Papers

We explore the homogeneous structures of Albanese morphisms and maximal rationally chain-connected fibrations of algebraic varieties with nef anticanonical divisors. Furthermore, we show that there exists a natural group homomorphism associated with a maximal rationally chain-connected fibration, which induces the Chevalley decomposition of $\operatorname{Aut}^0$ up to an isogeny. This result is analogous to a similar theorem of Nishi and Matsumura for Albanese morphisms. In addition, we reprove Nishi and Matsumura's theorem using the modern language of algebraic geometry.

Many variants of the multiple zeta values have been studied in recent years. There are a few among them that remain to be real numbers, such as the multiple mixed values defined by the authors as level-two generalizations, which include both Hoffman's multiple $t$-values and Kaneko–Tsumura's multiple $T$-values. A central question is when such a value descends to level one, that is, when it can be expressed as a $\Q$-linear combination of multiple zeta values. We call these values unramified. In this paper, we further consider the alternating version of the above variants and identify five families of unramified alternating multiple mixed values using the descent theory of Brown and Glanois. We conjecture that all unramified truly alternating multiple mixed values are given in this paper.

The celebrated theorems of Shilov, Arens–Royden, and Forster give direct descriptions of the first three integral cohomology groups of the Gelfand spectrum of a commutative complex Banach algebra. In his 1974 ICM address, Taylor asked whether the higher cohomology groups admit descriptions in terms of the underlying ring. We give a solution to this question in even degrees: The étale (aka Lichtenbaum) Chow group in every codimension is canonically isomorphic to the corresponding even integral cohomology group of the Gelfand spectrum.

We study compact Kähler varieties admitting int-amplified endomorphisms from the viewpoint of positivity of tangent sheaves. Our main result shows that the tangent sheaf of such a variety is weakly positively curved; in particular, it is pseudo-effective in a strong sense. This establishes a new link between complex dynamics and the positivity theory of tangent sheaves, and provides an alternative to equivariant MMP techniques in the compact Kähler setting. As an application, we prove a Kähler analogue of Yoshikawa's structure theorem: for a compact Kähler klt variety admitting an int-amplified endomorphism, after an equivariant quasi-étale cover, it admits an equivariant flat MRC fibration with irreducible fibres onto a complex torus; in the smooth case, the fibration is smooth whose periodic fibres are of Fano type. To this end, we prove the existence of minimal models in the case of numerical dimension zero canonical divisors for non-projective Kähler varieties. We further study rationally connected manifolds with pseudo-effective tangent bundle, focusing on their relation to Fano-type properties, almost homogeneity, and fundamental groups. In the proof, we also show that every surjective endomorphism of a compact klt Kähler variety lifts to a suitable maximally quasi-étale cover.

Fix integers $k,d\geq2$. Let $\mathrm{End}_d^k$ denote the parameter space of endomorphisms of $\mathbb{P}^k$ of algebraic degree $d$. We prove that there exists a dense Zariski open subset $U_{d,k}\subset\mathrm{End}_d^k$, defined over $\mathbb{Q}$, such that, for every $f\in U_{d,k}(\mathbb{C})$, every dominant rational self-map of $\mathbb{P}^k$ commuting with an iterate of $f$ is itself an iterate of $f$. The key intermediate result classifies dominant rational semiconjugacies to iterates of $f$ from normal projective varieties of dimension $k$. The proof uses finite-level monodromy and its action on the rooted preimage tree of a point outside the branch locus of an iterate of $f$. An analogous centralizer theorem holds generically for regular polynomial endomorphisms of $\mathbb{C}^k$. These results extend several theorems of Pakovich to higher dimensions. We give two applications. We first classify periodic dominant correspondences between a general endomorphism and an arbitrary endomorphism of degree at least two. We then show that, for a general $f\in\mathrm{End}_d^k$, an irreducible hypersurface of $\mathbb{P}^k$ is $f$-special in the sense of Ghioca–Tucker and DeMarco–Mavraki if and only if it is $f$-preperiodic.

