Papers

We introduce virtual classes in the sense of Behrend-Fantechi, explaining how these directly generalize Fulton's localized top Chern class. We derive Siebert's formula for a virtual class on any Deligne-Mumford stack with a perfect obstruction theory having a global resolution.

We study density functions associated with filtrations of graded ideals in standard graded domains. Extending earlier work, we develop a general theory for arbitrary filtrations and establish their fundamental properties, including existence, continuity, log-concavity, and integral representations. We show that every density function is locally uniformly approximable by piecewise polynomial density functions arising from truncated Noetherian filtrations. We prove differentiability properties of saturation density functions and give a numerical characterization of when the symbolic powers of a graded radical ideal in a polynomial ring coincide with the integral closures of its ordinary powers. Finally, we establish several inequalities for the mixed multiplicities of equigenerated ideals.

Let $X\subset \mathbb{O}^n$ be a nodal hypersurface of degree $d$ with an ordinary $m$-fold point. Let $\delta_X$ be the largest value of $\delta$ such that $X$ is contained in the closure of the Severi variety of degree $d$ hypersurface in $\mathbb{P}^n$ with $\delta$ nodes. After recalling how the semicontinuity of the spectrum yields an upper bound for $\delta_X$, we produce various lower bounds for $\delta_X$. These bounds are given by polynomials in $m$ of degree $n$, with different leading coefficients. This improves previous bounds given by Ciliberto-Galati.

We study the geometry of Shimura varieties associated with a Hermitian form of signature $(p,q)$ over an imaginary quadratic field $E$, where $2\leq p\leq q$. We prove that when $p\geq w_E$ and $(p,q)\neq(2,2),(2,3)$, where $w_E:=\#\mathscr{O}_E^\times$, there exists a toroidal compactification with at worst canonical singularities. As an application, combining this singularity analysis with Arthur's multiplicity formula, we prove that only finitely many pairs $(p,q)$ satisfying the above conditions and $p+q\equiv1\pmod{w_E}$ give rise to unitary Shimura varieties that are not of general type. Our method also improves the singularity bound of Gritsenko–Hulek–Sankaran (Invent.\ Math., 2007) for $\mathrm{O}^+(2,n)$ to the range $n\geq6$ and shows that this bound is sharp.

In this paper, we prove Koszul duality for symmetric algebras of connective complexes consisting of locally free sheaves of finite rank, which is not necessarily bounded below. As an application, we construct an equivalence between derived categories of derived loop spaces and the symmetric algebra of the tangent complex for arbitrary separated schemes of finite type over a field $k$ of characteristic zero. This can be regarded as a generalization of the classical duality between derived loop spaces and cotangent bundles by Kapranov in the smooth cases. We also give a geometrical interpretation of our equivalence in terms of derived formal stacks.

For an algebraically closed field $k$ of characteristic $p>0$, we study unramified $\mathbf{Z}_p$-towers $\{X_n\}_{n\ge 0}$ of smooth, projective and connected curves over $k$. Our main result provides an Iwasawa-theoretic description of the Newton polygons of their Jacobians, establishing that the slopes of the new part at level $n$ are eventually affine functions of $p^{-n}$ with rational coefficients.

We study toric vector bundles on complete toric varieties whose total equivariant Chern classes are trivial. Our approach is tropical, using the notion of tropical toric vector bundles as piecewise linear maps introduced by Kaveh and Manon. We prove that any toric vector bundle of rank $r$ with trivial Chern class and affinely independent equivariant Chern roots is equivariantly isomorphic to a toric vector bundle pulled back from one of a finite set of varieties with dimension at most $r-1$ after twisting by a character. This extends a theorem of Payne about toric vector bundles of rank $r\leq 3$ with trivial Chern class. As an application, we construct examples of complete toric varieties of dimension $n$ that admit no nontrivial toric vector bundles of rank $r\leq n+1$ with the aforementioned properties. We also introduce combinatorial gadgets we call flag decorations of permutohedra, whose convexity properties are key for our results.

