Papers

Chen and Ngo showed that the Hitchin morphism for Higgs bundles on a projective manifold $X$ factors through a closed subscheme of the Hitchin base called the spectral base, and the induced map is called the spectral morphism. In this paper, we investigate local Cohen-Macaulay obstructions to the surjectivity of spectral morphisms via a refined spectral correspondence. We show that every affine normal variety with a local ring that admits no rank one maximal Cohen-Macaulay modules can be geometrically realized as a local obstruction to the surjectivity of spectral morphisms, and that these local obstructions can be compactified via the Kawamata covering trick. As an application of these constructions, we provide counterexamples to the Chen-Ngo conjecture on the surjectivity of spectral morphisms for $GL_r$-Higgs bundles in dimensions $n\geq 9$. Moreover, we construct counterexamples to the Chen-Ngo conjecture for $SL_2$-Higgs bundles in every dimension $n\geq 2$ and for semistable $GL_3$-Higgs bundles on a smooth projective surface.

We construct a smooth projective family of rational surfaces over $\mathbb{G}_{m,\mathbb{Z}}$. The Mori dream property of a fiber is determined by the torsion of the normal bundle of an anticanonical cycle. Over $\mathbb{C}$, the locus of Mori dream fibers is Zariski dense. For every prime $p$, every geometric fiber over a closed point of the reduction modulo $p$ is a Mori dream surface, whereas the geometric generic fiber is not a Mori dream space. In either setting, no restriction to a nonempty open subset is a Mori dream morphism. We also prove that, over any algebraically closed field, a projective fibration becomes a Mori dream morphism after shrinking the base whenever the set of points with Mori dream fibers is not contained in a countable union of proper closed subsets.

We prove that a general smooth complete intersection of positive dimension and codimension at least 2 over an algebraically closed field has trivial linear automorphism group, except in the case of two quadrics. The proof uses incidence varieties and treats separately prime-to-characteristic and unipotent automorphisms.

The Thom-Levine theorem is a classical method for proving the triviality of deformations by integrating suitable vector fields. In this note, we show that the converse of this criterion does not hold in the bi-Lipschitz category. More precisely, we give an example of a smooth one-parameter deformation of a real function germ in two variables which is bi-Lipschitz trivial but not strongly bi-Lipschitz trivial. The central germ has an isolated critical point, though it is not finitely determined.

Let M be a plumbed integral homology sphere 3-manifold associated with a connected negative definite plumbing graph $\Gamma$. One of its most important invariants is its multivariable Poincaré series (or zeta function) $f_\Gamma(\mathbf {t})$. Several numerical invariants can be read from $f_\Gamma(\mathbf{t})$, or even from its `polynomial part' $ \operatorname{Pol}_{\Gamma}(\mathbf{t})$. For example, the `normalized' Casson's invariant equals $ \operatorname{Pol}_{\Gamma}(1)$. In this note we provide several splice (surgery) formulae for $f_\Gamma$ and $\operatorname{Pol}_\Gamma$ (reduced to the node variables, or to the node variables of connected sub-graphs). In this way, these global invariants can be recovered from a collection of certain smaller graphs. Recall that some (integral homology sphere) 3-manifold invariants are additive with respect to the splice decomposition (like the Casson's invariant). However, some invariants need some `splice correction terms'. In our formulae the correction terms are easily computable one variable Alexander polynomials.

