Papers

A $\mu$-constant deformation of isolated complex hypersurface singularities of dimensions different from 2 preserves the diffeomorphism class of the Milnor fiber. If it would be known that $\mu$-constant deformations have no vanishing folds as defined in by O'Shea, this result is even true in all dimensions since the members in the family would have a common Milnor ball-radius. In this note a symplectic version of this is shown which in particular applies to weighted homogeneous polynomial singularities.

Let $S_1$ and $S_2$ be rational elliptic surfaces with section over an algebraically closed field of characteristic zero, and let $X=S_1\times_{\mathbb{P}^1}S_2$. We determine the local equations produced by pairs of Kodaira fibers, including non-reduced fibers, and construct crepant resolutions for the admissible local models. For every construction we record whether it is projective and compute the number of irreducible components and the Euler characteristic of the resolved central fiber. Non-existence is asserted only in the normal Gorenstein cases covered by the terminal factorial obstruction; the remaining entries are stated as open for the methods of this paper. Under an explicit global compatibility hypothesis, the local data give formulas for the Hodge numbers of a smooth projective resolved fiber product. We also study Kummer quotients by fiberwise involutions. Their fixed curves are analyzed through the monodromy-orbit description of normalized fiber products of multisections. The final section records what the same constructions imply after reduction in characteristics $5$ and $7$, without interpreting characteristic-$p$ Euler and Picard data as complex Hodge numbers.

By studying wall crossings of Hassett moduli spaces of weighted pointed curves, we prove a recursive formula that reduces the computation of the cohomology of $\overline{\mathcal M}_{g,n}$ to that of the moduli space $\overline{\mathcal M}_{g,(0^+)^n}$ of stable pointed curves of minimal weights. For $g=1$, we further prove an explicit formula for the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,(0^+)^n}$. Combining these, we prove formulas that enable us to compute the cohomology of $\overline{\mathcal M}_{1,n}$ efficiently.

We establish the abundance theorem for log surfaces without assuming $\mathbb{Q}$-factoriality or log canonicity.

Let $f:X\to S$ be a smooth projective family, and fix a semisimple flat bundle on a fiber $X_0$. Its isomonodromic deformation determines a holomorphic section $\sigma_{\mathrm{dR}}:S\to M_{\mathrm{dR}}(X/S)$ of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a section $\sigma_{\mathrm{Dol}}:S\to M_{\mathrm{Dol}}(X/S)$, whose value at each $s\in S$ is the Higgs bundle corresponding to the flat bundle $\sigma_{\mathrm{dR}}(s)$. Unlike $\sigma_{\mathrm{dR}}$, the section $\sigma_{\mathrm{Dol}}$ is in general only real analytic. We study the geometric consequences of its holomorphicity along a complex analytic subvariety. First, we prove an absolute lifting theorem: the relative isomonodromic Higgs bundle admits an absolute Higgs lift, and the fiberwise harmonic metrics can be modified to solve the Hitchin–Simpson equation on the total space. Second, we prove that the spectral one-form on every resolved irreducible component of the relative spectral scheme is Gauss–Manin flat. As an application, we prove the nilpotent rigidity conjecture of Hu–Sun–Yang–Zuo: if the initial Higgs field is nilpotent, then the Higgs field remains nilpotent along the entire holomorphic isomonodromic locus.

Every cluster variety admits an action of its cluster dilation group. We prove that, in the case of braid varieties for simple Lie groups, this action always extends to a regular action on each of the brick compactifications. We explore two applications of this result. First, we show that any closed Richardson variety admits a faithful action of a torus of rank the Kazhdan-Lusztig $d$-invariant, answering affirmatively a recent question of E. Gorsky–S. Kim–M. Sherman-Bennett. The same result holds for projected Richardson varieties. Second, we show that the braid variety is a torus if and only if for each of its brick compactifications, the polar dual of the moment polytope for this action realizes the corresponding subword complex. The braid words satisfying this property turn out to be precisely the double root free words of V. Pilaud and C. Stump. This provides a novel approach to the longstanding open question of the polytopality of spherical subword complexes asked by A. Knutson and E. Miller, and in particular gives infinite families of subword complexes admitting polytopal realizations in dimension higher than the rank of the corresponding Coxeter group. As a common consequence of these two applications, we classify all Bruhat intervals in finite crystallographic Coxeter groups which are isomorphic to face lattices of convex polytopes via certain double root free words.

