Papers

We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary $\partial\bar\partial$-lemma and prove that it admits no holomorphic connection. This gives, in particular, a negative answer to a question posed by Cao–Deng–Matsumura. In contrast, for compact simply connected complex manifolds $X$, we prove that the answer is affirmative if $H^1(X,\mathcal O_X)=0$, and hence if the Frölicher spectral sequence of $X$ degenerates at $E_1$.

Fix two positive braid words $\beta_+,\beta_-$, and let $\text{Conf}(\beta_+,\beta_-)$ be the corresponding (type A) double Bott-Samelson variety. Let $\beta$ be a double braid word containing $\beta_+,\beta_-$ as the top, bottom words. We consider the open cluster torus $T(C_\beta)$ associated to a triangulation $C_\beta$ in $\text{Conf}(\beta_+,\beta_-)$ from arXiv:1904.07992, and we identify these with weighted plabic fences, where the usual local moves on planar bipartite graphs naturally correspond to change of torus coordinates in double Bott-Samelson variety. Moreover, $T(C_\beta)$ can be parametrized explicitly by certain matrix products, and also has monomial coordinates given by the cluster variables, which are matrix minors. We interpret these minors as a generalized notion of perfect matchings on weighted plabic fences, which may not be reduced plabic graphs. Using this interpretation, we show that the cluster variables are given by "generalized minimal matchings", extending the minimal matchings from arXiv:1606.08383. Lastly, we derive a Chamber Ansatz formula for double Bott-Samelson varieties via face alternating products from dimer theory. In general, plabic fences are not reduced plabic graphs, yet we are able to extend and apply the standard tools such as local moves on plabic graphs, trips, and minimal matchings to them.

Let $f\colon X\to X$ be a polarised endomorphism of a complex projective manifold, and let $D\subset X$ be a reduced totally invariant hypersurface. We prove that every irreducible component of the singular locus of $D$ has codimension one in $D$. In particular, if $D$ is normal, then it is smooth. As an application we prove the linearity of totally invariant divisors for endomorphisms of $\mathbb P^4$.

Within the framework of classical contact geometry, the first author introduced the notion of contact symplectic structure –the data of an exact symplectic fibration with a contact base– and proved, under mild conditions, that the total space of such a fibration inherits a contact structure compatible with the fibration map, thereby establishing the contact Thurston theorem. This paper provides a derived contact version of that result by incorporating our prior work on the derived symplectic Thurston theorem. In this paper, we prove, under certain conditions, that if a morphism $\pi: X \rightarrow S$ of derived stacks has a shifted exact symplectic fibration structure and the target stack $S$ admits a shifted contact structure, then one can construct a shifted contact structure on the source stack $X$, compatible with $\pi$ in a sense similar to the smooth case. Our framework relies on the theory of relative shifted structures; hence our result, called the derived contact Thurston theorem, in fact establishes a relative-to-absolute type construction. As an application, we present examples of our relative-to-absolute construction formalism in the derived contact setting, including conormal stacks, quotient mapping stacks, and affine exact symplectic fibrations.

The $a$-number of a function field $F$ is the dimension of the kernel of the Cartier operator acting on the holomorphic differentials of $F$. We compute the action of the Cartier operator on the holomorphic differentials on the Hermitian function field $\mathcal{H}$, function fields of Fermat type $\mathcal{F}_{n,m}$ and the Hurwitz function field $\mathcal{H}_n$. From this, we obtain various results about the $a$-number of these function fields, including a generalisation of the results on the Fermat function field from Kodama and Washio.

We formulate a conjecture predicting the existence of exotic isomorphisms between Igusa stacks associated to different Shimura data whose underlying groups are pure inner forms of each other, and prove it in many cases of interest. In combination with the spectral action of Fargues–Scholze, this allows us to relate the cohomology groups of the two Shimura varieties, giving a geometric incarnation of the Jacquet–Langlands correspondence. An important ingredient in our proofs is the work of Xiao–Zhu on exotic Hecke correspondences between the perfect special fibers of different Shimura varieties, which we reinterpret in terms of exotic isomorphisms of perfect Igusa stacks.

