Papers

We show that the nilpotent cone associated to a simply connected semisimple algebraic group, together with all its crepant projective partial resolutions, can be constructed as Hamiltonian reductions of the affine closure of the cotangent bundle of base affine space for suitable choices of stability parameter of a maximal torus. We also realize the base change of the universal Poisson deformation of each of these crepant projective partial resolutions as the GIT quotient of the affine closure of the cotangent bundle of base affine space at the same stability parameter.

We study the positivity of the tropical multidegrees of a tropical variety contained in a product of real vector spaces. These multidegrees are obtained by stably intersecting the tropical variety with pullbacks of positive tropical divisors. We introduce projection-purity and facet-selectability, two conditions under which positivity is determined by the dimensions of the natural projections, and the support of the tropical multidegrees is precisely the set of lattice points of a polymatroid base polytope. This extends He's theorem for translation-admissible tropical varieties. We also show that these conditions alone do not force the corresponding tropical volume polynomial to be Lorentzian. By contrast, for the augmented Bergman fan of any polymatroid, the positive multidegrees are supported precisely on the lattice points of the polymatroid base polytope, and the tropical volume polynomial is Lorentzian for every sequence of positive tropical divisors.

Suppose we are given an arbitrary line bundle $\mathcal{L}$ on a complex algebraic variety $X$, not necessarily smooth. For a positive integer $N$, suppose there exists a global section $s \in \Gamma(X, \mathcal{L}^{N})$ that defines an effective Cartier divisor $D$. If we denote $\pi: Y \rightarrow X$ to be the $N$-fold cyclic covering of the divisor $D$ resulting from the global section $s$, we describe the cyclic covering in terms of Morihiko Saito's theory of mixed Hodge modules.

$X$ is a Nakajima quiver variety. In this note we report the observation that the space parameterizing quasimaps from $\mathbb{P}^1$ to $X$, sending infinity to a given point, may be presented globally as the critical locus of an explicit function.

Let $X$ be an $n$-dimensional generalized Grassmannian not isomorphic to $\mathbb{P}^n$. We prove that $k(X)\le n-1$, where $k(X)$ denotes the maximal integer such that every uniform bundle on $X$ of rank at most $k(X)$ is homogeneous. In particular, for smooth quadrics $\mathbb{Q}^n$, we have $k(\mathbb{Q}^n)=n-1$ for odd $n$, and $n-2\le k(\mathbb{Q}^n)\le n-1$ for even $n$. We classify uniform rank $n$ bundles on $\mathbb{Q}^{n}$ for $n=3$, $5$. Furthermore, we characterize projective spaces among generalized Grassmannians in terms of uniform bundles.

We present a framework for studying the moduli space of linear determinantal representations for flat families of complex projective cubic surfaces across singularity boundaries. First, we deal with the classical deformation theory where a surface S_0 possesses a rational double point (RDP) of type E_6. By establishing a global simultaneous resolution over a finite ramified Galois covering of the global parameter slice, we realize the relative moduli space H_N as a diagonal Weyl quotient (h x R)/W(E_6). Second, we extend this classification to the critical boundary configuration where S0 contains a unique isolated simple elliptic singularity of type E~_6. Because the local monodromy group becomes infinite, the classical simultaneous resolution framework completely breaks down. We bypass this obstruction by constructing a geometric substitute Z derived from the filtration of Looijenga's invariant algebra of affine Weyl groups. We also show that the vector bundle Z decomposes into a direct sum of line bundles over the structural elliptic curve E, with degrees explicitly matching the negative Coxeter marks of the highest root of E_6. Utilizing Riemann's extension theorem and vector bundle rigidity over elliptic curves, we settle our isomorphism result, proving that the moduli space H-bar representing the semiuniversal family is globally isomorphic to the total space of Z.

We construct certain special rational elliptic surfaces with four reduced singular fibres that are naturally attached to the platonic solids and belong to a group of surfaces complementing the well-known semi-stable Beauville surfaces. We describe the basic algebraic geometric properties of these surfaces, their associated Picard-Fuchs operators, integer sequences, Laurent polynomial representations and Apéry constants. On the arithmetic side, we discover the need for a refinement of the Dwork-expansion used to find the Euler factors of the fibres of these fibrations. This is explained by comparing the $q$-coordinates coming from the elliptic curve and the $q$-coordinate of the Picard-Fuchs equation and is also directly reflected in the shape of the monodromy matrices in the Frobenius basis.