We introduce a matrix-valued finite difference formalism for the genus zero BPS resummed local Gromov-Witten potential of a threefold flop. Factoring the central difference as \[ \Delta_\eta=\nabla_\eta^2, \qquad \nabla_\eta=T_{h\eta/2}-T_{-h\eta/2}, \] we prove that the mixed differences \( \mathbf H=(\nabla_{\eta_i}\nabla_{\eta_j}F)_{i,j} \) admit an exact rank one signed \(q\)-Gram decomposition. Primitive BPS classes are therefore detected by rank one coefficient matrices, and a matrix-valued Möbius inversion reconstructs all multiplicities on a primitive ray. Under a simple threefold flop the Gram forcing is covariant away from the flopped ray and acquires the universal logarithmic anomaly \[ \kappa_C(C^+\otimes C^+)\log r, \qquad \kappa_C=\sum_{d\ge1}d^3n_{dC}. \] After a natural renormalization it becomes flop covariant, while one classical logarithmic derivative recovers the cubic correction in the crepant transformation formula. We interpret \(\kappa_C\) as the third moment of the GV width distribution: Toda's noncommutative width is the second moment, and contraction algebra BPS invariants give a finite cohomological width series whose numerical realization contains all these moments. For root supported BPS theories, simple root Gram coefficients reconstruct the signed Cartan matrix. Applied to the standard ADE foldings, the theory distinguishes \(B_n\) from \(C_n\), detects the \(F_4\) double bond and \(D_4\) triality, and yields explicit global \(q\)-Gram discriminants.

We give a partial generalization of the Kurdyka theory of arc-analytic functions and arc-symmetric sets on Nash manifolds to the setting of arbitrary locally closed semialgebraic sets.

We expand upon a previous study conducted by the author on the behavior of punctured log Gromov-Witten theory under log étale modifications $\widetilde{X} \rightarrow X$, giving expressions for log Gromov-Witten classes on $\widetilde{X}$ in terms of log Gromov-Witten classes on $X$, facilitating a complete reduction of the punctured log Gromov-Witten theory of any modification $\widetilde{X}$ of an snc log scheme $X$ to the punctured log Gromov-Witten theory of $X$. We apply this result in two settings. First, we prove a log-orbifold correspondence equating the logarithmic invariants of the canonical wall structure of Gross and Siebert with a class of orbifold invariants considered in the relative quantum cohomology ring of Tseng and You, and more generally construct a family of algebra homomorphisms from the intrinsic mirror algebra $R_{(X,D)}$ of Gross and Siebert to appropriate power series rings. Second, we show the punctured log Gromov-Witten classes of split toric bundles are effectively reconstructed in terms of the punctured log Gromov-Witten classes of the base. Additional input for the second application is the introduction and study of double ramification cycles with target log variety, generalizing the double ramification cycles with target variety investigated by Janda-Pandharipande-Pixton-Zvonkine.

We study polynomial maps in three variables with nilpotent Jacobian over a field of characteristic zero. The proposed classification reduces maps with linearly independent components to a family determined by a univariate polynomial evaluated at a quadratic coordinate. The geometric part of the argument produces two algebraically dependent constant linear combinations of the components. A derivation argument then yields the normal form over the original field. We obtain explicit polynomial inverses and tame factorizations. We then study higher-dimensional maps in which all but the last three components depend on three variables. For this class, we give separate normal forms for the two three-variable blocks and deduce stable tameness from the residue-field criterion of Berson, van den Essen, and Wright.

Schimpf's connected reformulation of Pandharipande's pole conjecture for stable pairs predicts the following restriction: in class \(\beta\), connected descendent coefficients have nonzero \(p\)-poles only where \(-p\) is an \(m\)-th root of unity with \(1\leq m\leq\divis(\beta)\). We give a toric counterexample. The threefold is a smooth quasi-projective toric Calabi–Yau threefold whose compact torus-invariant curves form a chain \(C_1\cup C_2\). For the primitive class \(\beta=2[C_1]+[C_2]\), the coefficient of \(Q^\beta z^3\) in the connected series for the insertion \(\ch_z([v]_T)\), where \([v]_T\) is the equivariant class of \(v=C_1\cap C_2\), has Laurent expansion \[ {\frac{9}{4}}\frac{1}{p-1}+O(1) \] after restricting to the one-parameter subtorus with tangent weights \((1,2,-3)\) at \(v\). Since \(\divis(\beta)=1\), Schimpf's condition allows only the nonzero pole \(p=-1\). The pole at \(p=1\) is therefore forbidden.