This is a paper about contraherent cosheaves on non-semi-separated schemes. We prove two theorems, a negative one and a positive one. On the negative side, let $X$ be a quasi-compact, quasi-separated scheme that is not semi-separated. We present a locally cotorsion contraherent cosheaf on $X$ that does not have an admissible monomorphism into any locally injective locally contraherent cosheaf. The construction and proof follow the arguments of Slavik and Stovicek in arXiv:1902.05740. On the positive side, let $X$ be a Noetherian scheme of finite Krull dimension. We prove that every $\mathbf W$-locally contraherent cosheaf on $X$ has an admissible monomorphism into a locally cotorsion $\mathbf W$-locally contraherent cosheaf. Moreover, the cokernel is a flat contraherent cosheaf. The proof is based on the theorem of Raynaud-Gruson about the projective dimensions of flat modules and Enochs' classification of flat cotorsion modules.

We generalise Yin's motivic integration with section to the Hensel-minimal framework, with and without volume forms, building on Stout and Vermeulen's work on a Hrushovski-Kazhdan style motivic integral for $1$-h-minimal valued fields. In particular, we get a bijective motivic integral \emphà la} Hrushovski-Kazhdan between certain Grothendieck groups of definable sets on the valued field and the coarse Grothendieck groups on $\mathrm{RV}$ in the Denef-Pas language, with angular component. In the second half of the article, we build a Cluckers-Loeser-like framework and show that our constructions indeed give back Cluckers-Loeser motivic integral. Thus Cluckers-Loeser motivic integration derives from an isomorphism of Hrushovski-Kazhdan motivic integration, but in the Denef-Pas language with a section of $\mathrm{RV}$.

We study cohomological Hall algebras of curves and their actions on the homology of Quot schemes. We introduce the virtual homology of Quot schemes and show that it is preserved by both creation and annihilation actions. We prove that the torsion CoHA is isomorphic to a shuffle algebra and, equivalently, to a braided symmetric algebra associated with a Yang-Baxter operator. We use this description to determine the ideal of tautological relations for punctual Quot schemes and obtain a new basis for their cohomology rings. Finally, we introduce a universal way to double the torsion CoHA and show that it acts naturally on the virtual homology of Quot schemes of arbitrary type.

We study surjectivity of the geometric Sen map for perfectoid $G$-torsors over smooth rigid analytic spaces over $\mathbb{C}_p$, where $G$ is a compact $p$-adic Lie group. We construct an affinoid perfectoid $\mathbb{Z}_p^2$-torsor over a smooth curve whose Sen map is not surjective, thereby disproving Rodríguez Camargo's conjectured implication from perfectoidness to surjectivity. We prove that the Sen map of every perfectoid $G$-torsor over a smooth curve is nevertheless surjective on a Zariski open dense subset. Finally, we show that locally analytic decompletion implies Sen surjectivity for diamondian affinoid perfectoid torsors.

Let $X$ be an affinoid rigid analytic space over a $p$-adic field, equipped with a suitable étale map to a unit polydisk. We provide a formalism of imperfect relative period rings over $X$, which lie inside the corresponding perfect relative period rings constructed by Kedlaya–Liu. Then we establish a relative version of the Fontaine–Cherbonnier–Colmez equivalence between $p$-adic local systems on $X$ and étale $(\varphi, \Gamma)$-modules over such imperfect relative period rings, generalizing previous works of Andreatta–Brinon and others. Using this equivalence, we construct $p$-adic differential equations attached to de Rham local systems, carrying both geometric and arithmetic differential operators. This generalizes the work of Berger to the relative geometric setting. Along the way, we study a relative Fontaine–Sen theory on the decompletion of $\Gamma$-modules over the relative $\mathbf{B}_{\mathrm{dR}}^+$-period rings.

In this paper, we give a complete description of the Hilbert schemes of rational curves up to degree $6$ on a prime Fano threefold $X$ of degree $22$. One of the key ingredients in the geometry of these Hilbert schemes is the geometry of bisecant conics associated with rational curves on $X$. As applications, we describe additional geometric features of these moduli spaces and derive Donaldson–Thomas (DT) type invariants.