We study negative spheres obtained from exceptional sections of Gurtas Lefschetz fibrations. Starting with the known system of $4n$ disjoint $(-1)$-sections, we show that a marked double sum contains $4n$ symplectic $(-2)$-spheres and $2n$ smooth $(-4)$-spheres obtained by pairwise tubing. By carrying the same sections through a fourfold sum, we instead obtain $4n$ disjoint symplectic $(-4)$-spheres and hence symplectic rational blowdowns. We then consider the Lefschetz fibrations on knot-surgered elliptic surfaces. Their branched-cover description gives $2n$ branch sections together with simultaneous geometric duals arising from node resolution. In the knot-surgery double these sections glue to symplectic $(-4)$-spheres, while the duals make the complement of every subcollection simply connected. The resulting rational blowdowns provide simply connected exotic symplectic four-manifolds under the parity condition stated in the main theorem. We also record the boundary-multitwist relation determined by the Gurtas sections.

In this paper, we study supersingularity of modular Jacobians $J_0(N)$ and abelian varieties $A_f$ of $\mathrm{GL}_{2}$-type from both the vertical perspective (where the prime $p$ is fixed, and the level $N$ varies) and the horizontal perspective (where the newform $f$ is fixed, and the prime $p$ varies). Vertically, we prove that $J_{0}(N)$ is supersingular modulo a given prime $p$ for only finitely many levels $N$. Horizontally, we show that for sufficiently large $p$, the reduction of $A_f$ modulo $p$ is supersingular if and only if it isogenous to a power of a supersingular elliptic curve.

For a fixed prime $p$, let $\mu_p$ denote the $p$-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) $\mu_p$-equidistribution of the $p$-th Hecke eigenvalues over the Atkin–Lehner subspaces $S_k^\sigma(N) \subseteq S_k(N)$ and $S_k^{\operatorname{new}, \sigma}(N) \subseteq S_k^{\operatorname{new}}(N)$. We then highlight five applications of this explicit equidistribution result. For the first application, we generalize Kim's vertical analog of the Atkin–Serre conjecture to the Atkin–Lehner setting. For the second application, we obtain explicit bounds on the number of newforms $f \in S_k^{\operatorname{new}, \sigma}(N)$ for which $p$ is extremal over $S_k^{\operatorname{new}, \sigma}(N)$. For the third application, we prove explicit asymptotics for the number of $\mathbb{F}_{p^r}$-points on the modular Jacobian $J_0(N),$ as well as on its factors $J_0^{\operatorname{new}}(N),$ $J_0^\sigma(N),$ and $J_0^{\operatorname{new}, \sigma}(N)$. We also make explicit an asymptotic result of Serre concerning point counts of the modular curves $X_0(N)$. For the fourth application, we generalize lower bounds due to Murty and Sinha on the sizes of large $\mathbb{Q}$-simple factors of $J_0(N)$ to analogous bounds for $J_0^\sigma(N)$. Finally, for the fifth application (the details of which are given in a separate paper), we use our explicit equidistribution result to prove that only finitely many modular Jacobians are supersingular modulo any fixed prime.

We study the geometry of parameter spaces of homogeneous ideals in k[x,y,z] minimally generated by three forms of the same degree, stratified by the degree of their base scheme and by the initial degree of their syzygies. We realize these strata through Quot and Hilbert schemes, construct universal families of Bourbaki schemes, and prove smoothness and irreducibility results for the free loci. We then compare these spaces with the loci of gradient ideals and plane curves. Using slope semistability of the associated syzygy bundles, we give a vector-bundle interpretation of the refined du Plessis–Wall bounds and obtain the same numerical restriction for arbitrary three-generated ideals. Finally, in the case of triples of cubic forms, we determine all nonempty degree strata, prove that they are smooth and irreducible, and identify their gradient loci in terms of the Bourbaki stratification of reduced quartic plane curves.

We show how a counterexample to the Jacobian Conjecture can be found by elementary mathematics. The example we find is, apart from linear coordinate changes the example found by Levent Alpöge.

In this paper, we study logarithmic Donaldson-Thomas invariants for local log Calabi-Yau $4$-folds with simple normal crossing divisors. Using the family version of shifted Lagrangian classes announced by Khan-Kinjo-Park-Safronov, we construct relative and family logarithmic $\mathsf{DT}_4$-theories for logarithmic Hilbert schemes of curves, and prove a degeneration formula. For logarithmic Hilbert schemes of points, we construct virtual classes and prove a degeneration formula in Chow groups; in particular, it is independent of shifted Lagrangian classes. Finally, combining our degeneration formula with logarithmic cobordism, we explicitly compute the zero-dimensional logarithmic $\mathsf{DT}_4$-invariants for local surface snc pairs.