Let $X=\mathbb{P}^3$ and let \[ \alpha_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\mathbb{Q}) \] with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\mathcal{O}_X(1))$. We prove that every Gieseker semistable sheaf on $X$ with Chern character $\alpha_n$ is stable and uniquely of the form $\iota_{P*}\mathcal{I}_Z$ for a plane $P\subset X$ and a length-$n$ subscheme $Z\subset P$. Hence the coarse stable-sheaf moduli space is the relative Hilbert scheme of points on the universal plane over the dual projective space $X^{\vee}$. Using the standard perfect obstruction theory for stable sheaves on a Fano threefold, we identify the obstruction bundle with a relative Carlsson–Okounkov twisted tangent bundle and obtain a closed product formula \[ \sum_{n\ge 0} \Gamma_n q^n=\prod_{m\ge 1}(1-q^m)^{-7} \] for the natural point-inserted two-dimensional Donaldson–Thomas invariants on $\mathbb{P}^3$. We then develop the analogous divisor-supported theory for a smooth divisor $D$ in a smooth projective Fano threefold, distinguishing moving and rigid divisors. For a rigid divisor satisfying $H^1(D,\mathcal{O}_D)=0$, the divisor-supported moduli component is a Hilbert scheme of points on $D$, its obstruction bundle is the twisted tangent bundle $\mathsf{T}_D^{[n]}(N_{D/X})$, and its generating series is \[ \prod_{m\ge 1}(1-q^m)^{-(c_2(X)\cdot D+D^3)}. \] We work out the examples of the exceptional divisor in $\operatorname{Bl}_p\mathbb{P}^3$ and of a rigid section in a Fano $\mathbb{P}^1$-bundle over $\mathbb{P}^1\times\mathbb{P}^1$.

Consider the ring of complex-valued continuous functions on a compact Hausdorff space $X$. Cortiñas–Thom computed its nonpositive algebraic $K$-theory, confirming Rosenberg's expectation. We give a motivic refinement of their result: For $i\geq 2j\geq 0$, the $i$th integral motivic cohomology group of weight $j$ is naturally isomorphic to the $i$th Betti cohomology group of $X$.

Let $q=p^h$ be an odd prime power and let $X/\F_{q^2}$ be a maximal curve of genus at least two. We classify the curves for which the full geometric automorphism group is transitive on the set of $\F_{q^2}$-rational points. We prove that, if $h>1$, then $X$ is the Hermitian curve. If $h=1$, the only additional possibility occurs for $q=5$: the unique $\F_{25}$-maximal genus-three curve, namely the maximal $S_4$-model of the Klein quartic. The proof separates tame and wild actions. In the tame case, genus bounds, signatures of quotient maps, specialization to characteristic zero, low-genus automorphism classifications, and a Hasse–Witt computation reduce the problem to the Klein quartic. In the wild case, the rational points are identified with the Sylow $p$-subgroups of the automorphism group. Noncyclic Sylow subgroups are treated using a finite-group classification theorem together with Henn's large-automorphism classification, while cyclic Sylow subgroups are excluded by local ramification and the Riemann–Hurwitz formula. Combined with the known characteristic-two result, this yields the classification for all prime powers.

We study function fields in one variable over the field with one element $\mathbf{F}_1$. It is proved that the Galois extensions of $\mathbf{Q}$ are isomorphic to the curves over $\mathbf{F}_1$ being understood as the Deitmar schemes. Specifically, one gets explicit formulas linking the genus and the number of cusps of an algebraic curve over the extension $\mathbf{F}_{1^m}$ of order $m\ge 1$ of the field $\mathbf{F}_1$ and the $m$-th roots of unity of the corresponding number field. It follows that elliptic curves with one cusp over $\mathbf{F}_1$ are either the cyclotomic fields or the maximal abelian unramified extensions of the quadratic number fields. Our proof depends on representaion of the Drinfeld modules by the bounded linear operators on a Hilbert space and the crossed product structure of the Cuntz-Krieger algebras.