We prove Morita invariance of the Campbell–Lind–Malkiewich–Ponto–Zakharevich restriction-system trace after passage to perfect modules. It therefore defines an intrinsic integral restriction trace for an exact endofunctor of a small idempotent-complete stable $\infty$-category. On $\pi_0$, the $m$-th ghost is the laced trace of the $m$-fold iterate, compatibly with Frobenius. For lattice-graded algebras, the ghost targets have twisted cocenters in degree zero; over the open parameter torus, this applies to the cyclic bimodule of Dinkins–Karpov–Krylov. For finite monomial endomorphisms of toric varieties, we construct motivic restriction classes whose ghosts are sums over cones fixed by the iterates. In one example, two classes have the same first ghost, while their second ghosts differ after rational Betti realization.

Let $D$ be a smooth divisor in a smooth projective complex variety $X$. For connected curves of arithmetic genus one with prescribed signed contact orders, we prove that the refined punctured cycle of Battistella–Nabijou–Ranganathan and the negative-contact cycle of Fan–Wu–You agree after pushforward to the common moduli space of stable maps with divisor evaluations. Thus the pushed-forward refined punctured cycle is the constant coefficient of the pushed-forward root-stack virtual class, normalized by one power of the root order for each negative contact. The key step is a comparison for the universal target. After restricting to finite-type open substacks determined by the fixed pair $(X,D)$ and numerical data $\Gamma$, we prove that the positive BNR space maps finitely and with generic degree one onto Crumplin's main component. Using Crumplin's genus-one component description and degree formulas, we identify this component's fundamental cycle with the constant coefficient of the universal orbifold virtual class under comparison of root orders. Refined zero-section pullback recovers the negative contacts, and compatible virtual pullbacks and root-forgetting pushforwards transfer the resulting identity to $(X,D)$.

Cheong proved that if $S$ is a simple Lie group and $P$ is a parabolic subgroup, then $S/P$ admits a unique equivariant compactification of the unipotent radical $N$ of $P$, up to isomorphism, provided that $S/P$ is not isomorphic to a projective space. We generalize this result to a smooth projective horospherical variety $X$ with Picard number 1. Let $G = \mathrm{Aut}(X)$, and let $H$ be the isotropy subgroup at a point in the open $G$-orbit. We describe a unipotent subgroup $N$ of $H$, which is not necessarily the unipotent radical of $H$ when $X$ is not homogeneous. Moreover, we prove that $X$ admits a unique equivariant compactification of $N$, up to isomorphism.

A general $4\times3\times3$ tensor defines three determinantal varieties in $\mathbb{P}^3$ and $\mathbb{P}^2$ and $\mathbb{P}^2$ where it drops rank: a cubic surface and two length 6 subschemes. We study the Geometric Invariant Theory (GIT) quotient of $\mathbb{P}(\mathbb{C}^4\otimes\mathbb{C}^3\otimes\mathbb{C}^3)$ by $\operatorname{SL}(4)\times\operatorname{SL}(3)\times\operatorname{SL}(3)$, a moduli space of linear determinantal representations of cubic surfaces. We give a geometric description of the (semi)stable locus and some of the geometry of this GIT quotient. This GIT quotient admits a surjective rational map to the GIT moduli space of cubic surfaces, undefined at only one point.

We prove a version of genus-0 mirror symmetry for toric Fano surfaces at the level of big quantum cohomology. The mirror Landau-Ginzburg model $(\check{X}, W_{\infty})$ of such a surface $X$ can be defined in terms of counts of Maslov index two tropical or holomorphic discs with bulk insertions. While these mirror LG models are not well-defined, the Jacobian ring $\operatorname{\mathsf{Jac}}(W_{\infty})$ of the superpotential $W_{\infty}$ is well-defined. We show that $\operatorname{\mathsf{Jac}}(W_{\infty})$ is isomorphic over the Novikov base to the big quantum cohomology of $X$ using tropical methods, providing a purely algebraic analogue of the cobordism argument used in works of Fukaya-Oh-Ohta-Ono.