Different instances of the Riemann-Hilbert correspondence relate initial value spaces of Painlevé equations to affine varieties known as monodromy surfaces, built from monodromy invariants for associated linear ODEs. We show that these affine varieties admit realisations as embedded affine del Pezzo surfaces, characterised by their degree and a prescribed divisor at infinity, first in this paper for cases associated with Painlevé equations $\rm{VI},\rm{IV},\rm{II},\rm{I}$. We prove that the monodromy groups of these monodromy surfaces form the finite parts of the affine Weyl symmetry groups of the corresponding Painlevé equations. We realise the monodromy group in each case: analytically as permutations of lines induced by continuation along loops in parameter space, Galois-theoretically in terms of the function field of the incidence variety of lines, and combinatorially via their intersection graph. This in particular yields an interpretation of the parameter spaces of Painlevé equations, modulo symmetries, as moduli spaces of categories of embedded affine varieties. Further, it shows that, despite the affine Weyl group symmetries becoming trivial when conjugated by the corresponding Riemann-Hilbert map, the finite Weyl group part survives as monodromy of the monodromy surface.

We prove that the Bernstein-Sato polynomials of ideals satisfy the multiplicative Thom-Sebastiani property. This affirmatively settles a question of Shi and Zuo that extends a question independently formulated by Budur and Popa.

Let $X$ be a complete toric variety and let $e$ be the identity of its open torus. The Cox ring of ${\rm Bl}_eX$ is described by saturated powers of the toric-point lattice ideal $I_X$. We give a finite criterion for generation in Rees multiplicity one in terms of analytic spreads of projected lattice ideals and zero divisors in the associated graded ring. We apply this criterion first to fake weighted projective spaces, proving that multiplicity-one generation is equivalent to $I_X$ being a complete intersection. For projective toric surfaces, we prove that multiplicity-one generation holds if and only if it holds for every Picard-number-one toric surface dominated by $X$.

We introduce two new arithmetic invariants associated with Hermitian line bundles on arithmetic varieties, which play the role of the Bergman distortion function in the arithmetic setting. As a first step, we show that these two functions are asymptotically equivalent. This leads to the introduction of an intrinsic measure $ \widehat{\mu}^{\sup}_{\mathrm{eq}},$ which we show to be a natural arithmetic analogue of the equilibrium measure. We prove that $\widehat{\mu}^{\sup}_{\mathrm{eq}}$ serves as a detector of arithmetic positivity. In particular, we determine it for weakly nef Hermitian line bundles and, in full generality, in the toric case. Moreover, $\widehat{\mu}^{\sup}_{\mathrm{eq}}$ governs the arithmetic volume function and its variational properties. More precisely, we establish weak integral representations of the arithmetic volume and of its first variation in terms of $\widehat{\mu}^{\sup}_{\mathrm{eq}}$. These formulas provide a measure-theoretic interpretation of the volume derivative of Yuan–Zhang and reveal a structural parallel with variational formulas in complex pluripotential theory. Our approach differs fundamentally from previous treatments based on Harder–Narasimhan filtrations or arithmetic Okounkov bodies. It places the arithmetic volume at the center of the theory and suggests that asymptotic volume invariants, rather than heights, provide the natural analytic framework for problems in equidistribution and arithmetic dynamics.

Ellingsrud, Göttsche, and Lehn proved a remarkable universality result for tautological integrals on Hilbert schemes of points on surfaces. We prove a relative form: over a base $B$, the tautological pushforwards are governed by universal power series paired with relative $\kappa$-classes. As applications, we focus on Segre classes on families of curves and surfaces. For curve families, we determine all universal coefficients for total Segre pushforwards. These formulas answer a question of Oprea–Pandharipande, and the resulting recursions are surprisingly related to monotone Hurwitz numbers. For surface families, motivated by Marian–Oprea–Pandharipande's work on Lehn's conjecture, we obtain a codimension-one relative form of the conjecture, with explicit formulas for every universal series.

