Papers

We classify the smooth projective Fourier–Mukai partners and determine the derived autoequivalence group of the moduli space $U_C(r,d)$ of stable bundles of coprime rank $r\geq2$ and degree $d$ on a smooth complex projective curve, assuming $g(C)\geq3$ and $\Aut(C)=1$. When the Jacobian $J(C)$ has endomorphism ring $\Z$, the partners are indexed by $(\Z/r\Z)^\times/\{\pm1\}$. We also prove that two coprime moduli spaces of vector bundles over complex curves of genus at least four are derived equivalent if and only if they are isomorphic.

We study the geometry of the three simplest known formulas for roots of a general sextic polynomial. In contrast with work of Green on the quintic, we show that no "simultaneous uniformization" of the normal forms associated to the solutions exists. This demonstrates an intrinsic qualitative complexity of sextic equations that is not present in quintic equations. Additionally, we give a formula for the roots of a general sextic in terms of Hilbert modular cusp forms and the uniformizing differential equation of the associated Hilbert modular surface.

For a (quasi)projective variety $X$ and an automorphism $f$ of $X$, we define moduli spaces of twisted endomorphisms of sheaves on $X$. If the automorphism $f$ has finite order, we construct a sheaf of noncommutative algebras $\mathcal{A}_0(f)$ and show that sheaves with a twisted endomorphism can be described as $\mathcal{A}_0(f)$-modules. We compute the topological Euler characteristics of Hilbert schemes of points of the algebra $\mathcal{A}_0(f)$, generalizing the answer for the case where $f$ is the identity automorphism.

We show that the cohomology rings of toric Richardson varieties in the Grassmannian are finite truncations of the ring of quasisymmetric functions. We exhibit an affine paving of each such variety whose cell closures give rise to the basis of fundamental quasisymmetric functions. We similarly interpret the homology of these varieties in terms of the ring of noncommutative symmetric functions and show that the expansion of the homological class of any torus-invariant subvariety into the affine paving basis agrees with the expansion of a corresponding generalized noncommutative ribbon function into the ribbon basis. By taking the direct limit of all toric Richardson varieties, we obtain an ind-variety equipped with a weak $H$-group structure whose cohomology is the Hopf algebra of quasisymmetric functions. We conjecture that it is isomorphic to a similar $H$-group constructed by Baker–Richter. As a byproduct, we deduce that the $f$-vectors of shard polytopes are log-concave, making the first progress on a question of Ferroni–Schröter for matroid base polytopes.

We initiate a `Peterson program' that seeks to extend Dale Peterson's theory for the quantum cohomology of flag varieties to arbitrary Schubert varieties, and that incorporates a mirror-symmetric approach. In this first paper we introduce a generalisation of the (localised) Peterson variety associated to an arbitrary Schubert variety $\check X_{w,P}$ inside a partial flag variety $\check G/\check P$. This generalisation is possibly a non-reduced affine scheme that lies in the Langlands dual full flag variety $G/B$. Additionally, we construct a Lie-theoretic `superpotential' associated to $\check X_{w,P}$, generalising earlier ones for the flag varieties $\check G/\check P$ from [Rie08], and we show that its relative critical point locus recovers the generalised Peterson variety. We make the key conjecture that for smooth Fano Schubert varieties the coordinate ring of the generalised Peterson variety recovers the quantum cohomology ring of $\check X_{w,P}$ localised at the quantum parameters. For smooth Fano Schubert varieties $X_{w,B}$ in $\check G/\check B$ we furthermore map our generalised Peterson variety to the cotangent bundle $\mathcal T^*(T)$ of the maximal torus $T$ of $G$ and construct a partial compactification that we conjecture models the full quantum cohomology ring of $\check X_{w,B}$, in analogy with the Givental-Kim presentation of $QH^*(\check G/\check B)$ via the degenerate leaf of the Kostant Toda lattice of $\check G$. These conjectures are verified for all smooth Schubert divisors in the complete type $A$ flag variety, and we show that our superpotential agrees with that introduced for Grassmannian Schubert varieties by Rietsch and Williams, after a suitable isomorphism of the domains.