Motivated by a question of Corvaja and Zannier, we investigate whether the projections on the real torus of three real algebraic plane curves admit infinitely many triple intersections. We analyze the case of three lines in detail, obtaining a complete classification. For higher-degree curves, we derive several finiteness results using Siegel's and Levin's theorems on integral points on curves. On the other hand, we construct explicit examples of higher-degree curves with infinitely many triple intersections.

Dinh and Nguyên asked whether the determinant of a Griffiths positive matrix of $(1,1)$-forms belongs to the Hodge–Riemann cone. For every $n\geqslant4$ and $2\leqslant k\leqslant n-2$, we present counterexamples constructed by AI: Nakano positive $k\times k$ matrices of constant $(1,1)$-forms on $\mathbb{C}^n$ whose determinants have singular Lefschetz maps in bidegree $(1,n-k-1)$. Under the simultaneous diagonalizability (SD) condition, we prove the Hodge–Riemann property in every bidegree $(p,q)$ with $p+q=n-k$ and $\min(p,q)\leqslant1$. The proof uses the generalized Alexandrov–Fenchel inequality of Ross, Süß, and Wannerer. For every $n\geqslant6$ and $2\leqslant k\leqslant n-4$, we also present SD examples whose determinants have singular Lefschetz maps in bidegree $(2,n-k-2)$. Their underlying idea comes from Ross–Toma's construction. The remaining cases $n=k\geqslant4$ and $n=k+1\geqslant5$ are open for general Griffiths positive matrices. They reduce to the top-Chern case of Griffiths' positivity question, equivalently to Finski's double mixed discriminant problem. We also ask whether a $2\times2$ counterexample on $\mathbb{C}^4$ can be both Nakano positive and dual Nakano positive.

We give Hodge-theoretic characterizations of characteristic-zero varieties of open $F$-nilpotent and open weakly $F$-nilpotent type, extending a result of Srinivas and Takagi beyond isolated singularities. We prove one direction unconditionally and the converse assuming the weak ordinarity conjecture.

Deciding whether a system of multivariate quadratic equations over $\mathbb F_2$ has a solution is a classical NP-complete problem, and remains so for square systems, with as many equations as variables. The hardness of this problem is one of the cornerstones of nowadays post-quantum cryptography. Let $\MQ_0(n)$ and $\MQ_1(n)$ denote the sets of square quadratic systems in $n$ variables having respectively no solutions and exactly one solution. $\cup_{n\geq 2} \MQ_0(n)$ is a coNP-complete language, while $\cup_{n\geq 2} \MQ_1(n)$ lies in DP. It is known that $\lim_{n\to \infty} |\MQ_1(n)|/|\MQ_0(n)|=1$. Here we prove the explicit finite-$n$ bounds \[ |\MQ_0(n)|<|\MQ_1(n)| \le \left(1+\frac{1}{2^n-1}\right)|\MQ_0(n)|, \] More generally, let $Q_d$ be the space of polynomial functions $(\FF_2)^n\to\mathbb F_2$ of degree at most $d$, and let $\alpha_k$ count square systems in $(Q_d)^n$ having exactly $k$ solutions. Then \[ \alpha_0<\alpha_1 \le \left(1+\frac{1}{2^n-1}\right)\alpha_0\,, \qquad 2\le d\le n \,. \] The proof combines matroid and coding-theoretic methods. We interpret $(\FF_2)^n$ as the ground set of the evaluation matroid of $Q_d$, express $\alpha_0$ and $\alpha_1$ through characteristic polynomials, and use a Whitney-type sign-reversing involution to show that the only terms that can push $\alpha_1-\alpha_0$ below $\alpha_1/2^n$ come from the elements of a matroid port. These are identified with minimal-support words of the Reed–Muller code $\RM(n-d-1,n)=\RM(d,n)^\perp$; the required estimate then follows from the MacWilliams identity, the minimum-distance bound $2^{d+1}$, and the even-weight structure of the code.

We develop a theory of logarithmic $A_{\mathrm{inf}}$-cohomology with coefficients for a class of $p$-adic log formal schemes that are “sufficiently log smooth”, where the coefficients are given by relative log BKF modules. Then we establish comparison isomorphisms with étale, de Rham, and crystalline cohomology, and also extend these results to the derived setting. As an application, we give a new proof of the $C_{\mathrm{st}}$ conjecture for semistable local systems. The proof also uses the prismatic interpretation of semistable local systems established by Du–Liu–Moon–Shimizu.