In the moduli space of abelian varieties, Prym loci parametrise Prym varieties associated to covers of curves with fixed degree and monodromy data. We derive and implement an algorithm computing the tautological projections of classes of Prym loci. The key input is a comparison formula relating three different Hodge bundles on moduli spaces of admissible covers. We present computer and hand calculations for double and triple covers, with the locus of hyperelliptic Jacobians as a special case. Two applications follow. First, we calculate the degrees of the Prym maps in the codimension-zero cases, unifying a series of previously ad hoc results. Second, we prove in low genera the non-divisibility of the Jacobian class by the Prym class.

We introduce bicomplex fiber bundles and develop a framework for bicomplex characteristic classes and almost bicomplex structures on smooth manifolds. In the first part, we present the basic definitions and properties of these bundles, including their fundamental idempotent decomposition and the correspondence between their isomorphism classes and homotopy classes of maps into the bicomplex infinite Grassmannian. We then define bicomplex characteristic classes and show that, in general, they are not determined by the Chern classes of the underlying complex bundle. We also construct the bicomplex Chern character and discuss its algebraic properties. Finally, we apply the resulting theory to almost bicomplex manifolds, providing examples and obstructions to the existence of such structures.

We compute the geometric monodromy factorization of Xiao's genus-two Lefschetz fibration directly from its branched-cover construction, following Moishezon's braid-monodromy method. The complete quadrangle determines the motion of the six branch points and gives four nonseparating and three separating vanishing cycles. At each of the three $3+3$ degenerations, the branch motion determines the spherical mapping class, and Xiao's local holomorphic model determines its genus-two lift. After fixing a distinguished system of paths, we obtain an ordered positive factorization of type $(4,3)$ and give Artin coordinates for the seven factors.

Overconstrained rational single-loop linkages are efficient, compact, and low-cost custom mechanisms, yet their deployment in industrial settings is limited. In simulations, rigid body formulations fail due to redundant constraints. This study presents a flexible multibody modeling framework based on the floating frame of reference formulation, and delivers an overall accuracy analysis of assembled linkages prototypes. The approach is validated against 3D-printed PLA prototypes of a Bennett four-bar mechanism, including variants with intentional joint-axis misalignment, which theoretically, from the rigid body point of view, cannot be assembled. A supplementary contribution is delivered in the form of a kinematic parameter identification methodology suited for this type of mechanism with ill-conditioned Jacobian. The experimental and simulation results are compared and reveal that these overconstrained mechanisms exhibit a self-assembling tendency – structural compliance drives the assembly toward the ideal geometric configuration, distributing constraint stress throughout the structure. Additional qualitative demonstrations using cardboard tubes and bamboo sticks as link building blocks confirm that functional mechanisms can be realized from low-cost, unconventional materials with limited manufacturing accuracy. The proposed modeling pipeline is fully algorithmic and enables design optimization in the future.

We study the numerical geography and tangent geometry of very amply polarized Calabi–Yau threefolds $(X,H)$ through the positivity of the first jet bundle $J^1(H)$. Writing $d=\int_X H^3$, $c=\int_Xc_2(X) H$, and $e=\int_X c_3(X)$, we exploit two different positivity properties of this single bundle. Mixed intersections on $\mathbb{P}(J^1(H)^*)$ give $e\ge-5d-c-c^2/(4d)$, while a volume estimate for a perturbed tautological class gives $e\ge40d\left[(1-\frac{c}{10d})^{3/2}-1\right]$; in particular $e+6c\ge0$, improving Sun's inequality $e+10c\ge0$. As consequences, we obtain the uniform Hodge bounds $-4d-80\le h^{1,1}(X)-h^{2,1}(X)\le173d/66$, the lower bound $\mathrm{deg}X^\vee\ge78$ for the dual hypersurface, and, in the critical case $X\subset\mathbb{P}^6$, the upper bound $d\le34$. We also prove that, for every $m\ge2$, the tangent-incidence morphism associated with $|mH|$ is the normalization of the tangent variety, conjecture tangent birationality for complete embeddings $X\subset\mathbb{P}^N$ with $N\ge7$, and verify it for several families.