We study quasi-Galois points, which are a generalization of Galois points. We characterize smooth quartic surfaces admitting a quasi-Galois point in terms of K3 surfaces with a certain involution, and provide a criterion for quasi-Galois points for quartic surfaces. Moreover, we also determine all quasi-Galois points of the Fermat quartic surface and show that it has exactly 28 quasi-Galois points.

Let $K$ be a $p$-adic field. We study the arithmetic theory of abeloid varieties over $K$. Our aims are twofold. First, we study the theory of rigid analytic 1-motives, which will be viewed as a tool to describe degeneration of abeloid varieties, similarly as in the classical algebraic setting. Our key new results are the equivalence between formal (resp. log formal) 1-motives over $\mathcal{O}_K$ and rigid analytic 1-motives with good (resp. semi-stable) reduction over $K$, and the Néron-Ogg-Shafarevich criterion for the good (resp. semi-stable) reduction of rigid analytic 1-motives. In particular, we construct log formal 1-motives and log $p$-divisible groups over $\mathcal{O}_K$ from semi-stable abeloid varieties over $K$. Next, we study the conjugate uniformization of an arbitrary abeloid variety $A$ over $K$. This is a type of $p$-adic uniformization initiated by Iovita–Morrow–Zaharescu in case of abelian varieties with good reduction. Our approach here is based on Fargues' theory of $p$-divisible rigid analytic groups. In fact, we view the theory of conjugate uniformization as a study of rational points of dualizable $p$-divisible rigid analytic groups in terms of their classification Hodge–Tate triples. Along the way, we construct $p$-divisible rigid analytic groups from rigid analytic 1-motives.

The blowup equations of Huang, Sun and Wang are bilinear relations satisfied by the refined topological string partition function of a local Calabi–Yau 3-fold. We prove the blowup equations for the local Hirzebruch surfaces \(\operatorname{Tot}_{\mathbb F_\ell}K_{\mathbb F_\ell}\), \(0\leq\ell\leq2\), as identities of torus-equivariant symmetrized \(K\)-theoretic stable pair invariants; for \(\ell=2\) the invariants are localized indices. The proof identifies the stable pair vertex sum, after division by the fibre contribution, with the equivariant Euler characteristic of \((\det\mathcal V)^\ell\) on the moduli space of framed rank \(2\) sheaves on \(\mathbb P^2\). The blowup formulas of Nakajima–Yoshioka for framed sheaves then yield the unity and vanishing equations. For local \(\mathbb P^2\) we obtain the blowup equations conditionally on two explicitly stated conjectures. We also state the Huang–Sun–Wang conjecture for general local Calabi–Yau 3-folds in the language of stable pairs, and formulate stable pair conjectures for the local rational elliptic surface involving the \(E_8\) lattice.

Let $X$ be a smooth projective curve and let $G$ be a reductive group over an algebraically closed field of characteristic zero. We compute the image of arbitrary finite-rank local systems on $\mathrm{Bun}_G(X)$ under the geometric Langlands equivalence, confirming a conjecture of V. Lafforgue in the case of the constant sheaf.

Generalized Grassmannians form a fundamental class of flag manifolds associated with Lie groups. The purpose of this paper is to classify self-maps of generalized Grassmannians of nonzero degree in terms of their induced actions on cohomology. We prove a rigidity theorem showing that, despite the rich and intricate structure of their cohomology rings, the induced cohomology endomorphisms fall into only two natural types: Adams operations determined by the degree and Dynkin symmetries arising from automorphisms of the Dynkin diagram. Combining the geometry of root and weight systems, the actions of Weyl and Dynkin symmetries on Schubert classes, and Bott–Samelson desingularizations of Schubert varieties, our approach applies uniformly to generalized Grassmannians of all Lie types.