Let $k$ be an algebraically closed field of characteristic $p>0$, $n\ge1$, and let $X\subset \mathbb P^n_k$ be a finite nonempty set of distinct points with the defining ideal $I=I(X)$. We give a proof of Demailly's inequality in positive characteristic. The argument is based on the Frobenius–Hasse derivative method used in this context by Hà and Sivakumar. The key additional observation is a strict-growth lemma for the initial degrees of symbolic powers of a finite set of affine points: $$ \alpha(J^{(t)})\ge \alpha(J^{(t-1)})+1\qquad(t\ge1). $$ In characteristic $p$, if a minimum-degree polynomial has a nonzero first ordinary derivative, this follows by differentiation; if all first ordinary derivatives vanish, perfectness gives a $p$th root and an induction on the symbolic exponent. The remainder of the proof uses the $q=p^e$ Frobenius decomposition and a maximal Hasse derivative.

Consider a compact leaf $\mathcal{L}$ of a holomorphic foliation $\mathcal{F}$ such that all the leaves of the restriction of $\mathcal{F}$ to some neighborhood $U$ of $\mathcal{L}$ are closed in $U$. We show that there exists an invariant germ of analytic set $V$ in a neighborhood of $\mathcal{L}$, of dimension higher than $\dim (\mathcal{F})$, consisting of compact leaves, such that the volume of the leaves in $V$ is uniformly bounded by above. We also provide a globalization of this result if the ambient manifold is projective. In order to study closed leaves foliations, and via the holonomy representation, we introduce a new concept, of independent interest, for subgroups $G$ of germs of holomorphic diffeomorphisms, the so called torsion locus. We show that it is non-trivial if $G$ has finite orbits.

We prove that the Hodge filtration on local cohomology is contained in the symbolic Ext filtration, and hence in the usual Ext filtration, answering a question of Mustata and Popa. Our main observation is that the Ext filtration admits a filtered-duality interpretation via Hartshorne's algebraic de Rham complex and its infinitesimal filtration. Comparing this construction with the filtered Du Bois complex yields the containment. The same mechanism proves the higher injectivity conjecture of Popa–Shen–Vo and several variants.

The super Schottky ideal defines the image of the superperiod map, which takes a family of genus $g$ supercurves to its Jacobian. We prove that if $g \geq 5$, then $d=g$ is the minimal number such that $I_{Sch,g}^d \subset I_{s-Sch,g}$, where $I_{Sch,g}$ is the Schottky ideal and $I_{s-Sch,g}$ is the super Schottky ideal.

The study of super Riemann surfaces and their moduli led Witten and Felder, Kazhdan and Polishchuk to ask what is the smallest $d$ such that the $d$-th power of the Schottky ideal is contained in the super Schottky ideal. Felder, Kazhdan and Polishchuk proved that $d=g$ in odd genus $g\geq5$ and that $d\in\{g-1,g\}$ in even genus $g\geq4$; Y.~Shen subsequently proved that $d=g$ in even genus $g\geq6$. Thus the only remaining open case was $g=4$. Here we close this gap by showing that in genus $4$ too, the smallest power of the Schottky ideal contained in the super Schottky ideal is $d=g=4$. The question is equivalent to one concerning the codifferential of the super period map. The quadratic odd contribution sends a conormal vector to a bivector, represented by a skew-symmetric $(2g-2)\times(2g-2)$ matrix. The question is to find such a vector for which, at a generic point, this matrix has maximal rank $2g-2$, or $6$ in our case. Our computation uses the standard exact sequences on $C\times C$ attached to the sheaves $\OO(a,b,c)$, together with a degeneration to a vanishing theta-null and some facts about the Szegő kernel. At a curve with a vanishing theta-null, a regularized version of the codifferential can be described by an explicit multiplication map, allowing an easy computation of the rank. This rank turns out to be $4$. A first-order calculation on an explicit deformation of a cyclic trigonal vanishing theta-null differentiates the global Gaussian-map identity for the regularized conormal bivector. Including the variation of the Gaussian map gives a nonzero first normal symbol. The resulting first-order form is nonzero on the null space of the limiting skew-symmetric matrix, so the rank jumps to $6$ on nearby curves, showing that $d=4$.