We study Chow-quotient compactifications of the moduli space of degree-$d$ rational self-maps of the projective line. We compute the degree of every three-dimensional orbit cycle in the Clebsch–Gordan compactification in terms of the main component and the hole depths. As consequences, we obtain the generic orbit class and a rational map on Chow quotients induced by iteration. We also introduce a marked Chow quotient designed to record collections of rescaling limits, describe it by decorated trees, and extend the fixed-point multiplier map to the normalization of the unmarked Chow quotient.

We compute the Wedderburn decomposition of a finite twisted skew group algebra over a finite abelian group $G$ and a finite field $K$. To do so, we prove that this twisted skew group algebra is a homomorphic image of a finite group algebra. We then characterise its Shoda pairs, as well as propose an algorithm for computing a complete set of them.

Let $G$ be a connected reductive complex algebraic group and let $q\in\mathbb{C}^{\times}$ be not a root of unity. We prove that every $(G,q)$-connection on a formal punctured disc admits a $(G,q)$-oper structure.

We study the action of the Serre functor of smooth proper dg categories at their Hochschild-Serre algebra, and the invariant sub-aglebra of the Serre functor. As applications, we prove some theorems of categorical Torelli. Namely, let $\Ku(\X)$ be the Kuznetsov component of degree $d$ smooth hypersurface in weighted projective space $\mathbb{P}(a_0, a_1, \cdots, a_n)$, where the common maximal divisor $\gcd(d, \sum^{n}_{i=0}a_{i})=1$. We show the categorical Torelli for $\Ku(\X)$. We show that the $\mathbb{C}^{\ast}$ equivariant matrix factorization category associated with a quasi-homogeneous polynomial function $f$ that has an isolated singularity together with a twisted functor $\{1\}$ reconstructs $f$ up to an isomorphism.

We first prove a version of the Demailly-Paun theorem for compact weakly Kaehler spaces. Secondly, for a big and nef Bott-Chern class on a compact normal Kaehler space, we prove that the restricted non-Kaehler locus, defined using Kaehler currents with weak analytic singularities, is equal to the usual non-Kaehler locus.

For every integer $d\geq2$ and every field of characteristic different from $2$, we prove that on the moduli space $\mathcal{M}_d$ of degree-$d$ rational maps, the multiplier spectrum morphism formed from the periodic points of periods one and two is birational to the closure of its image. Consequently, over every algebraically closed field of characteristic different from $2$, it is generically injective, which proves a recent conjecture of Ji and Xie in characteristic zero. The proof uses a fixed-index normal form, a non-archimedean degeneration of two-cycles, and birational reconstruction of an affine fixed-point configuration from pair invariants.

Let $k$ be an algebraically closed field of positive characteristic $p$, and let $C_p$ act linearly on a finite-dimensional $k$-vector space $V$, with indecomposable Jordan summands $V_{d_i}$ of dimension $d_i$. We study projective (crepant) resolutions of $V/C_p$ by constructing a model using the invariant weighted blowup. For $V=V_2^{\oplus n}$, our model is smooth and is the normalization of a blowup of $V/C_p$. It has a unique exceptional divisor of discrepancy $n-p$, such that our model is a crepant resolution when $n=p$. In almost all other cases when the quotient is canonical but not terminal, the invariant weighted blowup model does not produce crepant resolutions, but gives the unique nontrivial projective crepant birational model of the quotient. Consequently, up to trivial summands, we classify the $p$-cyclic linear quotients admitting a projective crepant resolution.

The Bott-Samelson varieties provide natural desingularizations of Schubert varieties, and their generalizations, called generalized Bott-Samelson varieties, were constructed by Nicolas Perrin as towers of locally trivial fibrations with fibers isomorphic to Schubert varieties. In this article, we give a complete characterization of Fano and weak Fano generalized Bott-Samelson varieties. As a consequence, we recover the corresponding results for ($G$-)Bott-Samelson varieties and minuscule generalized Bott-Samelson varieties.