Papers

In this paper, we discuss recent developments and open problems concerning the geometry of compact Kähler varieties whose tangent sheaves are pseudo-effective in a strong sense. As our main result, we prove that, after passing to a finite quasi-étale cover, any compact Kähler variety with quotient singularities and pseudo-effective tangent sheaf admits a flat fibration onto a complex torus, with rationally connected fibers. We also obtain an analogous structure theorem for klt compact Kähler varieties, assuming the existence of minimal models in numerical dimension zero; consequently, the result holds unconditionally in the projective setting and for low-dimensional compact Kähler varieties.

In this paper, we study cyclic spectral data on smooth varieties over an algebraically closed field in characteristic zero. We show that the Hitchin fiber of any nonzero cyclic spectral data is non-empty. We also construct sections of the Hitchin morphism for cyclic spectral data and tautological (flat) families of cyclic spectral covers.

We prove an explicit lower bound of optimal quadratic order for the anticanonical volume of every complex $\epsilon$-log canonical weak del Pezzo surface, with no restriction on its Picard number or its singularities. More precisely, the volume is at least $\min\{c_{exc},\epsilon^2/3032064\}$, where $c_{exc}>0$ is an explicit absolute constant. Quotients of the projective plane show that the exponent two is optimal. We also prove cubic upper bounds for the cyclic determinant mass and for the Picard number of a smooth rational log Calabi–Yau pair. We give geometric examples showing that these two auxiliary exponents are optimal in their respective settings.

Recently, Huang and Satriano determined the sharp dimension threshold for Batyrev's non-negativity conjecture on stringy Hodge numbers: the conjecture holds in dimensions at most four and fails in every dimension at least five. We show that their counterexamples fit into a single broad three-parameter family $Y_{m,d}:=V(\sum_{i=0}^m s_i g_i(x)+h(x))\subset\mathbb{P}^{m+d+1}$, where each $g_i$ is homogeneous of degree $d-1$ and $h$ is homogeneous of degree $d$, whose singular locus is a linear space $\Lambda\cong\mathbb{P}^m$. Blowing up $\Lambda$ gives a log resolution with one smooth exceptional divisor of discrepancy one. We compute the Hodge-Deligne polynomials of both the resolution and the exceptional divisor from their projective-space fibrations over $\mathbb{P}^d$, derive a master formula for $E_{\mathrm{st}}(Y_{m,d}\times(\mathbb{P}^1)^n)$, and prove the structure formula $h_{\mathrm{st}}^{p,q}(Y_{m,d}\times(\mathbb{P}^1)^n)=A_{p,q}+B_{p,q}-C_{p,q}$, where the three terms are non-negative. Thus the only possible source of negativity is $-C_{p,q}$, which is governed by the primitive middle cohomology of $Z_{m,d}=V(g_0,\ldots,g_m)\subset\mathbb{P}^d$. Off the diagonal, every negative entry equals $-C_{p,q}$, a binomially weighted sum of shifted primitive Hodge numbers of $Z_{m,d}$. On the diagonal, the behaviour is governed by the parity of $m+d$. We also obtain a complete classification for $0\le m\le d-1$, $d\ge3$, and $n\ge1$. The only non-counterexamples are $(m,d)=(0,3)$ for every $n\ge1$, and $(m,d)=(2,3)$ for $n\ge4$.