For a linear map on a matrix space, we study the projective schemes obtained by intersecting its projectivized kernel with determinantal rank loci. For matrix nets, that is, three dimensional matrix spaces, we classify every positive dimensional intersection with the rank one Segre variety in arbitrary rectangular size. The possibilities are a ruling plane, a smooth conic, two Segre lines from opposite rulings, a reduced Segre line, or a Segre line with one reduced or embedded residual point. The smooth conic case is contained in a $2\times2$ compression. For $a,b\ge3$, the corresponding locus in $\mbox{Gr}(3,M_{a,b})$ has exactly three irreducible components, whose geometry and intersections are determined explicitly. For nets in $M_3(\mathbb C)$, every finite rank one scheme has length at most three; the adjugate identity gives an intrinsic determinantal obstruction to the length four case allowed for general systems of plane quadrics. As an application, a four parameter family arising from the merging construction admits exact positivity and decomposability criteria. After normalization, positivity is the unit square and decomposability is the quarter disk. The remaining region is atomic and carries explicit PPT entangled states of birank $(5,5)$ and Schmidt number two. The point of this application is that the exact phase boundary is realized on a smooth conic determinantal kernel stratum selected independently by the projective classification.

We define a derived stack $\mathscr{Rep}_n(X)$ which generalizes the assignment $A\leadsto \operatorname{Rep}_n(A)$ to an algebra of its derived $GL_n$-representation variety of arXiv:1112.1449 from algebras $A$ to perfect stacks $X$ over characteristic 0 fields. In the case of a quasi-projective classical scheme $X$, we show that $\mathscr{Rep}_n(X)$ admits a subfunctor $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)\subset \mathscr{Rep}_n(X)$, which is in fact represented by a derived scheme almost of finite type. We further construct a Fourier-Mukai integral transform between the derived categories of quasi-coherent sheaves on $X$ and $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ which induces a trace map at the level of Hochschild homology generalizing the trace morphism constructed in arXiv:1112.1449 (for finitely presented commutative algebras). We show that the subfunctor $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ is a derived enhancement of the framed locus of the Quot scheme of points and that this derived scheme is a $GL_n$-torsor over the stack of coherent length $n$ torsion sheaves. Hence, this stack is an analog of the derived character stack for quasi-projective schemes, and we show that for smooth, Calabi-Yau $X$, it inherits a shifted symplectic structure in the sense of arXiv:1111.3209 from the one constructed in arXiv:1812.11913 on the moduli stack of perfect complexes with proper support. This structure is shown to give a generalization of the classical symplectic structure on character varieties of surfaces of genus 1 constructed by Goldman \cite{gold}.

Existing classifications of real plane cubic curves rely on sophisticated tools and require a lengthy exposition. This paper provides a concise yet elementary treatment of this classical topic. Using the centers of polynomials, we establish a correspondence between the algebraic structure of these centers and geometric properties of binary cubic polynomials, which yields a novel approach to Newton's canonical forms.

We construct Ulrich bundles on smooth intersections of two quadrics. We determine all possible ranks and prove the existence of indecomposable Ulrich bundles of every allowable rank. To our knowledge, this gives the first such construction for a family of nonhomogeneous varieties of arbitrarily large dimension. We also study the moduli spaces of these bundles and relate them to moduli spaces of vector bundles on curves. For intersections of two even-dimensional quadrics, our results prove a conjecture of Eisenbud and Schreyer.

We give a complete characterization of the weak Lefschetz property (WLP) for monomial complete intersections over a field of positive characteristic. Richard Stanley observed that in characteristic zero WLP, and in fact the strong Lefschetz property (SLP), holds for all degree sequences, as a consequence of the hard Lefschetz theorem, and the same result was explained by Junzo Watanabe using the representation theory of $\mathfrak{sl}_2$. In positive characteristic, many partial results are known, most notably the classification for constant degree sequences due to Brenner–Kaid and Kustin–Vraciu. Our approach is based on a cohomological and representation-theoretic interpretation of WLP. Combined with an analysis of cohomology characters, this leads to a complete numerical criterion for WLP, expressed by simple inequalities involving the $p$-adic digits of the exponents. We also give a new proof of the known classification of SLP using Renaud's algorithm for multiplication in the Green–Han–Monsky ring.

We prove that if $X$ is a semi-rigid normal rational affine algebraic variety with torus action of complexity one, with only constant invertible global functions and finitely generated divisor class group, then the identity component of the automorphism group of $X$ is a semi-direct product of a torus and the maximal unipotent-generated subgroup.

