Papers

The homotopy types of Oka manifolds are exactly those of countable finite-dimensional CW complexes. We construct realizations as open subsets of complex Euclidean spaces with explicit dimension bounds. Other realizations have good open covers by sets biholomorphic to $\C^N$, with prescribed finite nerves, or are unions of Fatou–Bieberbach domains whose nonempty finite intersections are again Fatou–Bieberbach domains. Every nonempty Stein manifold admits a proper holomorphic embedding into an Oka open subset of some $\C^N$, and this embedding is a homotopy equivalence. We obtain holomorphic representatives of ordinary and generalized cohomology classes, and realize every countable group as the fundamental group of an Oka domain in $\C^5$. The fundamental groups of algebraically elliptic manifolds are precisely the finite groups, and every virtually abelian Kähler group is the fundamental group of a projective Oka manifold. Finally, we classify holomorphic cofibrations in Lárusson's intermediate model structures and establish factorizations over aspherical targets into proper holomorphic embeddings and holomorphic fibre bundles with Oka fibre.

By virtue of their relation to Fulton–MacPherson compactifications, the cohomological field theory of Hilbert schemes of points on a surface, in extremal curve classes, is universally and effectively determined by the characteristic numbers of the surface and by the corresponding theory of Hilbert schemes of the affine plane. As a consequence, we compute the extremal genus 1 Gromov–Witten invariants of Hilbert schemes of all surfaces.

We give counterexamples to the conjecture of Dais-Henk-Ziegler on projective crepant resolutions of complete-intersection quotient singularities. For finite $G\subset\operatorname{SL}_4(\mathbb{Z})$, we determine whether $\mathbb{C}^4/G$ admits a projective crepant resolution, with the representations considered up to conjugacy in $\operatorname{SL}_4(\mathbb{C})$. Ten explicit criteria decide all cases. The examples also show that the Molien series and the age polynomial do not determine existence. We also discuss questions of Ito-Reid on maximal cyclic subgroups and the generation of cohomology in degree two.

We study the Solomon–Terao algebra and the Solomon–Terao complex of a central hyperplane arrangement $\mathcal A$ in $\mathbb C^\ell$. Let $\eta$ be a homogeneous polynomial of degree $d\geq2$ whose restriction to every flat of $\mathcal A$ has no critical point away from the origin. We prove that the Solomon–Terao algebra $\mathrm{ST}(\mathcal A,\eta)$ has top degree $|\mathcal A|+\ell(d-2)$ and a one-dimensional top graded component with an explicit generator, solving a conjecture of Abe, Maeno, Murai, and Numata. For $d=2$, we show that the total dimension of $\mathrm{ST}(\mathcal A,\eta)$ is bounded below by that of the Orlik–Solomon algebra. Both results hold without assuming that the arrangement $\mathcal A$ is tame. Finally, we prove that $\mathcal A$ is tame if and only if all positive-degree homology groups of the Solomon–Terao complex associated with $\eta$ vanish.

We prove that every Lawson surface $\xi_{k,l}\subset\mathbb S^3$ of genus $kl>1$ is non-algebraic: any real polynomial vanishing on $\xi_{k,l}$ vanishes identically on $\mathbb S^3$. This answers a question posed by Hsiang in 1967 and proves Lawson's 1970 conjecture for $\xi_{2,2}$. The proof uses a reflection principle for the monodromy of the associated family of flat connections.

This note proves that the Kuznetsov component of a quartic double solid or a special Gushel-Mukai threefold does not have non-trivial admissible subcategories.

We study Galois coinvariants of the geometric étale fundamental group and étale cohomology of varieties over a finitely generated field $k$. We establish finiteness properties of these coinvariants for (possibly infinite) Galois extensions $L/k$ such that $\operatorname{Gal}(L/k)^{\mathrm{ab}}$ has finite exponent. As an application, we obtain a finiteness result for étale fundamental groups of arithmetic schemes over number fields.

We prove the existence of minimal models for smooth co-rank one foliations of general type.

Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.

For an algebraic stack of finite type, we construct a finite stratification of an atlas such that every universally closed open substack pulls back to a union of strata. As a result, there are only finitely many open substacks admitting a proper adequate moduli space.

In this paper we show there does not exist any arrangement of degree $7$ consisting of $k\geq 1$ conics and $d\geq 1$ lines admitting certain naturally chosen ${\rm ADE}$-singularities.