We study the Brill–Noether theory of higher rank sheaves on surfaces $X$ with $H^1(X,\mathcal{O}_X) = 0$ and its interactions with the classical Brill–Noether theory of line bundles on curves lying on $X$. Denoting by $\mathrm{Spl}(v)$ the moduli space of simple sheaves on $X$ with Chern character $v = (r,c_1,\mathrm{ch}_2)$, we introduce the Brill–Noether loci $BN^k(v)^\circ \subseteq \mathrm{Spl}(v)$ of \emph{generically $k$-generated} sheaves and show, under assumptions on $c_1$ and $K_X$, certain well-behavedness results. In particular, if $c_1$ is indecomposable and $X$ is a K3 surface, we generalize a Brill–Noether theorem for sheaves on K3s due to Yoshioka. This is achieved by employing He's deformation theory of the moduli space of coherent systems and a globalization of the Lazarsfeld–Mukai construction, providing a correspondence between the Brill–Noether theories of generically $r$-generated sheaves on $X$ and that of line bundles on curves $C \in |c_1|$. The same techniques yield upper bounds on the dimension of $\mathcal{W}^{r,\mathrm{bpf}}_d(|c_1|) \setminus \mathcal{W}^{r+1}_d(|c_1|)$ consisting of basepoint-free line bundles in terms of the number of endomorphisms of associated Lazarsfeld–Mukai bundles, for arbitrary $c_1$. This generalizes a result of Aprodu and Farkas in the context of Green's conjecture beyond K3 surfaces. In the case $r = 1$ of pencils, we obtain smoothness and expected-dimension results under an effectivity condition on $-K_X|_C$ for $C \in |c_1|$.

Given a convex lattice polygon $\Delta\subset \mathbb R^2$, let $N_\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\Delta$ on the toric variety $\mathbb P_\Delta$ defined by $\Delta$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\Delta$. We perform calculations of $N_\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-Göttsche, and hence encodes higher-genus logarithmic Gromov–Witten invariants, by a theorem of Bousseau.

We prove a Frobenius-dilation inequality for Hilbert functions of ideals generated by powers of fixed homogeneous forms in characteristic zero. Via Emsalem–Iarrobino inverse system duality, it yields quantitative fat-point postulation estimates and a new proof of the recent theorem of Ha–Sivakumar resolving Demailly's conjecture for arbitrary finite point sets. A fixed-characteristic version controls tangent spaces to varieties of powers and, together with plane interpolation and secant-theoretic results, proves complete nondefectivity of $V^k_{2,d}$ for all $k\ge3$ and $d\ge2$; in particular, it settles the ternary Waring problem for generic $k$-rank conjectured by Lundqvist–Oneto–Reznick–Shapiro (suggested by Ottaviani). We also show that, for a prime power exponent $q$, any failure of Nicklasson's conjecture in three variables is confined to $2q-2$ consecutive degrees.

For every nonempty Hassett divisor $\mathcal C_d$ parameterizing special cubic fourfolds, we consider the quaternion class $\beta_d=(d/2,-3)$ in the two-torsion Brauer group. We prove that if a Hodge-general member of ${C}_d$ is rational then $\beta_d=0$ – or, equivalently, that Huybrechts' twisted-K3 condition $(**')$ holds. Consequently, a very general member of $\mathcal C_d$ is irrational if $\beta_d\ne0$. Thus, a very general cubic fourfold containing a smooth cubic scroll or a Veronese surface is irrational, and hence so is a very general Küchle fourfold of type $(\mathrm{c7})$. We also obtain an analogous obstruction for Hodge–special Gushel–Mukai fourfolds: a very general member in the discriminant–$d$ locus can be rational only if $d$ is a sum of two squares. Consequently, a very general Gushel–Mukai fourfold containing a cubic scroll is irrational.

Based on the conjectured (and checked for tree-type quivers) quiver Yangian/Coulomb branch algebra correspondence, we give a quiver Yangian interpretation of the monopole formulas for 3D $\mathcal N=4$ good $A$-type quiver gauge theories, also known as the $T_\rho(SU(N))$ theories. Using the algebra action on the $\frac12$-BPS vortices, we reorganize the generators of the truncated shifted quiver Yangian into boundary-adapted generators, and obtain the classical relations in terms of $U(N)$ Casimirs on the Slodowy slice $\mathcal S^{\mathfrak{gl}_N}_\rho$, whose coordinates are given by these boundary-adapted generators. We also show that the boundary-adapted generators and the Casimir relations have the correct fugacities as predicted by the Hall-Littlewood expression of the monopole formula for $T_\rho(SU(N))$, hence reconstructing it as the Hilbert series of the truncated shifted quiver Yangian.