We study a particular instance of the frame work from \cite{Grieve:CM:Line:Slope:Stab}, building on earlier work of Arezzo-et-al \cite{Arezzo:DellaVedova:LaNave}, Ross and Thomas \cite{Ross:Thomas:2007}, \cite{Ross:Thomas:2006} and others, which applies the theory of the Chow-Mumford line bundle to determine the Donaldson-Futaki invariant of deformation to the normal cone test configurations with respect to big and nef line bundles. In particular, here we consider the case of projective $n$-space blown-up along a linear subspace. Our main result arises as an application of \cite[Theorem 1.1]{Grieve:CM:Line:Slope:Stab}. It establishes $\K$-semistability of the deformation to the normal cone test configuration for the blow-up $\PP(\Osh_{\PP^s}^{\oplus r} \oplus \Osh_{\PP^s}(1)) \simeq \operatorname{Bl}_{\PP^{r-1}}(\PP^s)$ along the exceptional divisor and with respect to the tautological line bundle $\Osh_{\PP(\Osh_{\PP^s}^{\oplus r} \oplus \Osh_{\PP^s}(1))}(1)$.

We prove the Eisenbud–Ulrich conjecture on the powers of linearly presented ideals in characteristic zero. More generally, we study stabilization bounds for powers of m-primary ideals in a polynomial ring with partially linear resolutions, making further progress toward the more general Eisenbud–Huneke–Ulrich conjecture. We prove a global generation result on relevant higher syzygy bundles, which is a crucial ingredient to the proof of the Eisenbud–Ulrich conjecture. We also establish lower bounds for the number of minimal generators in two settings: ideals generated by quadrics and ideals that are virtually linearly presented. Motivated by these results, we formulate a conjecture predicting sharp lower bounds for the number of minimal generators of such ideals.

We prove the Tate conjecture in every codimension on every power of an abelian variety of dimension at most four over a finite field. For geometrically simple fourfolds, we use Broe's theorem and the classification of Frobenius relations. For dihedral pairs of abelian surfaces, we construct algebraic classes using families of abelian threefolds, the Gross–Schoen height formula, and polarization contractions. We also prove the Hodge conjecture for every power of a complex CM abelian fourfold, using Markman's theorem on Weil classes and Milne's criterion. The finite field results imply standard conjecture~D and the expected pole orders of the zeta function on every power.

We study a deformation of the central heat trace on $\mathrm U(N)$ obtained by deforming Schur polynomials to Jack polynomials, and prove that it admits an asymptotic expansion to arbitrary order. Its coefficients are governed by elliptic $b$-Hurwitz numbers, which are genus-one counterparts of the $b$-Hurwitz theory of Chapuy and Dołęga \cite{ChapuyDolega22}. We construct the associated elliptic $b$-Hurwitz theory by means of generalized coverings on a torus and identify it with the genus-one closure of the genus-zero simple $b$-Hurwitz theory. At $b=0$ the construction recovers ordinary elliptic Hurwitz theory, while at $b=1$ it gives an automorphism-weighted geometric interpretation of the connected and disconnected twisted elliptic Hurwitz numbers of Hahn–Markwig \cite{HahnMarkwig26}. Our results extend the topological expansion obtained in the classical case in \cite{LemMai25,LM2} and take the form of a coupling between chiral and antichiral elliptic $b$-Hurwitz generating functions.

For a DM-type morphism of algebraic stacks $f \colon X \to Y$ with a relative perfect obstruction theory $\mathbb{E} \to \mathbb{L}_{X/Y}$ having a global resolution, we give a formula for the induced virtual pullback in terms of $\mathbb{E}$ and $f$. We apply our formula to prove functoriality properties of the virtual fundamental class in Gromov-Witten theory.