We describe an algorithm for computing the zeta function of a proper, smooth curve over a finite field $k$ of characteristic $p$, when the curve is given together with some auxiliary data, including a lift $C$ to the valuation ring in a finite extension of $\Q_p$. The algorithm is denominator-free if the ramification is at most $p$. Our method computes the matrix of the action of a semilinear Frobenius on the first de Rham cohomology group of the curve by means of Poincaré duality, using cup products that can be computed from local expansions of a globally defined lift of Frobenius. Its complexity is softly cubic in the field degree for (general) smooth, planar curve, for which we work out our general estimates in more detail. We make explicit how to compute a suitable basis of the first de Rham cohomology group of $C$, base on 1-forms with `locally integrable polar parts', in both the general case and when the curve is smooth planar. We show the crystalline Frobenius preserves the first de Rham cohomology group of $C$ if the ramification is at most $p$, improving upon known results. In an appendix we prove a well-known formula for the cup product, and a formula by Serre, on the first de Rham cohomology group for a curve in characteristic zero, for which no reference seems to exist.

A standard question in computational real algebraic geometry is to compute all real solutions to a system of polynomial equations with real coefficients. One classical and promising approach is to track along a connected component of a real curve defined by a Newton homotopy, which is dependent upon the selected start point. As the start point varies, different subsets of real solutions may be obtained. This yields a partition of the space of start points into cells, and it is important to understand the structure of this partition in order to develop efficient algorithms based on Newton homotopies. The structure of the boundary of such cells and the number of cells in the corresponding partition are investigated for bivariate systems. Several examples are included to demonstrate the results.

Let $F$ be a totally real field of degree $n$ and discriminant $\Delta_F$, and let $X_1(\eta)$ be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by $\mathcal O_F$, together with a torsion point having annihilator $\eta$. Let $L=K_{\overline X_1(\eta)}+D$ be the log-canonical bundle on a smooth toroidal compactification, and let $H_L$ be an associated multiplicative height. We prove that rational points on $X_1(\eta)$ become sparser as $|\mathrm{Nm}(\eta)|\to\infty$, with $(\eta,\Delta_F)=1$. More precisely, for every number field $K$, set $$ N_{\eta,K}(B)=\#\{x\in X_1(\eta)(K):H_L(x)\leq B\}. $$ If $|\mathrm{Nm}(\eta)|\ge 5^n$ and $(\eta,\Delta_F)=1$, we prove $$ \limsup_{B\to\infty}\frac{\log\max\{1,N_{\eta,K}(B)\}}{\log B} \leq\delta_{\eta,K,n},\qquad \delta_{\eta,K,n}\ll_{[K:\mathbb Q],n}|\mathrm{Nm}(\eta)|^{-1/(2n)}. $$ In particular, $\delta_{\eta,K,n}\to0$ uniformly when $n$ and $[K:\mathbb Q]$ are bounded and $|\mathrm{Nm}(\eta)|\to\infty$. The main geometric result is a uniform lower bound, growing with the level, for the log-canonical degree of subvarieties of $X_1(\eta)$. Combining this estimate with recent progress derived from determinant-method, due to Ellenberg–Lawrence–Venkatesh and Brunebarbe–Maculan, we obtain the sparsity result above. We also prove that, for sufficiently large $|\mathrm{Nm}(\eta)|$, every subvariety of $X_1(\eta)$ is of general type, and establish a higher-dimensional generalization of the geometric torsion theorem of Bakker–Tsimerman. Namely, for a family of abelian varieties with real multiplication over a quasi-projective base of arbitrary dimension, we bound the torsion subgroup of its Mordell–Weil group in terms of the canonical volume of the base, uniformly in the totally real multiplication field of fixed degree.

We introduce a new secondary operation on Lawson homology of smooth complex projective varieties. It refines the Lawson intersection product by retaining null-homotopy data that are invisible at the level of the graded Lawson intersection algebra. Intrinsically, the operation is the Toda bracket in multiplicative morphic cohomology transported through Friedlander–Lawson duality; in an associative cochain model it is represented by the classical Massey formula. For every defined triple, the affine bracket modulo its full indeterminacy gives a well-defined secondary invariant. The new structure is genuinely nontrivial and strictly finer than its singular-homology realization. On a smooth projective eightfold we construct integral Lawson–Massey brackets whose quotient classes are nonzero, while specified values have exact order two and map to zero under the Lawson cycle map to singular homology. A square-zero perturbation makes all three associated singular-homology classes nonzero without changing the secondary values. Infinitely many such values remain independent even after quotienting by the sum of all their indeterminacies, over one fixed singular-homology triple. Projection-formula transfers preserve the phenomenon under projective bundles and odd-degree generically finite morphisms, and cyclic triple covers give examples with ample canonical bundle in every dimension at least eight. The same defining systems yield nonzero motivic Massey values, scalar-indecomposable higher Chow torsion, and obstructions to integral and two-local formality.