Laza and the second author have classified all possible symplectic automorphism groups of smooth cubic fourfolds. There are $34$ such groups, and by Koike, there are $48$ families of cubic fourfolds with speficied symplectic automorphism group. For $42$ among those families, we complete the classification of all possible automorphism groups. The approach of this paper is mainly lattice theoretic, starting from known results about the period map of cubic fourfolds by Voisin, Hassett, Looijenga and Laza. We use a lattice-enumeration algorithm implemented in OSCAR.

Using some recent examples of Anghel in dimension $2$, we prove that for any $n \ge 3$ there exist many smooth $n$-dimensional varieties $X$ with a very ample line bundle $H$, such that there is no Ulrich bundle for $(X,H)$. Moreover, we find many new examples in dimension $2$.

We prove that a general surface in $\mathbb{P}^3$ of degree at least $18$ contains no rational or elliptic curves, strengthening the classical result of Clemens by replacing the original very general assumption by a genuine Zariski-open condition. Previously, nonexistence results in the “general” setting were known only in much higher degrees. Combining this with established algebraic degeneracy results for entire curves, we deduce the Kobayashi hyperbolicity of a general surface in $\mathbb{P}^3$ of degree at least $18$, thereby resolving a question asked by Demailly–El Goul. Our proof uses foliations induced by $2$-jet differentials. Two independent such differentials give rise to a multi-foliation tangent to all rational and elliptic curves. We establish a Poincaré-type bound for its algebraic leaves, yielding a mechanism to upgrade very general statements to general ones. Our method also applies to complements of plane curves. In particular, we prove that the complement of two general cubic curves in $\mathbb{P}^2$ is hyperbolically embedded.

We study transversely projectively flat regular foliations on compact Kähler manifolds.

For a constructible sheaf $\mathcal{F}$ on a smooth scheme $X$ over a perfect field, the singular support $SS(\mathcal{F})$ and the characteristic cycle $CC(\mathcal{F})$ are defined on the cotangent bundle. In analogy with the construction of Kashiwara–Schapira via microlocalization in the transcendental setting, we construct the microlocal versions ${ SS}_\mu(\mathcal{F})$ and $CC_\mu(\mathcal{F})$ of $SS(\mathcal{F})$ and $CC(\mathcal{F})$ respectively, and prove \[ SS_\mu(\mathcal{F})= SS(\mathcal{F}),\qquad CC_\mu(\mathcal{F})= {\rm cl}CC(\mathcal{F}), \] where $\operatorname{cl}$ is the cycle class map. As a consequence of the second equality, we show that the cohomological characteristic classes are the cycle classes associated with the corresponding characteristic cycles, as conjectured in (Invent.~Math.~207: 597–695, 2017). As further applications, we give new proofs of the index formula and Bloch's conductor formula in the proper case, independently of Abe's homotopical approach via $\infty$-categories.

For irreducible components $C_\sigma$ of Lusztig's nilpotent varieties of type $A$ with graded dimension $(1,2,\dots,n,n-1,\dots,1)$ arising from permutations $\sigma\in S_n$, we characterize the property that $C_\sigma\oplus C_\sigma$ is an irreducible component combinatorially in terms of $\sigma$. The permutations that occur, which we call pseudosmooth, are described by a recursion on direct sums and deleting suitable corners, whose terminal cases are the permutations obtained from \[ 3412,\quad 4231,\quad 35142,\quad 42513,\quad 45312,\quad 426153,\quad 463152,\quad 526413 \] by inflating the entries into consecutive decreasing blocks. The main new input is a hereditary property valid for decompositions of multisegments with disjoint extreme points.

Let $G$ be a finite group acting properly by lattice automorphisms on a complete simplicial fan $\Sigma$. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology $H^*(X_{\Sigma})$ of the associated toric variety $X_{\Sigma}$ is isomorphic to a permutation representation of $G$. We prove that Stanley's question has an affirmative answer for all smooth projective toric varieties without the properness assumption on the action. The proof is inspired by toric mirror symmetry.