Given a collection of algebraic numbers $\mathcal{S}\subset \overline{\mathbb{Q}}$ we study the varieties $V$ in $\mathbb{A}^m_\mathbb{C}$ such that $V(\mathbb{C})\cap\mathcal{S}^m$ is Zariski-dense in $V$. We show that for many classical families of algebraic numbers $\mathcal{S}$—such as the family of roots of generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, for a finite collection of $\alpha\in \mathbb{Q}$—an unlikely intersections theorem holds. For example, in the case $m=2$, we prove that an irreducible curve in $\mathbb{A}^2_\mathbb{C}$ has infinitely many points from $\mathcal{S}^2$ if and only if it is of the form $x_1=x_2,$ or $x_1=s$, or $x_2=s$ for a fixed $s \in \mathcal{S}$. This is an analogue of the classical theorems of Ihara, Serre, and Tate, treating the case of $\mathcal{S}$ consisting of the roots of unity, and of the Manin–Mumford conjecture. We also show that a similar result holds almost surely for roots of a collection of random polynomials of growing degree and bounded height. The proofs rely on a uniform Galois-theoretic criterion ensuring the unlikely intersection property.

The question of whether a 3-by-3 magic square of squares with distinct integer entries exists has been open since the 18th century. We study the geometry of the algebraic surface parameterizing 3-by-3 magic squares of squares, which is a singular complete intersection of six quadrics in projective 8-space. We compute its geometric automorphism group via an argument involving Gale duality. We compute the basic topological invariants and Hodge diamond of its resolution. We provide an explicit rank 518 sublattice of its geometric Picard group, close to the Hodge-theoretic upper bound of 544. Finally, we study the geometry and arithmetic of del Pezzo, K3, and Enriques surfaces that arise as coordinate projections.

We provide a complete classification of Fano fourfolds of Picard rank 1 and index 1 obtained as zero loci of completely reducible homogeneous vector bundles in generalized Grassmannians, i.e. Grassmannians of Dynkin type different from A. This extends the classification done by Küchle in standard Grassmannians. In doing so, we exhibit three new families of Fano fourfolds of Picard rank 1 and index 1. For each of the studied Fano fourfolds, we compute several numerical invariants, such as volume and Hodge numbers.

We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity–fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of $\operatorname{Fl}(1,2;V_n)$ embedded by $\mathcal{O}(1,1)$: the only defective cases are $\sigma_2(\operatorname{Fl}(1,2;V_3))$ and $\sigma_3(\operatorname{Fl}(1,2;V_4))$, both of defect one.

In this paper, we study non-perturbative vortex partition functions of four-dimensional N=1 supersymmetric gauge theories on the space-time geometry $T^2 \times D^2$, and we propose that these partition functions offer a notion of quantum elliptic cohomology. For particular U(1)-gauge theories we calculate these partition functions explicitly by applying equivariant localization methods to Handsaw quiver varieties that realize for this particular class of U(1)-gauge theories the moduli spaces of the non-perturbative vortex sectors. The determined vortex partition functions are annihilated by difference operators, which are interpreted as Ward identities among N=(0,2) BPS surface defects. Compared to lower dimensional gauge theories with four supercharges, anomalies play an essential role for a consistent formulation of the four-dimensional partition functions. In our proposal towards a mathematical formulation of the vortex partition functions in terms of equivariant elliptic cohomology, the gauge theory anomalies relate to geometric properties of the Thom sheaves corresponding to the relevant quasimap moduli spaces. Motivated by the explicit computations we reflect on the existence of a `virtual structure sheaf' on the moduli space of quasimaps for a general mathematical theory of quantum elliptic cohomology.

Higher genus Gromov-Witten invariants of Calabi-Yau threefolds are encoded in a generating function which is an asymptotic series in a formal parameter $\lambda$. Using resurgence, analytic functions in this formal parameter were uncovered. In this paper we focus on the resolved conifold and study the enumerative meaning of the strong-coupling asymptotic expansion, in powers of $ 1/\lambda $, of the resurgent analytic functions. We show that this expansion contains both a closed curve-counting contribution in dual variables and a contribution governed by relative Gromov-Witten invariants, naturally interpreted in logarithmic geometry.

Simple affine vertex operator algebras $L_k(\mathfrak{g})$ at non-integral admissible levels $k$ are generally neither $C_2$-cofinite nor rational. Despite the absence of these standard finiteness conditions, we prove that the sheaf of coinvariants (and, dually, of conformal blocks) of ordinary modules in the category $\mathcal{O}_k$ forms a vector bundle over the moduli space $\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed genus-zero curves. The proof relies on establishing the finite-dimensionality of genus-zero coinvariants, a vanishing theorem that isolates ordinary admissible modules at the boundary, and a smoothing theorem enabled by partial strong identity elements in the mode transition algebra. Consequently, admissible affine vertex operator algebras supply a rich class of non-rational, non-$C_2$-cofinite examples in which vector-bundle behavior is preserved for a well-behaved subcategory of modules.