Let $C$ be a connected nodal curve of genus $g$ over an algebraically closed field $k$ of characteristic zero. Let \[ A(C)=\Big(J(C), {\overline{P^{g-1}_C}}, \Theta(C) \Big) \] be the stable semi-abelic pair (in the sense of Alexeev) associated with its canonical compactified Jacobian $\overline{P^{g-1}_C}$ in degree $g-1$. The canonical triple of $C$ consists of the closure of its canonical image together with two sets of distinguished points encoding the information of the images of the nonseparating nodes and of the branch data of the canonical map. We prove a functorial reconstruction: given connected nodal curves $C$ and $C'$ over $k$, every isomorphism $A(C)\cong A(C')$ of stable semi-abelic pairs canonically induces a projectivity between the corresponding canonical triples of $C$ and $C'$. Passing to isomorphism classes, this recovers the reconstruction statement in the compactified Torelli theorem of Caporaso-Viviani. Our proof extends Andreotti's Gauss-map strategy. The branch locus of the projection from the normalization of the Gauss graph recovers the nonlinear canonical components and branch data by projective biduality; boundary strata of the compactified Jacobian recover the node-images; and incidence arguments recover the linear components. Along the way, we repair technical gaps in the classical treatments of Andreotti's proof of Torelli's theorem for smooth curves.

Let G be a non-trivial finite subgroup of SL_2(C) and let p : X \to C^2/G be the minimal resolution. We prove the C^*-equivariant Cohomological Crepant Resolution Conjecture for the ADE orbifold [C^2/G]. More precisely, after specializing the quantum parameters at suitable roots of unity, we prove that the Bryan-Gholampour transformation induces an isomorphism between the C^*-equivariant quantum cohomology of X and the C^*-equivariant Chen-Ruan cohomology of [C^2/G]. Our proof is direct and is based on the McKay correspondence, ADE root systems, and character theory. Following the character-theoretic form of the Bryan-Gholampour change of variables, we use the McKay correspondence to diagonalize the transformation and to reduce the compatibility of the products to a uniform root-system identity. In type A, this reduction admits a discrete Fourier realization. In type D, the cyclic subgroup of the binary dihedral group leads instead to finite sine transforms, while the exceptional cases E_6,E_7,E_8 are treated by exact matrix computations over the corresponding cyclotomic fields. As a preliminary result, for a symplectic complex vector space V and a finite subgroup G of Sp(V), we give an explicit presentation of the C^*-equivariant Chen-Ruan product of [V/G] and identify the resulting algebra with the Rees algebra of the center of the group algebra with respect to the age filtration. In particular, the equivariant Chen-Ruan algebra interpolates between the center of the group algebra and its associated graded algebra, the latter recovering the ordinary Chen-Ruan cohomology.

Explicit computations of generic infinitesimal Newton–Okounkov bodies are difficult even for varieties with simple product structure. We give an explicit formula in arbitrary dimension for positive-degree external tensor products on products of smooth projective curves. Writing $d^\downarrow=(d_1^\downarrow,\ldots,d_n^\downarrow)$ for the decreasing rearrangement of the degree vector and setting $d_{n+1}^\downarrow=0$, the body admits the explicit Minkowski decomposition $\sum_{j=1}^n(d_j^\downarrow-d_{j+1}^\downarrow)S_j^{(n)}$, where the $S_j^{(n)}$ are the embedded simplices defined below. Using this description, we give a sharp criterion for equality in the Minkowski inclusion. A simultaneous relabeling argument also allows finitely many such bodies to be realized on a common very general locus of flags after independent decreasing rearrangements of their degree vectors.

We describe transversal singularity types of the singular locus of the $A$-discriminant for $A\subset\mathbb Z$, and deduce a Whitney type theorem for a coordinate projection of a surface defined by a general polynomial equation with a given Newton polytope $N$: under mild combinaorial conditions on $N$, all multisingularities are stable (folds, cusps, and double folds). We then enumerate the multisingularities in terms of $N$. The results rely on the analysis of degeneracy of relevant Vandermonde/Schur type matrices, which may be of independent interest.

Let $p>3$ be a prime number. We prove that every principally polarized superspecial abelian surface over $\mathbb{F}_{p^{2}}$ with $p^{2}$-Frobenius $[-p]$ admits a separable polarized isogeny to its Frobenius conjugate with multiplier at most $(p^{3}/2)^{1/5}$. This generalizes a recent result of Aubry, Oyono, and Vincent (2026, arXiV:2607.14624) who proved that every supersingular elliptic curve defined over $\bar{\mathbb{F}}_{p}$ admits an isogeny to its Frobenius conjugate with degree at most $(p/2)^{1/3}$.