Let $G$ be the topological fundamental group of a nonsingular complex projective surface. Troncoso and Urzúa proved that the Chern slopes $c_1^2/c_2$ of minimal surfaces of general type $S$ with $\pi_1(S)\simeq G$ are dense in $[1,3]$, and left $[1/3,1)$ open. We prove that they are dense in $[1/2,3]$, an interval that cannot be enlarged without contradicting either a theorem of Mendes Lopes and Pardini or Reid's conjecture. We prove more: the slopes of such surfaces with $K_S$ ample are dense in $[1/2,2]$, the first case in which the conjecture of Troncoso and Urzúa on ample canonical classes is established. The tool is an exact ampleness criterion for their product construction, which shows in particular that their own surfaces never have ample canonical class, whatever the defining sections.

Let $k$ be a field of characteristic zero, let $R$ be a commutative $k$-algebra, let $I$ be a proper ideal of $R$, and assume that $A=R/I$ is Noetherian. We prove that if $a=[f]$ in $A$ is nilpotent and $d_{A/k}a=0$, then $I:f$ is not contained in $(I,f)$. If $R$ is local with maximal ideal $\mathfrak n$, then $I:f$ is not contained in $(I,f)+\mathfrak n(I:f)$. As applications, we prove the isolated hypersurface case in arbitrary dimension and extend the conclusion to non-isolated critical loci on smooth varieties. Finally, we exhibit a five-variable isolated singularity satisfying $J_f:f\subseteq\overline{J_f}$, thereby disproving the integral-closure strengthening.

This article arose out of notes for a CIME summer school mini-course in Cetraro. In it we develop differential-geometric constructions on the moduli space ${\cal M}_{g,n}^{\rm spin}$ of smooth spin curves, with particular emphasis on their interpretation in hyperbolic geometry. We relate the locally constant sheaf arising from a hyperbolic spin structure to the corresponding holomorphic sheaf, extending this known relationship to spin curves with Ramond marked points. We use the hyperbolic description to construct characteristic forms and spin measures on moduli spaces of hyperbolic surfaces with geodesic boundary, providing a framework for applying Teichmüller-theoretic methods to their volumes and volume recursions. Finally, we study separately the even and odd components of spin moduli space and find symmetries between their volume contributions which are not apparent from the contributions of individual boundary strata.

We provide a spectral geometric description of the KdV variational equation around cnoidal waves in terms of two interacting spectral curves.

Hessenberg varieties $\mathcal{H}\mathrm{ess}(\mathsf{X},\mathbf{h})$ are subvarieties of the flag variety parameterized by a Hessenberg function $\mathbf{h}: [n] \to [n]$ and a matrix $\mathsf{X} \in \mathfrak{gl}_n(\mathbb{C})$. In recent work, Goldin and the second author showed the existence of flat degenerations of Hessenberg varieties to nilpotent Hessenberg varieties over the minimal sheet. This implies that all Hessenberg varieties over the minimal sheet have the same dimension. Our main result generalizes this dimension result to arbitrary sheets. Specifically, we prove that for a fixed Hessenberg function $\mathbf{h}:[n] \to [n]$, all Hessenberg varieties $\mathcal{H}\mathrm{ess}(\mathsf{X},\mathbf{h})$ defined in the type $A$ flag variety by linear operators $\mathsf{X}$ from the same sheet of the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$ have the same dimension.

Standard neural architectures often fail to generalize to longer inputs for dynamic programming (DP) targets. We investigate what makes this hard geometrically. Every finite min-plus DP is a shortest path on a DAG, which is equivalently a tropical polynomial whose extended Newton polyhedron encodes the decision boundary of which path wins. We prove these three descriptions (graph, polynomial, polyhedron) form isomorphic semirings at two levels — formal polynomials and their computed functions — connected by operations that characterize all structural redundancies. We then address the length-generalization question geometrically: does the decision boundary at length $T$ decide the boundary at $T+1$? We present two structural negatives. The semiring's two native ways to reduce dimension (setting a variable to each identity) are neither injective nor always closed within the DP. Series and parallel composition fail to construct all DAG topologies from smaller sub-DAGs, and even all terminal-only operations do not capture all DP compositions.