In this paper, we show that the border Waring rank of $x_1x_2\cdots x_n$ over fields of characteristic zero is exactly $2^{n-1}$. As a consequence, the classical polarization identity is an optimal Waring decomposition even if we allow limits. As a symmetric tensor, this monomial is identified with the $n\times n$ permanent tensor. However, the lower bound is proved via the higher-order Koszul flattening of the $n\times n$ determinant tensor, which is not symmetric.

Let u be a morphism of algebraic varieties. Given m integer or infinite, we prove that the set of points whose fiber has size m is a constructible set.

We show that the Tate module of the $\ell$-adic unramified cohomology $H^3_{\nr}(X, {\Q_\ell}/{\Z_\ell})$ of a smooth projective surface $X$ over a local or strictly local field $k$ of residue characteristic prime to $\ell$ vanishes under suitable conditions. We apply this to answer some questions concerning the left and right kernels of the Brauer–Manin pairing for open subvarieties of $X$. To prove the vanishing theorem, we establish Bloch's formula for the Chow group of codimension-two cycles on a semistable model of $X$, and derive applications to the restriction map for such cycles to the closed fiber of the model.

Classically, the places of $\mathbb{Q}$ are often regarded as a one-point compactification of $\mathrm{Spec}(\mathbb{Z})$, with the real place corresponding to a formal “prime” added at infinity. Re-examining this picture from a topos-theoretic perspective reveals a subtler geometry: while the non-Archimedean places are identified with singletons indexed by the non-zero prime ideals of $\mathbb{Z}$, the Archimedean place is represented by the space of upper reals $\overleftarrow{[0,1]}$, which may be informally thought of as the unit interval equipped with a non-Hausdorff topology. On a technical level, our analysis brings together geometric logic and descent techniques from topos theory, distinguishing standard descent from lax descent toposes both at the level of sheaves and of the geometric theories they classify. More broadly, this paper brings into conversation two parallel distinctions: on the number-theoretic side, between Archimedean and non-Archimedean phenomena, and on the topos-theoretic side, between standard and lax descent. Looked at from a high level, these perspectives begin to converge on a common theme: how should the connected and the disconnected interact?

In their work on Brill-Noether theory, Eisenbud and Harris established the geometry of the universal parameter space of linear series over curves, proving that for even genus $g$ and degree $d = g/2 + 1$, the projection to the moduli space of curves is a finite cover of degree equal to the Catalan number $C_{g/2} = \frac{1}{g/2+1}\binom{g}{g/2}$. In this paper, we construct the tropical counterpart of this universal family: a polyhedral cone complex $\mathcal{G}_{g \to 0, d}^{\mathrm{trop}}$ parametrizing degree-$d$ tropical morphisms from genus-$g$ metric graphs to metric trees. For even $g$ and $d = g/2 + 1$, we prove that the forgetful projection $\Pi \colon \mathcal{G}_{g \to 0, d}^{\mathrm{trop}} \to \mathcal{M}_{g}^{\mathrm{trop}}$ is a branched cover of degree $C_{g/2}$ equipped with natural determinantal multiplicities. We compute this degree by showing that on caterpillars of loops the morphisms are in bijection with ballot sequences, and we establish its global invariance across $\mathcal{M}_{g}^{\mathrm{trop}}$ via a tropical balancing condition across codimension-$1$ walls. Via deformation and path lifting, this yields an effective method to construct Catalan-many gonality-witnessing maps for any generic metric graph, establishing that the tree gonality of any genus-$g$ metric graph is at most $\lceil g/2 \rceil + 1$.

This is the first part of a program to construct hyperholomorphic families of boundary conditions for the Kapustin–Witten B-twist of the Langlands QFT, otherwise known as (BBB)-branes. We define the category of quasi-algebraic sheaves over the Deligne moduli stack, which serves as an analog of the twistor space of Hitchin's moduli stack. This allow us to construct a representation of a simplified version of the Moore–Tachikawa category which is motivated by the relative Langlands program in the sense of Ben-Zvi–Sakellaridis–Venkatesh.

We classify complete toric threefolds $X$ such that the automorphism group $\text{Aut}(X)$ acts on $X$ with an open orbit whose complement does not contain a divisor. The latter condition means that for any ray $\rho$ of the fan $\Sigma_X$ there is a Demazure root of $\Sigma_X$ associated with $\rho$. We also find among these varieties those $X$ for which the group $\text{Aut}(X)$ is transitive on the smooth locus $X^{\text{reg}}$. The classifications are based on Gale-dual interpretations of these properties.