We develop a local-global theory for visibility of lattice points on families of parameterized curves. We introduce a notion of $p$-adic visibility and ask whether a lattice point is globally visible precisely when it is visible at every prime. We prove that this local-global principle holds for a broad class of families whose parametrizations are homogeneous with respect to positive weights. When the points on these curves fill the entire positive integer lattice, we compute the local and global densities of visible points and show that the global density is the product of the local densities. We then consider polynomial families of the form $y=qP(x)$, with $q\in\mathbb{Q}_{>0}$, and show that visibility can be detected prime by prime exactly when $P$ is a monomial. For non-monomial polynomials the local-global principle can fail, but the set of points where it fails has density zero; for separable polynomials we also obtain a quantitative bound for the number of non-visible points, improving the previously known bound. We further consider families whose lattice points lie on a proper lower-dimensional algebraic subset of the ambient space and show that their visibility densities can behave differently from those of the full lattice. Finally, we extend the theory from visibility from the origin to visibility from one lattice point to another and show that the corresponding local-global principle continues to hold for weighted homogeneous families.

We compute the Fourier expansions of the Kudla-Millson theta lifts of vector-valued Hilbert cusp forms of parallel weight. These expansions are with respect to 0-dimensional cusps of non-compact orthogonal Shimura varieties defined over any totally real number field, and computed using a mixed model of the adelic Weil representation under a partial Fourier transform. As a consequence, we obtain the injectivity of the theta lifts under some additional hypothesis on the variety. Furthermore, we show that the theta lifts are square-integrable harmonic differential forms, and deduce lower bounds for dimensions of cohomology groups of such varieties.

We define numerical adelic characteristic classes for vector bundles on projective varieties over number fields. The metric data are carried by the tautological quotient line bundle on the associated projective bundle. The construction uses the established intersection theory of adelic line bundles. Higher characteristic classes are multilinear intersection functionals. They are continuous for simultaneously controlled model sequences that are Cauchy for the supremum norm. We construct the resulting tautological numerical intersection algebra. We also prove controlled numerical Bogomolov–Gieseker inequalities. We first treat the case of a curve over $K$, where the numerical inequality is combined with the Deligne pairing and the adelic Hodge index theorem. We then pass to higher dimension, using Moriwaki's model-level dimension induction before taking the controlled Zhang limit. The curve theorem includes equality and uniform-gap criteria.

We provide first examples of border rank lower bounds obtained using border apolarity beyond “weak border apolarity”. Border apolarity introduced by Buczyńska-Buczyński characterizes tensors of border rank greater than $r$ as those whose annihilator does not contain an ideal with certain properties depending on $r$ and the ambient space of the tensor. Previous applications of border apolarity depended on the algorithm developed by Conner-Harper-Landsberg. For a tensor with a large symmetry group it lists a set of ideals that satisfy some of the properties mentioned above. We give first examples where the list is non-empty but we are able to prove that no ideal on this list satisfies all necessary conditions coming from border apolarity. The new lower bound examples include tensors of interest in complexity: the first Schönhage triple sum of matrix multiplication tensors where the border rank was unknown, and the direct sum of two copies of a tensor that could potentially be used to prove the exponent of matrix multiplication is two. A third example of interest in group theory related to work of A. Leitner is also analyzed.

We prove finiteness and Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems on proper $p$-adic rigid-analytic spaces in the framework of Banach–Colmez spaces and establish their optimal cohomological vanishing bounds. As a consequence, we also obtain ordinary finiteness and duality for their arithmetic pro-étale cohomology. We deduce these results from their analogs for perfect complexes on the Fargues–Fontaine curve, which in turn reduce to Poincaré duality for perfect complexes over period rings. We give a simple proof of the latter duality via the same diagrammatic argument as in our previous work for finite coefficients. Along the way, we establish optimal $v$-descent for perfect complexes over certain period rings on affinoid perfectoid spaces.

Let $Y$ be a smooth complex algebraic variety. We develop a tropical framework for studying algebraic loops, i.e., free homotopy classes in $Y^{\mathrm{an}}$ associated with formal loops $\mathrm{Spec}\, \mathbb{C}(\!(t)\!)\to Y$. When $Y$ is very affine, every formal loop determines an integral point of $\mathrm{Trop}(Y)$. For schön varieties, we prove that the associated algebraic loop is determined by this tropical point together with a connected component of the corresponding initial variety. Consequently, when all initial varieties are connected, integral tropical points classify algebraic loops. We apply this theory to the case when $Y$ is a hyperplane arrangement complement. In this case, we identify the initial varieties with complements of graded arrangements and obtain explicit representatives for algebraic loops as products of relative full twists. In type $A$, these representatives recover pure algebraic braids. For root arrangements, our construction gives a tropical interpretation of the split root valuation data of Goresky, Kottwitz, and MacPherson. Finally, we prove that the cohomology of braid varieties associated with positive algebraic braids depends only on their root valuation data.