Conner, Gesmundo, Landsberg and Ventura (2019) determined the largest stabilizer dimension of concise $n\times n \times n$ tensors that are binding, and they determined the corresponding maximizing tensors to be the null algebra tensors. They left as an open problem to extend this to all concise $n \times n \times n$ tensors (i.e. dropping binding). We solve this problem: We prove that the largest stabilizer dimension of concise $n\times n\times n$ tensors is $n^2 + 1$ and the maximizers are the null algebra tensors (as in the binding case) and the skew symmetric tensor $e_1 \wedge e_2 \wedge e_3$. As part of our approach we obtain upper bounds on the stabilizer dimension of matrix tuples under left-right action (generalized Kronecker quiver representations), which we think are of independent interest.

We use an analogue of the Springer resolution to describe the $G$-module structure on the ring of regular functions on the universal cover $\widetilde{\mathcal{O}}$ of any nilpotent orbit for $G = SL_n$. Building on previous work on the extended Springer resolution, we construct a variety $\widetilde{\mathcal{M}}$ that is finite over the cotangent bundle of a partial flag variety $G/P$, and proper and birational over the affinization $\mathcal{M}$ of $\widetilde{\mathcal{O}}$. We use techniques in birational geometry to show that $\widetilde{\mathcal{M}}$ has rational singularities, which provides the cohomology vanishing needed to describe the ring of functions on $\widetilde{\mathcal{O}}$ as an induced representation from a Levi subgroup of $G$. Our results also yield a description of the structure of $R(\widetilde{\mathcal{O}})$ as a graded $G$-module. We describe the minimal embedding of $\mathcal{M}$, study the lifting of characters of the component group of $\widetilde{\mathcal{O}}$ to parabolics and Levi subgroups, and make a more general vanishing conjecture.

Lara, Srinivas and Stix showed that the étale fundamental group of a proper scheme over an algebraically closed field is topologically finitely presented. Building on this, we show that the étale fundamental group, as introduced by Noohi, of a proper algebraic stack with finite inertia over an algebraically closed field is topologically finitely presented.

We prove the cohomological Gopakumar-Vafa/Pandharipande-Thomas correspondence for the local plane and quadric in every effective curve class and Euler characteristic. The strict supports of the simple perverse constituents of the stable pair vanishing cycle direct images are closures of loci of transverse unions of smooth connected curves. Wall-crossing and duality leave a finite range of Euler characteristics to consider. In this range, a commutation relation for point modifications on bounded stacks of surface pairs excludes all other supports. Applying the family Macdonald formula on the reduced locus gives the correspondence as an identity of semisimplified perverse direct images on the Chow variety.

We extend Behrend's theory of complementary polyhedra to include Jordan-Hölder theory. This provides a uniform approach to describing the Jordan-Hölder filtrations and S-equivalence in moduli stacks of bundles, following Zhang. As an application, we give a description of Jordan-Hölder reductions and S-equivalence in the semistable locus of parahoric torsors and parahoric Higgs torsors, coming from their good moduli spaces constructed by Alper-Halpern-Leistner-Heinloth and Réga. This application uses the Rees construction to identify filtrations of moduli stacks with reductions of bundles, and induces a Jordan-Hölder stratification of the moduli stacks.

We construct relative Du Bois and relative intersection Du Bois complexes for smooth-factorizable morphisms to a smooth complex base using relative de Rham complexes of mixed Hodge modules and Grothendieck duality. For a smooth morphism, these complexes recover the relative Kähler differentials. Comparisons with the relative Du Bois complexes of Kovács and Ning remain open. We construct finite filtrations on the absolute Du Bois and intersection Du Bois complexes whose graded pieces are the corresponding relative complexes tensored with differential forms from the base. We also study restriction to fibers and its relation to symbolic $m$-Du Bois and weak $m$-rational singularities.

We establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into a projective variety and a family of hypersurfaces in subgeneral position. The above bound of the total defect obtained from our result is better than that of the previous results. Moreover, in our result, the truncation level of the counting functions is estimated explicitly and independently of the number of hypersurfaces. Especially, we do not need the assumption that the family of hypersurfaces must satisfy the Bezout properties as imposed in some earlier studies, but only a weak Bezout property.