Assuming a smoothness property for mixed-characteristic shtuka spaces and hyperbolic localization for twisted sheaves, we determine the inertial restriction of the Fargues-Scholze $L$-parameter for irreducible depth-zero representations. Let $E$ be a non-archimedean local field of residue characteristic $p$, let $G$ be an unramified connected reductive group over $E$, and let $\pi$ be an irreducible smooth depth-zero representation of $G(E)$ over $\overline{\mathbb{F}}_{\ell}$ where $\ell$ is a prime number which does not divide $p|\pi_0Z(G)|$. Let $\varphi_{\pi}^{\operatorname{FS}}$ be the Fargues-Scholze $L$-parameter associated to $\pi$. We describe the restriction of $\varphi_{\pi}^{\operatorname{FS}}$ to the inertia subgroup in terms of Deligne-Lusztig theory. We prove this by computing the inertia action on the cohomology of shtuka spaces with depth-zero level structure. As an application, we prove that the Fargues–Scholze $L$-parameters agree with DeBacker–Reeder $L$-parameters over $\overline{\mathbb{Q}}_{\ell}$.

Extending the methods previously established for plane sextics and quartic surfaces, we develop a unified and systematic framework for the equisingular deformation classification of projective models of $K3$-surfaces of arbitrary polarization. We treat all polarizations of degree $2n$ with $n\le 4$, including the higher-degree cases of sextic surfaces in $\mathbb{P}^{4}$ and octic surfaces in $\mathbb{P}^{5}$. The computational machinery developed here is degree-independent and provides a complete tool for enumerating and distinguishing the deformation classes arising from singular projective models of $K3$-surfaces. This framework lays the foundation for future investigations in higher degrees, particularly the geometry of equisingular strata, extremal configurations, real structures.

We establish lower bounds on the orbit cardinalities of $\text{SL}(2,\mathbb{Z})$ on the $\mathbb{F}_p$-points of the generalized Markoff surfaces given by $x^2 + y^2 + z^2 - xyz = \tau+2$. We show that outside of the reductions of a certain exceptional finite set of $\overline{\mathbb{Q}}$-points found by Dubrovin and Mazzocco, every orbit has cardinality $\ge p$. Together with a recent result of Nathaniel Kingsbury-Neuschotz, this establishes, for all but finitely many primes $p$, the classification conjectures of McCullough-Wanderley (2013), and gives a complete classification of the connected components of the Hurwitz stack of $\text{SL}(2,\mathbb{F}_p)$-covers of elliptic curves, only branched above the origin. This extends earlier work of Bourgain-Gamburd-Sarnak and the author in the Markoff case $\tau = -2$. Our method studies certain noncongruence modular curves attached to a $\text{SL}(2,\mathbb{Z})$-orbit. These modular curves parametrize elliptic curves with a $\text{PSL}(2,\mathbb{F}_p)$-cover. We use this moduli interpretation to show that the noncongruence modular curves involved have bad reduction, as a cover of the $j$-line, at some place above $p$. This implies that the degree of the cover is $\ge p$, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point $\text{PSL}(2,\mathbb{F}_p)$-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.

This paper studies the closures of the double Bruhat cells of symmetric spacess in its equivariant embeddings. We call them large double Bruhat varieties. Our aim is to establish the double Bruhat stratification as a natural setting for extending the geometry of Richardson varieties to symmetric spaces. For the wonderful compactification of the associated adjoint symmetric space, we show their scheme-theoretic intersections with the unique closed orbit are reduced unions of projected Richardson varieties. This connection with Richardson geometry on partial flag varieties supports large double Bruhat varieties as a natural class of subvarieties in symmetric-space embeddings. In positive characteristic, we prove that every equivariant embedding admits a Frobenius splitting compatible with all large double Bruhat varieties and all boundary divisors. These varieties are strongly F-regular, and globally F-regular when the ambient embedding is projective. In characteristic zero, they are of strongly F-regular type, and of globally F-regular type in the projective case; in particular, they are normal, Cohen-Macaulay, and have rational singularities.

Let $C$ be a smooth projective curve of genus $g$, and let $T_{n,L}$ denote the tautological line bundle on its $n$-th symmetric product $C_n$. We prove effective vanishing criteria for Koszul cohomology $K_{p,q}(C_n,T_{n,B};T_{n,L})$, extending two classical theorems of Mark Green. The main ingredient is a weight-one vanishing theorem proved by induction on $n$, using a filtration of the kernel bundle of $T_{n,L}$ to reduce the required cohomological vanishings to lower symmetric products. As a consequence, when $g\geq1$ and $n\geq2$, $T_{n,L}$ satisfies property $(N_p)$ for $\mathrm{deg}(L)\geq(n+1)g+p$. This bound is sharp for elliptic symmetric squares and also gives evidence for a folklore conjecture on adjoint syzygies.