We follow a $D$-module approach to the construction of canonical differential equations for Feynman integrals directly from their Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems. Starting from the Schwinger representation, we identify the associated generalized Euler integral and its GKZ system, as an alternative to the Lee-Pomeransky representation. The generalized Euler formulation allows reductions associated with facets of the Newton polytope to be identified directly, reducing the number of variables in the resulting differential systems. We construct the associated Pfaffian systems using Frobenius bases, which also provide direct access to the singular loci. After a suitable rationalizing change of variables, the systems are brought into canonical $\epsilon$-form. A key feature of the construction is that the dimension of the canonical basis is determined by the holonomic rank of the GKZ system and, for the examples considered, it is smaller than or equal to the number of master integrals obtained from integration-by-parts, thus providing a reduced description of the differential system.

Let $X$ be an irreducible reduced projective subvariety of a Chow variety. On a generic smooth parameter locus, normal motions of resolved components define a quadratic Wasserstein metric, even when the cycles are singular, reducible, or carry multiplicities. Its metric completion is canonically $X^\nu$: resolution fibers collapse, whereas distinct normalization branches over the same Chow cycle remain separated. The completed metric admits an exact ambient action formula. For degree-$d$ hypersurfaces, the inner $W_q$ geometry is compact and induces the projective topology for every $1\le q\le2$. For $d\ge2$, the uniform Hölder exponent $1/d$ is attained and sharp for each such $q$; when $d=1$, the comparison is Lipschitz. Elliptic quartics furnish an explicit model of the boundary branching.

We construct the KLR algebra (in type $A$) as affine sections of a coherent sheaf $\mathcal{E}$ on a product of projective spaces $\mathbb{P}^n$. It was shown by Elias-Qi that the Lie algebra $\mathfrak{sl}_2$ acts on such KLR algebras; our construction geometrically explains this action as stemming from the $\mathfrak{sl}_2$-action on the twisting sheaves $\mathcal{O}(k)$ of $\mathbb{P}^1$ and gives the maximal finite-dimensional $\mathfrak{sl}_2$-submodule of KLR as global sections of $\mathcal{E}$. We furthermore extend the Elias-Qi action to a Witt action. Kleshchev-Loubert-Miemietz showed that the KLR of type $A$ has a graded affine cellular structure; our construction also gives a geometric incarnation of this.

The Suzuki and Ree curves are two classical Deligne–Lusztig curves. They are maximal over suitable finite fields and their full automorphism groups are the Suzuki and Ree groups. In this paper we determine explicit generators of the fixed fields of these full automorphism groups in the corresponding function fields. Motivated by Dickson's classical study of invariants of finite linear groups, we use the natural projective representations of the two groups. In the Suzuki case the proof uses restricted Dickson invariants, whereas in the Ree case it relies on an invariant bilinear form of the seven-dimensional representation.

We show that the tautological ring of $M_{25}$ is not Gorenstein, disproving a conjecture of Faber from 1993. This also implies that the Faber-Zagier relations are the complete list of tautological relations on $M_{25}$.

We study the singular cohomology rings of toric varieties associated with several fans arising from uniform matroids. These rings generalize the Chow and augmented Chow rings of matroids. For the singular cohomology ring arising from the augmented Bergman fan of a uniform matroid, we construct an explicit basis derived from the retral basis for the singular cohomology ring of a uniform matroid introduced by the first author. We prove that the augmented Bergman fan does not yield a singular cohomology ring that satisfies the quasi-projective Strong Lefschetz property, whereas a suitable modification of the fan does. We then investigate the zeros of the corresponding refined Hodge–Poincaré polynomials. For uniform matroids, we prove that the refined Hodge–Poincaré polynomials associated with both the singular cohomology ring and the modified augmented singular cohomology ring are real-rooted. The former result resolves a conjecture of the first author. These results extend the real-rootedness theorem of Brändén and the second author for the Chow polynomials of uniform matroids. Finally, we relate the failure of real-rootedness for the augmented singular cohomology ring to the failure of Lefschetz properties.