In this article, we prove a Hochschild-Kostant-Rosenberg type theorem for the category of coherent matrix factorizations associated to a Landau-Ginzburg model (X,f), where X is a quasi-smooth derived Deligne-Mumford stack over an algebraically closed field of characteristic 0.

We construct an explicit smooth projective Fano fourfold $X$ over an algebraically closed field $k$ of characteristic two with $H^1(X,\mathcal{O}_X)\simeq k$. Moreover, $X$ does not admit any free rational curve, and so it is not separably uniruled. The example has been constructed using ChatGPT.

The degrees of the iterates of a projective endomorphism grow in two layers: an exponential rate measured by the dynamical degrees, and a polynomial correction carried by the peripheral Jordan blocks. Log-concavity constrains the first layer; we show that the Hodge index theorem already constrains the second, in every codimension and in arbitrary characteristic. Let $f$ be a surjective endomorphism of a normal projective $d$-fold over an algebraically closed field, with dynamical degrees $\lambda_i$, so that $\textrm{deg}_i(f^n)\asymp\lambda_i^n n^{\mu_i}$ for a unique integer $\mu_i\ge0$. We prove that these exponents are coupled across three adjacent codimensions: strict log-concavity of $(\lambda_i)_i$ at $i$ forces $\mu_i=0$, while equality gives \[ 0\ \le\ 2\mu_i-\mu_{i-1}-\mu_{i+1}\ \le\ 4 . \] Consequently, $\mu_i\le2i(d-i)$, and this is sharp for every $i$. For a zero-entropy automorphism, we prove that every mixed orbit function indexed by any weak composition $\gamma$ of $d$ is a multivariate quasipolynomial of even total degree at most $d^2-\sum_j\gamma_j^2$. For a zero-entropy holomorphic automorphism $g$ of a compact Kähler $d$-fold $Y$, the same method gives \[\left\|(g^n)^*|_{H^{p,q}(Y,\mathbb{C})}\right\| =O\bigl(n^{p(d-p)+q(d-q)}\bigr).\] Powers of elliptic curves attain all exponent bounds as well as the constant $4$.

We formulate a generalized Saito criterion for complex projective plane curves, building on a recent generalized Saito criterion for plus-one generated arrangements of hyperplanes due to Junyan Chu.

This paper deals with image-based visual servoing and pose estimation by observing four and five lines. Our main interest is to determine the relative configurations of the camera and the observed lines that lead to problems in control and stability. Since it is equivalent to finding the singularities of the corresponding Jacobian matrix, we use tools from computational algebraic geometry to seek configurations such that all of its minors vanish simultaneously. By choosing a suitable basis for this matrix, we revisit the problem in the case of three lines to show that one type of the singularities is when the camera lies on the hyperboloid of one sheet uniquely defined by the lines. This result is further exploited to prove that the one-dimensional singularities, if any, in the case of $n$ lines appear when the camera lies on the transversals to the observed lines. Thus, by forcing the transversals to be complex, we can avoid the aforementioned type of singularities in the case of four lines although the algebra shows that there can always be up to 10 inevitable singular locations of the camera for the other type of singularity. For five lines, we find out that there are no singularities in the generic case. The singularities are also characterized for four and five lines with orthogonality and parallelism constraints. Furthermore, a visual servoing library is used to conduct some simulated experiments to substantiate the theoretical results. As expected, we observe problems in control in the vicinity of a singularity as well as increased errors in pose estimation.

Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ and let $C\subseteq\mathbb{A}_{\mathbb{C}}^N$ be an irreducible curve. Suppose $C$ has an infinite intersection with the $f$-orbit of a point $x\in\mathbb{A}^N(\mathbb{C})$. Then the normalization of $C$ is isomorphic to either $\mathbb{A}^1$ or $\mathbb{G}_m$. We prove that $C$ is $f$-periodic in the latter case, as expected by the dynamical Mordell-Lang conjecture.