Let $G$ be a split real semisimple group of adjoint type. We prove that the cellular decomposition of the totally nonnegative part of its wonderful compactification is a regular CW complex, confirming a conjecture of Galashin, Karp, and Lam. In particular, the closure of every cell is homeomorphic to a closed ball. The proof combines a Bruhat atlas compatible with total positivity and a CL-labeling of the augmented face poset. The atlas models the strata locally with open Richardson varieties in the flag variety of a larger Kac-Moody group and identifies their totally nonnegative cells.

Frobenius morphisms play a crucial role in algebraic geometry and commutative algebra in positive characteristic. In particular, the Kunz theorem, characterizing non-singularity via Frobenius morphisms, is a monumental work and has a relative version, called the Radu–André theorem. In this paper, we extend the Kunz theorem and the Radu–André theorem to logarithmic algebraic geometry in positive characteristic.

This work studies the relationship between the multigrading in the cohomology of moment-angle complexes, introduced by V. Buchstaber and T. Panov in their computation of these cohomology groups, and the multigrading in the cohomology of the corresponding complements of coordinate subspace arrangements, arising from the mixed Hodge structure. It is shown that these multigradings coincide. This result refines a theorem from Yu. Eliyashev's 2013 dissertation. Furthermore, results on the mixed Hodge structure in the cohomology of complements of coordinate subspace arrangements are applied to the study of the same varieties over finite fields. In particular, various formulas are obtained for the number of $\mathbb{F}_q$-rational points and for the associated global $\zeta$-functions.

We show that the Stein degree of horizontal divisors on generalised log Calabi-Yau fibrations or certain vertical divisors of Fano type fibrations over arbitrary fields of characteristic zero is bounded by constants that are independent on the ground field.

We prove a Cohen-Lenstra type result commenting on the distribution of class group structures amongst function fields of superelliptic curves over finite fields. We also prove a result commenting on the sparsity of such superelliptic curves whose zeta functions vanish at a fixed complex number. These results are proven via point-counting techniques on certain Hurwitz schemes generalizing techniques of Ellenberg-Venkatesh-Westerland and Ellenberg-Li-Shusterman. To obtain these point counts, We prove a unitary big monodromy theorem generalizing big monodromy theorems of Yu and Achter-Pries by appealing to an arithmeticity theorem of Venkataramana.

We find a new presentation of the orbifold fundamental group of the moduli space of cubic surfaces, based on the Artin group of the Petersen graph. This group has two well-known quotients, which appear in the simplest possible way: killing the squares resp. sixth powers of our generators yields the Coxeter group $W(E_6)$ resp. a certain arithmetic group in $\operatorname{PU}(1,4)$. Along the way we present a certain central extension by $\mathbb{Z}$, of the genus zero mapping class group with any number of punctures. The $6$-puncture case is essential in this paper, and the general case may have independent interest.

Given a surface of genus $g \geq 2$, its inf spectrum is the ordered multiset of inf lengths of curves on the surface, the infimum being over all hyperbolic metrics. We show that the inf spectrum has exponential growth with coefficients that only depend on g, improving the earlier double exponential upper bound. The proof involves developing a number of tools including the process of "thickening" a surface and the notion of model surfaces. These tools may be of independent interest.

We prove the geometric halves of Hilbert's Nullstellensatz and of the real Nullstellensatz for submodules of finitely generated modules and for coherent sheaves, complementing our earlier result that semiprime (resp. real) submodules are intersections of prime (resp. real prime) submodules. First, we pass from prime to maximal submodules: over a Jacobson ring, a prime submodule $P$ of $M$ is an intersection of maximal submodules whenever $M/P$ is a torsionless $R/(P:M)$-module, in particular whenever $M/P$ is finitely generated (the latter case is due to Maani Shirazi and Sharif). Consequently, if $R$ is a finitely generated algebra over an algebraically closed field and $M$ is finitely generated, the smallest semiprime submodule containing a submodule $N$ of $M$ consists of all $f\in M$ such that, for every point $a$, the value $f(a)$ (i.e. the image of $f$ in the fibre $M/\mathfrak{m}_aM$) lies in $N(a)$ (i.e. the image of $N$ in $M/\mathfrak{m}_aM$). We prove the real analogues over real Jacobson rings; for finitely generated algebras over a real closed field they are determined by the real points. The module classifications and directional vanishing formulas yield both Nullstellensätze for coherent sheaves on schemes of finite type over a field. We also prove compatibility of semiprimeness, reality and both radicals with arbitrary localization, and the resulting stalk criteria.