In this note, we show that the Hilbert scheme $\text{Hilb}_{P_{d,4}(t)}(E_6/P_6)$ associated with the Hilbert polynomial $P_{d,4}(t)$ is disconnected by determining that it has exactly two connected components. This result adapts Seong's methods, successfully extending the disconnectedness of Grassmannians to the exceptional type $E_6$.

We prove that a nonconstant polynomial $f\in\mathbb{C}[x_1,x_2,x_3]$ is cylindrical (that is, $f\in\mathbb{C}[x_1,x_2]$ after an invertible linear change of coordinates) if and only if its bordered Hessian determinant vanishes identically. The same conclusion holds over $\mathbb{R}$. We also give explicit counterexamples to this criterion in every dimension $n\ge4$.

We consider the family of cyclic fourfold covers $w^4 = \prod_{j=1}^{7}(z-x_j)$ of the complex projective line branched at the eight points $x_1,\ldots,x_7,\infty$. The period map identifies the configuration space $X(2,8)$ of the branch points with a Zariski open subset of a quotient of the five-dimensional complex ball. The inverse of the period map is expressed projectively by $105$ automorphic forms $f_J$, which are proportional to the signed branch-point polynomials $\hat x_J$ with a common scalar factor. We determine this factor for the period $\eta$ of the differential $dz/w$: it is the product of the constant $-1/(2^{12}\Gamma(3/4)^{16})$ and the square of a quadratic form in $\eta$.

It is an important problem to know whether the monodromy on the cohomology of Milnor fibers associated to hyperplane arrangements is a combinatorial invariant. In this paper, we obtain a combinatorial vanishing criterion for certain eigenspaces of this algebraic monodromy in line arrangements. Combining with Hirzebruch inequality, which is a consequence of Bogomolov–Miyaoka–Yau inequality, we prove that for essential complex line arrangements, the eigenvalues of the monodromy have orders at most five. This is a partial progress toward Papadima–Suciu conjecture and proves Salvetti–Serventi connectivity conjecture. For essential complexified real line arrangements, the monodromy order is improved to at most four thanks to Shnurnikov's inequality. This confirms Papadima–Suciu conjecture for real line arrangements and also Yoshinaga's sharp pair conjecture.

Borisov and Yeung showed that the Cartwright-Steger surface $X$ is defined over ${\mathbb {Q}}$, hence complex conjugation gives an antiholomorphic involution on $X$. We show that the orbit space $Y$ is a closed oriented smooth four-manifold homeomorphic to ${\mathbb {CP}}^2\# (S^1 \times S^3)$, presenting $X$ as a double branched cover $X\to Y$, with branch locus diffeomorphic to the connected sum of three copies of the real projective plane.

We introduce a notion of $2$-homological category modelled on pointed homological categories in the sense of Grandis, using bizero objects whose null $1$-cells are zero objects in the hom-categories for formulating $2$-dimensional pointedness. We first prove directly that the $2$-category of di-exact homological categories, which generalize Puppe exact categories, functors preserving all kernels and cokernels, and arbitrary natural transformations is $2$-homological. Its normal subcategories are saturated thick subcategories, and its exact quotients are constructed by a complete three-arrow fraction calculus similar to the one known for Puppe exact categories. We then adapt this proof to prove that the $2$-category of triangulated categories is also $2$-homological; here, instead of the ternary fractions we use the well-known Verdier fractions. Abstracting the common quotient structure of these proofs yields a general criterion, using which we further establish that the $2$-categories of pointed, additive and abelian categories are also $2$-homological. The criterion also applies to categories enriched in semimodules over a fixed commutative rig, with a zero object, and to their full sub-$2$-category with finite biproducts. Their quotients are linear congruence quotients. Combining enrichment with Puppe exactness gives further examples whose quotients are exact linear localizations, including linear abelian categories as the finite-biproduct case. In the abelian and triangulated cases the normal subcategories and quotients are, respectively, Serre subcategories and Serre quotients, and thick triangulated subcategories and Verdier localizations.