We prove Torelli theorems for moduli of vector bundles in two complementary settings. In the first part, over an arbitrary field, we recover the coarse curve of a smooth, proper, geometrically connected, tame Deligne–Mumford curve of coarse genus at least two from its full fixed-determinant moduli stack in rank at least three. We also prove a fixed-degree version under an explicit compatibility hypothesis. The method is intrinsic to the stack: faithful special-linear linearizations recover Picard stacks, and determinants of tangent complexes recover the canonical principal polarization. In the second part, we study moduli spaces of twisted vector bundles on smooth projective curves. A twisted Hitchin construction recovers the curve in specified characteristic and degree ranges, with separate hypotheses for recovery of the twisting class. For rank two with trivial determinant, we recover the curve and the Brauer class from the moduli stack through an ordered Kummer cover and its polarized Jacobian torsor. For odd determinant on hyperelliptic curves in characteristic different from two, residual gerbes give a relative Desale–Ramanan construction without an actual determinant line bundle over the ground field. An intrinsic polarized Fano torsor recovers the curve and the class, and a corrected polarization obstruction recovers the determinant gerbe. We also give a twisted Newstead construction in genus two, with an explicit descent condition for its converse, and period–index applications.

This paper is the final installment in our series on the geometric theory of complex Monge-Ampere equations. We study singular Kahler metrics on compact normal Kahler spaces with klt singularities whose volume densities lie in $L^p$ for some $p>1$. We show that a lower bound for the Ricci current in the sense of pluripotential theory is equivalent to a synthetic Ricci lower bound in the sense of RCD theory. As a consequence, every Kahler class on a compact normal Kahler space with klt singularities admits an RCD structure, which makes differential geometric analysis available on singular complex spaces. As an application, we show that the fundamental group of a compact Kahler Calabi-Yau space with klt singularities is almost Abelian. We further establish compactness results for singular Kahler-Einstein spaces.

We prove the converse of the Serre-Grothendieck finiteness theorem for the cohomology of coherent sheaves, originally due to Lipman. This result characterizes proper morphisms as those morphisms whose derived pushforward preserves pseudo-coherent complexes. Although the derived pushforward does not reflect pseudo-coherence, we prove that it does after twisting by all perfect complexes. This provides a new characterization of properness.

We prove that a hyperkähler manifold of OG6 type is determined, up to isomorphism, by its integral second cohomology endowed with its Hodge structure and Beauville–Bogomolov–Fujiki form. The proof combines the monodromy and chamber theorems of Mongardi–Rapagnetta with the bimeromorphic Torelli theorem. In particular, bimeromorphic OG6 manifolds are biholomorphic.

To solve an overdetermined polynomial system numerically, one first makes it square, usually by replacing the given equations with as many random linear combinations as there are unknowns. This is a provably safe step, but it can substantially enlarge the supports. The alternative is to keep that many of the given equations themselves. Selection preserves sparsity but risks geometry: a genuine solution can cease to be an isolated point of the subsystem's zero set. We show that deciding whether a safe choice exists is NP-complete, already for an explicit family of systems of degree three with radical ideal and exactly three simple rational solutions. For strong selection with the nondegenerate rational solutions supplied explicitly, three is the exact threshold when degrees are polynomially bounded: one or two solutions reduce to matroid intersection, three already give NP-completeness. Even without a degree bound, an arbitrarily long list never takes the decision problem beyond NP. On the hard family, five natural notions of a faithful subsystem coincide, and every failing choice fails visibly: its zero set contains an affine subspace through one of the three solutions. A degree-four variant shows that cost information does not help: every candidate that could possibly succeed has mixed volume exactly three, and the problem is NP-complete still. The construction realizes Karp's three-dimensional matching problem as the selection of a square subsystem from the given equations.