Let $X$ be a smooth projective surface and let $C\subset X$ be a unicuspidal rational curve with one Puiseux pair $(p,q)$. We use the log minimal model program to prove a sharp upper bound on the self-intersection of $C$ conjectured by Weimin Chen in [C, Conjecture 1.15] in the course of his study of contact structures and their symplectic fillings. We then extend this argument to rational curves with arbitrary cuspidal singularities and compare the resulting algebraic bounds with the symplectic bounds of Golla and Starkston [GS]. Finally, we prove that there is no plane curve of degree $102$ and genus $10$ with a $(36,289)$-cusp, answering a question of Evans [E, Remark 7.3.5] about algebraic curves associated with the final post-Fibonacci step of the McDuff-Schlenk staircase.

We develop a geometric framework for constructing and organizing Kochen–Specker configurations in real four-dimensional space. The construction uses finite grids on pairs of smooth quadrics related by Euclidean polarity. Their incidence geometry directly produces orthogonal measurement contexts and parity proofs of quantum contextuality, yielding infinite families of configurations and a geometric interpretation and extension of a previously known cyclic construction. For a distinguished subfamily, the same geometry admits a canonical completion determined by secant lines. The first two instances of this completion recover the exceptional root configurations of types $F_4$ and $H_4$, while the smallest case also recovers the Cabello configuration and its embedding in the Peres configuration. Thus several prominent four-dimensional contextual configurations, previously obtained from different constructions, arise from a single projective-geometric mechanism.

We prove three results about refined $SU(N)$ VW invariants on Hirzebruch surfaces. First, we crucially simplify the modular anomaly equation satisfied by the generating series of these invariants by expressing it in terms of complementary generalized error functions, which requires extending these error functions to their singular loci. Second, we evaluate the variation of the modular completions of the generating series upon change of polarization. Finally, a representation of the generating functions in terms of indefinite theta series allows us to compute VW invariants at the walls of marginal stability, and to express them in terms of the invariants on the two sides of the wall and those of lower ranks. All the results exhibit a universal structure suggesting their wider applicability.

Equivariant localization in the $\Omega$-background is a powerful technique for the evaluation of the partition function of $\mathcal{N}=2$ supersymmetric field theory on a toric four-manifold. For the complex projective plane $\mathbb{CP}^2$, we apply equivariant localization to supersymmetric Yang-Mills theories with gauge groups $SU(2)$ and $SO(3)$ with $N_f$ massive hypermultiplets in the fundamental representation, and the $\mathcal{N}=2^*$ theory with the hypermultiplet in the adjoint representation. We evaluate equivariant correlation functions, which provide an equivariant extension of intersection numbers of moduli spaces of instantons on $\mathbb{CP}^2$, such as Euler numbers and Segre invariants. In the non-equivariant limit, we compare our results with the evaluation using low energy field theory and integration over the $u$-plane. Using this comparison, we identify a specific overall factor, which captures the difference between a partition function for gauge group $U(2)$ and $SU(2)$ or $SO(3)$ on a compact four-manifold. We also discuss our results in the context of $S$-duality of the $N_f=4$ and $\mathcal{N}=2^*$ theory, which includes the triality automorphism group of the flavor group for $N_f=4$.

The supermoduli stack ${\mathfrak M}_g$, parametrizing smooth unpunctured super Riemann surfaces of genus $g$, is known to be non-projected for $g \ge 5$. We extend this result to the even-spin component of ${\mathfrak M}_4$. This is done by exhibiting an appropriate compact curve in the spin moduli space ${\mathcal S\mathcal M}_4^{+}$, and computing the restriction to it of the class obstructing the projectedness of ${\mathfrak M}_4^{+}$. This restriction turns out to be non-zero, proving that ${\mathfrak M}_4^{+}$ is non-projected.

For a hyper-Kähler fourfold $X$ of generalized Kummer type, we construct an infinitesimal deformation of the tangent bundle $T_X$ with non-zero Yoneda square. This is the first example of a stable hyperholomorphic bundle with obstructed deformations.

We prove that the bracket width of the Lie algebra of vector fields on any smooth affine algebraic variety of dimension $n$ is at most $(n+1)^2$. We give improved bounds for some families of $\mathbb{C}^*$-varieties, in particular for $\mathrm{SL}_n(\mathbb{C})$ and for the Koras–Russell cubic threefold.