We prove the expected asymptotic in Waring's problem over $\mathbb F_q[T]$, with a power-saving error, whenever $n>2d$, the characteristic is greater than $(d-1)^2$, and $q$ satisfies an explicit lower bound. This range is sharp in general for the expected asymptotic uniformly in the target polynomial. Our main new input is an aggregate minor arc estimate: we count functionals according to the codimension of their associated singular loci and use intersection theory to bound the degrees of the resulting parameter spaces. In particular, if $n\ge (2+\varepsilon)d$, the required lower bound on $q$ is polynomial in $d$ of degree $2+4/\varepsilon$.

Let $\pi:Z\rightarrow Y$ be a smooth projective morphism of complex manifolds with connected rationally connected fibers. We prove holomorphic approximation on arbitrary compact sets and finite-jet interpolation on arbitrary closed discrete sets for continuous liftings defined on open Riemann surfaces and holomorphic near those sets. For smooth projective morphisms of smooth complex algebraic varieties and algebraic base maps from smooth affine curves, the approximating liftings can be chosen algebraic, with interpolation on any finite set. In both cases the resulting lifting is homotopic to the initial one through continuous liftings of the fixed base map. For connected smooth projective complex manifolds, this gives the equivalence between the algebraic Oka-1 property and rational connectedness. Every rationally connected smooth projective complex manifold is also Oka-1.

We prove that for every smooth well formed weighted complete intersection of general type of dimension $n>0$, the Hodge number $h^{0,n}(X)$ is positive; in other words, its Hodge level is maximal. We also obtain an explicit lower bound for the geometric genus $p_g(X)$. This implies that the only weighted complete intersection of general type that is not a numerical intersection with a linear cone with $p_g(X)=1$ is $X_{6,6}\subset\mathbb{P}(1,2,2,3,3)$. We also prove the Ambro–Kawamata effective non-vanishing conjecture for smooth well formed weighted complete intersections.

We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-$p$ multisection in $\Sigma_g\times S^2$, we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees. We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$. For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of $S^2\times S^2$. Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups. We also construct the twisted ruled analogue of the $(2,4)$ case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes $2S_3-F_3$ and $4S_2-2F_2$ in the nontrivial $S^2$-bundles over $\Sigma_3$ and $\Sigma_2$, respectively. More generally, for a square-zero degree-$p$ multisection in the nontrivial bundle the complement has first homology $\mathbb Z^{2g}\oplus\mathbb Z/(p/2)$. Suitable gluings in the twisted $(2,4)$ case have $b_1=0$, $b_2=2$, and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of $\mathbb CP^2\#\overline{\mathbb CP}^{\,2}$. We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology $S^2\times S^2$, obtained via knot surgery and twisted fiber sums.

Let $W=W(B_n)$ act diagonally on $\mathfrak{h}\oplus\mathfrak{h}^*$, let $S=\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^*]$, let $J\subset S$ be the ideal generated by the $W$-alternating polynomials and $\mathfrak{m}_S$ is the maximal ideal of the origin. For sufficiently large $m$ we compute $q,t$-Fuss-Catalan polynomial $Cat^{(m)}(B_n;q,t):=Hilb(\frac{J^m}{\mathfrak{m}_S J^m})_{det-part}$ and imply $Cat^{(m)}(B_n;1,1)=\binom{n(m+1)}{n}$. For proofs, we work with the $\Gamma$-equivariant Hilbert scheme $Y_n=n\Gamma$-$Hilb(\mathbb{C}^2)$, $\Gamma=\mu_2$ and Haiman-type Koszul complex that defines the punctual locus of $Y_n$. Our formula for $Cat^{(m)}(B_n;q,t)$ is derived from a localization computaion for the Haiman-type Koszul complex.