We generalize the Bondal–Orlov Reconstruction Theorem to smooth, proper, tame algebraic stacks with generically trivial stabilizer, whose canonical bundles are nowhere torsion: They do not become trivial after pullback along any finite morphism from a projective curve. Our main tools are coherent Tannaka duality as proved by Lurie and by Hall and Rydh, and Grothendieck duality for proper tame stacks as developed by Hall and Priver. We additionally use the notion of nowhere torsion line bundle on a scheme to prove a version of the Bondal–Orlov Reconstruction Theorem for finite type, separated, Gorenstein schemes which are not necessarily proper, which generalizes work of Ballard, Favero, Ito, and Matsui.

Starting from Alpöge's torus fibration with exactly three special fibres on a complex six-sphere, we construct a locally miniversal family of compact complex threefolds. Every member is Oka, and the projection over each relatively compact parameter disc is an Oka map. Deleting or simultaneously blowing up finitely many distinct points preserves the Oka property. We also determine the tangent cohomology, the smooth Kuranishi germ, the automorphism group, and the exact biholomorphism relation $X_a\simeq X_{a'}$ if and only if $a'-a\in2\mathbb{Z}$. The Oka arguments use explicit cyclic covers and Kusakabe's localization principle.

In this article, we extend the classical relationship between ADE singularities and their maximal Cohen-Macaulay modules to quotient singularities of types A and D. We identify projected hypersurfaces which allow us to construct explicit matrix factorizations, and in return, families of maximal Cohen-Macaulay modules. These hypersurfaces also yield the dual resolution graphs of the corresponding quotient singularities via Oka's method. Finally, we show that all special Cohen-Macaulay modules over the original quotient singularity can be recovered from those over the projected hypersurfaces.

We give an algebraic proof in characteristic zero of the Miyaoka–Mori criterion: every point of a curve of negative canonical degree on a smooth projective variety lies on a rational curve. Our jet technique gives, in addition, an effective numerical decomposition of the original curve class. For each prescribed point, the decomposition contains a rational curve through that point, with a positive coefficient independent of the point and with anticanonical degree at most $\dim X+1$. Together with BDPP cone duality, this recovers the projective uniruledness criterion over $\mathbb C$. The main result of this paper was obtained using the Pharos system. A detailed report on the use of Pharos and on the Lean~4 formalization of the main arguments is given in the Appendix B, written by Bin Dong, Guoxiong Gao, Zeming Sun, and Bin Wu.

We prove the existence, uniqueness, and partial regularity outside a proper analytic subset for the Kähler current solving the $J$-equation at the birational minimal slope. This confirms Datar–Mete–Song's conjecture 1.5 in \cite{DMS26}. We introduce an analytic threshold, and prove its equivalence to the birational threshold introduced by Datar–Mete–Song. One of the key tools is the approximation of subsolution by Bergman's kernels, which is motivated by the work of Demailly on the approximation of plurisubharmonic functions with analytic singularities. As an application of the Bergman kernel approximation, we combine the results of Fang–Ma \cite{FM26} to give an analytic characterization of the $J$-null locus of a semistable pair $(\alpha,\beta)$. This removes the technical assumptions in \cite{L26b}.

Given an abelian scheme $A$ over a $p$-adic ring $R$, Borger and Saha constructed a filtered module $\{\mathbf{H}_\delta(A) \supset \mathbf{X}_{\mathrm{prim}}(A)\supset \{0\}\}$ with a semilinear operator $\mathfrak{f}^*$ on $\mathbf{H}_\delta(A)$ using the theory of arithmetic jet spaces. The above object admits a canonical map $\Phi$ to the Hodge sequence $\{\mathbf{H}^1_{\mathrm{dR}}(A) \supset H^0(A,\Omega_A)\supset \{0\}\}$ of $A$ in the category of filtered modules. As a result, by restricting $\Phi$, we obtain a natural $R$-linear map $\Upsilon: \mathbf{X}_{\mathrm{prim}}(A) \rightarrow H^0(A,\Omega_A)$. In this paper, we show that the map $\Upsilon$ is an isomorphism of vector spaces over $K$, the field of fractions of $R$. As a consequence, we will show that for all abelian schemes $A$, the operator $\mathfrak{f}^*$ on $\mathbf{H}_\delta(A)_K$ is a bijection, and our object $\{\mathbf{H}_\delta(A)_K \supset \mathbf{X}_{\mathrm{prim}}(A)_K\supset \{0\}\}$ becomes a filtered isocrystal. In fact, the above results admit a generalization to the setting of semi-abelian schemes. The elements of $\mathbf{X}_{\mathrm{prim}}(A)$ are represented by primitive additive characters of the arithmetic jet spaces attached to $A$. Hence, our isomorphism given by $\Upsilon$ provides an interesting character-theoretic interpretation of $H^0(A,\Omega_A)$ in terms of primitive delta characters. As a result, to any $1$-form $\omega$, the above isomorphism associates a canonical numerical invariant that depends on deformation theoretic data of $A$. Furthermore, we also extend a comparison theorem between $\mathbf{H}_\delta(A)_K$ and the first crystalline cohomology $\mathbf{H}_{\mathrm{cris}}^1(A)_K$ in the general case when the elliptic curve $A$ is defined over the ring of integers of a $p$-adic field $K$ that is a finite extension of $\mathbb{Q}_p$.