The Cassels–Swinnerton-Dyer conjecture asserts that a cubic hypersurface contains a rational point if and only if it contains a point of degree coprime to $3$, or, equivalently, a zero-cycle of degree $1$. The case of smooth cubic surfaces in characteristic zero has been reduced by Coray (1976) and Voisin (2026) to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument to extend the result to smooth cubic surfaces over arbitrary fields. We further give a separate argument for the case of singular cubic surfaces, extending previous work of Coray over perfect fields. Altogether, this proves the Cassels–Swinnerton-Dyer conjecture for cubic surfaces.

Elements of the Brauer group $\operatorname{Br}(S)$ of a variety $S$ have geometric incarnations as étale-projective $S$-bundles, yet producing minimalist constructions of such bundles, which often power arithmetic applications, remains a difficult problem. When $S$ is a K3 surface with Picard rank $1$, we use the birational geometry of moduli spaces of twisted sheaves on $S$ to construct geometric realizations of nontrivial elements of $\operatorname{Br}(S)$. We recover many known geometric constructions of Brauer classes on K3 surfaces while providing a common moduli-theoretic framework for them. As a by-product, we give a new proof of the period-index theorem for very general K3 surfaces.

We solve the long-standing problem of determining the exact denominators of the coefficients of the logarithm $B(x)=x+\sum_{n\geq1}b_nx^{n+1}$, obtained from the universal formal group of complex cobordism by the modulus square construction: $$ b_n=\frac{C_n}{d_n},\qquad d_n=\frac{n+1}{2}\operatorname{lcm}(1,\ldots,2n+2), $$ where $C_n\in\Omega_{\mathrm U}^{-4n}$ is primitive and undecomposable. We prove that $C_n$ belongs to the coefficient ring $\Lambda$ of the universal two-valued law and to the subring $\Lambda_{\mathrm{St}}$ generated by quotient Stong manifolds, and give an explicit integral Stong manifold formula for it. The rational lift of $b_n$ to quaternionic cobordism modulo torsion has exact denominator $2d_n$. Consequently, $C_n$ has no integral quaternionic lift, whereas $2C_n$ does. Every Chern number of $C_n$ is divisible by $d_n$, and $c_{2n}(C_n)=d_n$. We introduce odd genera $\mathrm{Bc}_N$. For $N\geq n$, $\mathrm{Bc}_N$ detects the odd part of $d_n$. Its restriction to the Stong ring is integral exactly for $N\leq3$, while the universal odd genus $\mathrm{Bc}_{\infty}$ takes values there with only odd denominators. For every $n\geq5$, we construct undecomposable classes in $\Lambda^{-4n}$ with equal Ochanine genera and top Chern numbers but distinct $\mathrm{Bc}_3$ values. Finally, the oriented extension of $\mathrm{Bc}_3$ is integral on closed spin manifolds of real dimension below $24$ and on closed string manifolds of real dimension at most $24$. The spin bound is sharp: an Anderson-Brown-Peterson spin $24$-manifold has nonintegral $\mathrm{Bc}_3$ genus.

We prove that every rational proper holomorphic map from $\mathbb{B}^2$ to $\mathbb{B}^4$ has degree at most $5$, confirming D'Angelo's conjecture in this case. This sharp bound follows from Meylan's degree estimates and an intersection-theoretic argument ruling out degree $6$.

We prove that a Noetherian integral domain is regular if, and only if, the derived pushforward of the structure sheaf along every blowup is a perfect complex. For suitable local rings, we give a construction which realizes a shift of the residue field as a direct summand of the derived pushforward of the structure sheaf along an explicit blowup.