For a smooth locally reductive stack admitting an adequate moduli space, we show that there are finitely many open substacks preserving the adequate moduli space, and give a complete classification of such opens. For $\mathbb{G}_m$-actions, this becomes an explicit description in terms of the codimension-one Bialynicki-Birula strata. Moreover, for a stable adequate moduli space, we show that the number of preserving opens are bounded by 3 raised to the relative Picard rank. As an application, the number of fixed components for $\mathbb{G}_m$-actions is at least the relative Picard rank.

Let $X$ be a smooth compact irreducible hyperkähler manifold of dimension six. We prove that the second Betti number satisfies $b_2(X) \le 23$. Furthermore, we show that the third cohomology vanishes whenever $b_2(X) \ge 19$, and that all odd cohomology vanishes when $b_2(X) = 23$. If the second Chern class lies in the subalgebra generated by $H^2(X,\mathbb{Q})$, then $b_2(X) \in \{5,8,23\}$. The proof uses the Looijenga–Lunts–Verbitsky decomposition, curvature inequalities for Segre classes, and the Hodge–Riemann bilinear relations on the projective cotangent bundle.

We generalize Nakajima's theory of Hilbert schemes of points on surfaces to Hilbert schemes of points on orbisurfaces, and obtain initial factorization based on stratification by stabilizers. While the usual formula is retained for the trivial stabilizer stratum and for those strata of codimension $1$, genuine difficulties arise with stacky points strata. In the local picture of an isolated stacky point, the theory reduce to the study of $\Quot^\bfv[\CC^2/G]$ for a small finite subgroup of $GL_2(\CC)$. The generating series for their total cohomology is a geometric correction of the generating series for total cohomology of ordinary smooth surface schemes. In addition, we obtain an action of the Borel-Moore homology of the punctual fibre stacks on the total Hilbert cohomology by virtual correspondences, in analogy with the Fock space description in Nakajima's theory. The punctual fibre stacks are genuinely different from the symmetric stack, and has natural realizations via motivic integration.

It follows from the compactness of the connectedness locus that the locus of PCF polynomials in the moduli space of polynomials of degree d (at least 2) over the complex number is a countable union of zero-dimensional subvarieties. Over fields of characteristic p the analogous claim is certainly false, so it is natural to ask how large the components of the PCF locus can be. We show that the components of the PCF locus in the moduli space of polynomials of degree d over an algebraically closed field of characteristic p>0 have dimension at most d/p and that this bound is sharp, in the strong sense that for every p and d there are infinitely many distinct PCF components whose dimension is the greatest integer no greater than this bound (except in the case $p=d=2$).

We construct smooth irreducible plane curves whose complete amoeba contour is a smooth embedded curve equal to the amoeba boundary, although the curves are not torically equivalent to simple Harnack curves. Finite monomial coverings yield explicit infinite families with higher logarithmic multiplicity and nonmaximal normalized amoeba area. We classify the Newton polygons admitting such covering constructions through their exponent lattices, give an exact connectedness criterion in terms of monodromy, and treat primitive polygons by a separate phase-sensitive construction. Exact triangular and trapezoidal families are complemented by positive-genus investigations exhibiting bounded amoeba-complement components. These results show that a perfectly exposed amoeba contour does not determine the real structure, logarithmic sheet number, or Harnack type of the underlying curve.

We construct smooth plane curves whose logarithmic contours are smooth and embedded but strictly larger than their amoeba boundaries. The examples include connected irreducible curves, disconnected unions of nonlinear components with two-dimensional amoebas, nested families with arbitrarily many components, and a uniform construction realizing every positive genus. Exact critical-locus calculations distinguish the boundary-producing folds from compact or noncompact contour branches lying in the amoeba interior. These constructions show that smoothness of the complete contour does not force contour–boundary equality and that internal critical-value branches persist under monomial transformations, controlled perturbations, and finite monomial coverings.21 pages, 14 figures