In this article, we describe a new maximal function field $G$ over the finite field $\mathbb{F}_{11^2}$ with $11^2$ elements and show that it cannot be obtained as a subfield of the Hermitian function field. This provides the first known $\mathbb{F}_{p^2}$-maximal function field that satisfies this property. Further, we compute the automorphism group of $G$. The new maximal function field was found using AI.

Inspired by constructions of entire curves in Oka geometry, we construct rational curves on complex Fano manifolds analytically, by alternately deforming a holomorphic disk to reduce its area and enlarging its source. Positive Ricci curvature yields an area-decreasing deformation, while an affine-lift perturbation enlarges the source at a controlled area cost. The two operations change the area and the weighted derivative by amounts we estimate explicitly, and analytic compactness passes from the resulting disks to a holomorphic sphere. The main result produces a sphere that meets a prescribed compact fiber without being contained in it, with an explicit area bound.

We prove that a del Pezzo orbifold of degree less than 7, and del Pezzo orbifolds of higher degree with irreducible boundary are strongly Campana uniruled, and deduce that it is Campana rationally connected. This provides important non-toric examples in dimension two of a conjecture by Campana, in the form required by the weak approximation theorem, and yields weak approximation for Campana sections of Campana fibrations at all places whose general fibre is such an orbifold. In three appendices we classify the del Pezzo orbifolds of degree 6, 7 and 8 with irreducible boundary.

We classify commensurability relations between Deligne–Mostow–Thurston monodromy groups and the sixteen arithmetic Ghazouani–Pirio monodromy groups arising from moduli of flat cone metrics on the sphere and on the torus. In both settings, the monodromy group preserves a skew-Hermitian form coming from twisted homology. We use a degeneration method to compute their determinant classes in the genus one case. The defining CM fields and determinant classes of the skew-Hermitian forms determine the commensurability relations. In particular, each of the sixteen arithmetic Ghazouani–Pirio monodromy groups is commensurable with an arithmetic Deligne–Mostow–Thurston monodromy group.

We investigate the equal-characteristic case of the Moon–Taguchi bounded-conductor finiteness conjecture for mod $p$ Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field $K$ of characteristic $p$ and any integer $n \ge 1$, there are only finitely many isomorphism classes of continuous, semisimple, everywhere unramified, and geometric representations $\rho: G_K \to \mathrm{GL}_n(\overline{\mathbf{F}}_p)$ that admit an everywhere unramified characteristic-zero lift. Furthermore, we prove that the conjecture fails in general without this lifting hypothesis. Concretely, we construct infinitely many global function fields $K$ of characteristic $p$ admitting a continuous, surjective, absolutely irreducible, everywhere unramified, and geometric representation $\rho_r: G_K \twoheadrightarrow \mathrm{SL}_2(\mathbf{F}_{p^r})$ for each integer $r \ge 4$. As a corollary, we construct an everywhere unramified Galois extension \(L/K\), regular over \(\mathbf F_p\), with Galois group \[ \mathrm{Gal}(L/K)\cong \prod_{r\geq4}\mathrm{PSL}_2(\mathbf F_{p^r}). \] Combined with known cross-characteristic finiteness theorems, this completely resolves the question posed by Moon and Taguchi in dimension two: such extensions exist over global function fields of characteristic \(p\), whereas they cannot exist over global function fields of characteristic different from \(p\).

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato–Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

Generic vanishing conjecture proposed by Pareschi and Popa is about the relationship between the existence of subvarieties of codimension $>1$ of principally polarized abelian varieties representing minimal cohomology class and the positivity of their ideal sheaves twisted by the principal polarization. Inspired by their formulation, we prove a codimension $1$ analog of generic vanishing conjecture.

Recently, Jankowski introduced the Serre criteria for superschemes to define the super versions of normality and reducedness (called generic fermionic regularity and reducedness). We show that for a flat, proper morphism $f:X\rightarrow Y$, where $X$ is a superscheme and $Y$ is a Dedekind scheme of finite type over a field of characteristic $0$, the set of $y\in Y$ for which the fibers $X_y$ are normal, GFRR, regular, and Cohen-Macaulay is open in $Y$. In particular, if one has a degeneration over $\mathbb{A}_{\mathbb{C}}^{1|0}$, this shows that these properties can be lifted from the special fiber to the generic one.