbb {Q}}$-Fano varieties for which we can currently establish the Fano/Landau–Ginzburg correspondence. We construct 46 additional ${\mathbb {Q}}$-Fano threefolds with codimensions of new examples ranging between 19 and 5. Some of these varieties are presented as toric complete intersections, and others as Pfaffian varieties.

Let $(X, \Delta )$ be a klt threefold pair with nef anti-log canonical divisor $-(K_X+\Delta )$. We show that $\kappa (X, -(K_X+\Delta ))\geq 0$. To do so, we prove a more general equivariant non-vanishing result for anti-log canonical bundles, which is valid in any dimension.

We investigate the existence of 4-torsion in the integral cohomology of oriented Grassmannians. We establish bounds on the characteristic rank of oriented Grassmannians and prove some cases of our previous conjecture on the characteristic rank. We also discuss the relation between the characteristic rank and a result of Stong on the height of w1 in the cohomology of Grassmannians. The existence of 4-torsion classes follows from the results on the characteristic rank via Steenrod square considerations. We thus exhibit infinitely many examples of 4-torsion classes for oriented Grassmannians. We also prove bounds on torsion exponents of oriented flag manifolds. The article also discusses consequences of our results for a more general perspective on the relation between the torsion exponent and deficiency for homogeneous spaces.

In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let ${\mathcal {P}}$ be a perverse sheaf on $A$ and let $B=g B_0$ be a generic translate of $B_0$. Then our theorem implies $(-1)^{\operatorname {codim} B}\chi (A\cap B, {\mathcal {P}}|_{A\cap B})\geq 0$. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersections of Schubert cells. Such Euler characteristics are known to be the structure constants for the multiplication of the Segre–Schwartz–MacPherson classes of these Schubert cells.

Chow rings of flag varieties have bases of Schubert cycles $\sigma _u $, indexed by permutations. A major problem of algebraic combinatorics is to give a positive combinatorial formula for the structure constants of this basis. The celebrated Littlewood–Richardson rules solve this problem for special products $\sigma _u \cdot \sigma _v$, where u and v are p-Grassmannian permutations.

Building on work of Wyser, we introduce backstable clans to prove such a rule for the problem of computing the product $\sigma _u \cdot \sigma _v$ when u is p-inverse Grassmannian and v is q-inverse Grassmannian. By establishing several new families of linear relations among structure constants, we further extend this result to obtain a positive combinatorial rule for $\sigma _u \cdot \sigma _v$ in the case that u is covered in weak Bruhat order by a p-inverse Grassmannian permutation and v is a q-inverse Grassmannian permutation.

Given any toric subvariety Y of a smooth toric variety X of codimension k, we construct a length k resolution of ${\mathcal O}_Y$ by line bundles on X. Furthermore, these line bundles can all be chosen to be direct summands of the pushforward of ${\mathcal O}_X$ under the map of toric Frobenius. The resolutions are built from a stratification of a real torus that was introduced by Bondal and plays a role in homological mirror symmetry.

As a corollary, we obtain a virtual analogue of Hilbert’s syzygy theorem for smooth projective toric varieties conjectured by Berkesch, Erman and Smith. Additionally, we prove that the Rouquier dimension of the bounded derived category of coherent sheaves on a toric variety is equal to the dimension of the variety, settling a conjecture of Orlov for these examples. We also prove Bondal’s claim that the pushforward of the structure sheaf under toric Frobenius generates the derived category of a smooth toric variety and formulate a refinement of Uehara’s conjecture that this remains true for arbitrary line bundles.

Mirror symmetry for a semistable degeneration of a Calabi–Yau manifold was first investigated by Doran–Harder–Thompson when the degenerate fiber is a union of two quasi-Fano manifolds. They proposed a topological construction of a mirror Calabi–Yau by gluing of two Landau–Ginzburg models that are mirror to those Fano manifolds. We extend this construction to a general type semistable degeneration where the dual boundary complex of the degenerate fiber is the standard N-simplex. Since each component in the degenerate fiber comes with the simple normal crossing anticanonical divisor, one needs the notion of a hybrid Landau–Ginzburg model – a multipotential analogue of classical Landau–Ginzburg models. We show that these hybrid Landau–Ginzburg models can be glued to be a topological mirror candidate for the nearby Calabi–Yau, which also exhibits the structure of a Calabi–Yau fibration over $\mathbb P^N$. Furthermore, it is predicted that the perverse Leray filtration associated to this fibration is mirror to the monodromy weight filtration on the degeneration side [12]. We explain how this can be deduced from the original mirror P=W conjecture [18].

We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.

We give a short new proof of a recent result of Hanlon-Hicks-Lazarev about toric varieties. As in their work, this leads to a proof of a conjecture of Berkesch-Erman-Smith on virtual resolutions and to a resolution of the diagonal in the simplicial case.

The K-theoretic Schur P- and Q-functions $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $ may be concretely defined as weight-generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of stable Grothendieck polynomials and were introduced by Ikeda and Naruse for applications in geometry. Nakagawa and Naruse specified families of dual K-theoretic Schur P- and Q-functions $g\hspace {-0.1mm}p_\lambda $ and $g\hspace {-0.1mm}q_\lambda $ via a Cauchy identity involving $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $. They conjectured that the dual power series are weight-generating functions for certain shifted plane partitions. We prove this conjecture. We also derive a related generating function formula for the images of $g\hspace {-0.1mm}p_\lambda $ and $g\hspace {-0.1mm}q_\lambda $ under the $\omega $ involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author. Using these results, we verify a conjecture of Ikeda and Naruse that the $G\hspace {-0.2mm}Q$-functions are a basis for a ring.

The $\Delta $-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $\Delta $-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb {F}_q$ and partitioning the $\Delta $-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at $t=0$.

The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.