We prove a refinement of Hirzebruch's signature formula on the individual Hochschild diagonals of a smooth projective complex variety. Natural Hermitian forms, obtained by symmetrizing a Todd-corrected pairing, have explicitly computable positive and negative indices and are preserved by derived equivalences. On diagonals of the same parity as the dimension, their signatures separate the even and odd contributions to the Hochschild-Kostant-Rosenberg decomposition; on the other diagonals, their radicals and signatures are governed by the first Chern class. We deduce the derived invariance of $h^{n-2,0}$, $h^{n-1,1}$ and $h^{n-3,1}$ in any dimension, and of $h^{2,0}$ when the canonical bundle is trivial. Combining the invariance of $h^{2,0}$ with the multiplicative structure of Hochschild cohomology and Verbitsky's orthogonal action, we give a new short proof of the theorem of Huybrechts and Nieper-Wißkirchen that a derived partner of an irreducible holomorphic symplectic variety is again irreducible holomorphic symplectic. In dimension five with trivial canonical bundle, the signature invariants and the Libgober-Wood identity together establish the derived invariance of all Hodge numbers

We prove that the maximal derived length of a solvable subgroup of the plane Cremona group over $\mathbb C$ is equal to $5$. The main difficulty is the case of subgroups containing loxodromic elements. To treat this case, we prove that the maximal derived length of a finite solvable subgroup is $4$ and establish a centraliser theorem: the centraliser of a finite solvable subgroup of derived length $4$ contains no loxodromic element. The proof of the centraliser theorem relies on Blanc's classification of maximal algebraic subgroups and then uses equivariant Sarkisov theory, orbit estimates, superrigidity, and fixed curves of genus at least two.

Let $k=\mathbb F_q$. We classify, up to pre- and post-composition by $k$-Möbius transformations, all separable $k$-indecomposable rational functions $f\in k(X)$ of degree greater than one whose Galois closure has genus zero. The classification is valid in arbitrary characteristic and includes exact arithmetic conditions and class counts over the prescribed field. Semilinear Frobenius descent determines the finite-field forms and which geometric decompositions descend to $k$. For every separable $f\in k(X)$ of degree greater than one with Galois closure of genus zero and every $m\geqslant1$, we prove that $f$ permutes $\mathbf P^1(\mathbb F_{q^m})$ if and only if it is exceptional over $\mathbb F_{q^m}$, meaning that it permutes $\mathbf P^1(L)$ for infinitely many finite extensions $L/\mathbb F_{q^m}$. No indecomposability assumption or lower bound on $q$ is needed. The argument combines fixed-point averaging with ramification on the Galois-closure curve. If the full constant field is $\mathbb F_{q^d}$, these properties depend only on $\gcd(m,d)$. For each classified family we determine the permutation extension degrees explicitly and characterize polynomial representatives.

Symmetric powers of matroids were introduced by Lovász and Mason. The subject has recently attracted renewed interest through its connection with the realizability of tropical linear spaces by tropical ideals. Anderson proved that the tropical linear space associated with a matroid $M$ is the variety of a tropical ideal if and only if $M$ has a $d$th symmetric power for every positive integer $d$. He also asked whether the Vámos matroid $V_8$ has a second symmetric power. We answer this question in the negative by proving that $V_8$ has no second symmetric power. By Anderson's characterization, the associated tropical linear space cannot be realized as the variety of a tropical ideal.

Two combinatorial rules are known for the Grassmannian Grothendieck polynomial $G^{(\beta)}_\lambda$: a sum over set-valued tableaux of shape $\lambda$, due to Buch, and a sum over the efficient cells of a cellular decomposition of the Gelfand-Zetlin polytope $GZ(\lambda)$, due to E. Presnova and the author. All coefficients in both sums equal $1$. We construct an explicit bijection between the two indexing sets which matches the summands term by term, carrying the number of excess entries of a tableau to the dimension of the corresponding cell; in particular the two rules are equivalent, either being deducible from the other. The efficiency condition on cells turns out to be the column-strictness of tableaux. We then transport Yu's square-root crystal operators to the cells and find that they respect dimension, along a double $i$-string the cells alternate between two consecutive dimensions, but not incidence: consecutive cells of such a string need not share a point, already for $\lambda=(2,1,0)$.