We prove that the number of curves of a fixed genus g with n marked points over finite fields is a polynomial function of the cardinality of the field if and only if g = 0 or 3g + 2n is less than or equal to 24. This confirms a conjecture of Canning, Larson, and the authors.

The dimer spectral transform gives algebro-geometric action-angle coordinates for cluster integrable systems. Fock gave an explicit formula for its inverse. We study Fock's inverse map in the tropical limit where Harnack spectral curves degenerate to nodal curves. We determine the leading order asymptotics of the classical objects appearing in Fock's formula and show that they are governed by their tropical counterparts. This yields a tropical inverse spectral transform on tropical spectral data. We show that Fock's inverse spectral transform commutes with tropicalization, and also derive a tropical version of Fay's trisecant identity.

Schwartz-MacPherson classes of degeneracy loci of symmetric and skew-symmetric maps. Together with the known formulas for ordinary linear maps, this completes the stable SSM theory for the three classical types. We interpret these formulas probabilistically, in terms of the endpoint random partition of a stochastic rational six-vertex model: the skew classes are probabilities of Maya-dimer events, and the symmetric classes are Chebyshev moments of the random 2-core.

We prove that continuous $A$-bilinear pairings on the ring of Laurent series that are invariant under continuous automorphisms coincide, up to a constant, with the pairing given by the residue of a differential form over any commutative associative ring with identity.

In this paper, we prove Shokurov's conjecture in relative dimension $1$-case in sufficiently large characteristics. Depending on this result, we prove the boundedness of Stein degrees for horizontal boundaries in a log Calabi-Yau fibration in sufficiently large characteristics.

We construct good quotients for equivariant actions of group homomorphisms on morphisms of schemes. Using Geometric Invariant Theory, we obtain explicit open semistable loci in the source with Hilbert-Mumford descriptions admitting good quotients relative to a given good quotient of the target. In particular, we obtain quotients for reductive groups acting on projective-over-affine morphism. In the non-reductive case, we consider equivariant actions on affine morphisms which are 'graded' by a multiplicative group and satisfy certain unipotent stabiliser assumptions. This recovers known results in projective non-reductive GIT as a special case, which also proves the Hilbert-Mumford criterion in that setting. As applications, we consider moduli of unstable objects, representations of quivers with multiplicities and jets.

Let $X$ be a toric Fano surface. Let $\pi: \widehat{X} \rightarrow X$ be a toric blow up at a point. We use the Topological Vertex [AKMV] to prove a higher genus, higher winding, open-closed duality between the toric Calabi-Yau 3-folds $K_X, K_{\widehat{X}}$. For $g \geq 0, w \geq 1$, we show the equality $n_g(K_{\widehat{X}}, \pi^*\beta - wC) = (-1)^g N_{g, (w)}^{LMOV}(K_X/L, \beta)$, where $n_g(K_{\widehat{X}}, \pi^*\beta - wC)$ is the genus-$g$, Gopakumar-Vafa invariant of $K_{\widehat{X}}$ in curve class $\pi^*\beta-wC$, where $\beta \in H_2(X, \mathbb{Z})$ and $C$ is the exceptional curve, and $N_{g, (w)}^{LMOV}(K_X/L, \beta)$ is the genus-$g$, LMOV invariant of an outer Aganagic-Vafa brane $L \subset K_X$ in class $\beta$ and representation $(w)$, or the Young Tableau of a single row with $w$ boxes.

Let $p,q$ be positive braids with Demazure products $u,v$. The endpoint stratum $\mathrm{Conf}(p,q)_{u,v}$ of the double Bott-Samelson cell splits as $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$, a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on $\mathrm{Conf}(p,q)$, and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map $X(p) \times X(\Delta\Delta) \to X(p\Delta)$, where $\Delta$ is a reduced positive braid for $w_0$ and $\mathrm{dem}(p) = w_0$.