We report an explicit construction of a 27 x 27 x 27 symmetric tensor with rational entries that has tensor rank 55 over the rational numbers and symmetric tensor rank 56 over the complex numbers, providing a small counterexample to Comon's conjecture over the rational, real, and complex numbers. The construction follows the framework of symmetric adjoins introduced by Shitov. The primary technical contribution of this work is to prove a special case of Conjecture 6 appearing in Shitov's seminal 2018 work.

We study homotopy theories of higher groupoids in specified geometries, distinguishing geometric representability from realization and descent. Ordinary smooth and Banach-open sheaves admit full Brown categories of fibrant objects. For ordinary geometric groupoids, a crossing-axes obstruction motivates an empty-compatible incomplete Brown structure. Its split-Banach version requires a fixed chart-compatible plot site with local kernel-product closure. These results include all finite unique-horn bounds and their union. For represented derived models, finite matching and an outer-prism filtration give the structural Brown calculus. Nuiten's finite-geometric realization theorem yields the finite derived-smooth structure and its localization. Enriched realization base change on the original small Kan-enriched site gives the outer-unbounded hypercomplete structure, with positive-degree geometric horn conditions and Reedy fibrancy in every degree. For Banach domains, structured spectra, strict-open gluing and split-open hyperdescent establish geometric finite limits and hypersheaf representability. They give the represented Brown structure, using the same hypercompleted weak-equivalence class at every outer bound. A square-zero test detects nonordinary self-intersections of the chosen Banach charts, including infinite-dimensional charts. For connective associative affines, homotopy split horns, Pridham's finite effectivity theorem and a pointwise diagonal-fibration argument give categories of fibrant objects for his Artin and Deligne–Mumford classes. The associative theory uses quasi-isomorphisms of algebras and realization in presheaves.

Let $\mathfrak{C}$ be a projective stacky curve over an infinite field, together with a generating sheaf $\mathcal{E}$. We consider the moduli stack of $\mathcal{E}$-semistable vector bundles of fixed determinant and multiplicities on $\mathfrak{C}$, and show that the moduli stack of such bundles is $\mathbb{A}^1$-connected. As a consequence, we conclude that the moduli stack of parabolic $\alpha$-semistable vector bundles of fixed determinant and parabolic type on a connected smooth projective curve is $\mathbb{A}^1$-connected.

We study pairs consisting of a rational isosceles triangle and a rational triangle having a prescribed angle $\theta$, with equal perimeter and equal area. Writing $\rho=\cos(\theta)\in\mathbb{Q}$, we obtain a two-parameter description of the solutions and, for fixed $\rho$, a family of curves $C_\rho$ which are generically of genus $2$. We determine the two singular specializations: there is no genuine pair for $\rho=\frac{1}{2}$, while $\rho=\frac{47}{49}$ admits infinitely many genuine pairs. We completely parametrize two natural one-parameter families in which the $\theta$-triangle is isosceles, and prove several density results for values of $\rho$ admitting one or more genuine pairs. For thirty-five explicit values of $\rho$, Chabauty–Coleman computations determine $C_\rho(\mathbb{Q})$ completely and show that the corresponding pair is unique up to homothety. We also study multiplicity when one triangle is fixed; the condition for a scalene $\theta$-triangle to have two isosceles partners leads to an elliptic $K3$ surface and yields a dense set of such values of $\rho$. Finally, we construct infinite families with at least four genuine pairs and exhibit examples carrying at least five, and in two cases at least six, genuine pairs.