Papers

In this article, we present the results relating different rationality properties with $\mathbb{A}^1$-homotopy theoretic invariants.

For noncompact smooth Kähler-Ricci shrinkers, we prove a Matsushima theorem identifying the centralizer of the soliton field with the complexification of its Killing centralizer, uniqueness up to biholomorphism on a fixed complex manifold, and K-polystability of the associated polarized Fano fibration.

For a compact Riemann surface $X$, Simpson associated to each stable graded Higgs bundle $(E,\theta)$ a locus $W_{[(E,\theta)]}^1 \subset M_{\mathrm{dR}}(X,n)$, consisting of flat bundles that admit a Simpson filtration whose associated graded Higgs bundle is $(E,\theta)$. He conjectured that $W_{[(E,\theta)]}^1$ is Zariski closed in $M_{\mathrm{dR}}(X,n)$. We prove this conjecture in arbitrary rank. The proof is based on the Quillen geometry of the determinant-of-cohomology line bundle over the moduli of holomorphic bundles. To any holomorphic family of flat bundles, we associate a determinant-of-cohomology line bundle equipped with a canonical holomorphic determinant connection, and derive explicit formulas for its curvature. For a family of flat bundles in $W_{[(E,\theta)]}^1$, we then prove that the determinant connection form is exact. This exactness forces the associated determinant frame to extend as a nowhere-vanishing frame across any one-parameter degeneration, which controls the limiting filtration and yields the desired closedness.

Let $X$ be the moduli space of stable trace-free rank $r$ Higgs bundles over a smooth projective curve $C$ with fixed determinant $\Lambda\in\Pic(C)$ and $\gcd(r,\deg\Lambda)=1$. We prove that the $\cc^*$-equivariant quantum product of a divisor on $X$ is given by the Steinberg correspondence of the Lagrangian Steinberg cycle in $X\times X$. The Steinberg cycle is the image of the reduced virtual fundamental cycle of the $2$-pointed Kontsevich stable map space to $X$ under evaluation map. For $\Gamma=\Pic^0(C)[r]$, we give a character decomposition for $\Gamma$-invariant curve classes or finite orbit sums. In rank two and genus two we explicitly determine its fifteen endoscopic lines and the reduced spectral discriminant. %The resulting endoscopic support statement is cohomological, not a determination of the entire Chow cycle. For $\widehat X=[X/\Gamma]$, which is the moduli space of stable $\PGL_r$-Higgs bundles over $C$. We consider the orbifold quantum cohomology theory from a gerbe-twisted theory. The complex K-theory $\KU^*(X)$ and twisted K-theory $\KU^*([X/\Gamma],\alpha)$ by the lifting $\mu_r$-gerbe $\alpha$ of the universal projective bundle is an isomorphism, and is given by the Fourier-Mukai transformation. We prove that the Fourier-Mukai transformation is compatible with the quantum product by the divisors.

The paper contains three results. First, we give a global quotient presentation of the supermoduli space of genus $g>1$ super Riemann surfaces. Second, we prove that the moduli stack of marked super Riemann surfaces, which we call super Teichmüller space, is a complex analytic Deligne Mumford superstack. Third, we construct an étale atlas for super Teichmüller space whose underlying topological space is a disjoint union of $2^{2g}$ contractible unit balls.

We prove Mond's conjecture for $\mathcal A$-finite corank-one map germs $(\mathbb C^3,0)\to(\mathbb C^4,0)$: the $\mathcal A_e$-codimension is at most the image Milnor number, with equality for quasihomogeneous germs. For a quasihomogeneous germ we determine the derivations of its image modulo the conductor fields, in terms of the double-point surface, its cross-cap curve and its normalisation. This yields a closed formula, depending only on the weights and degrees, for the Hilbert series of the graded $\mathcal A_e$-normal space; its value at $t=1$ coincides with Ohmoto's formula for the image Milnor number, which comes from Segre–Schwartz–MacPherson Thom polynomials. For arbitrary germs we show that generic homogeneous corank-one germs of coprime degrees are $\mathcal A$-finite in every source dimension, and apply the reduction theorem of Fernández de Bobadilla, Nuño-Ballesteros and Peñafort Sanchis. In every pair of nice dimensions this reduces Mond's conjecture for corank-one germs to an inequality for generic homogeneous germs.

We develop an approach to study K-stability of Fano varieties by estimating the local stability threshold along a reducible zero-dimensional subscheme on a smooth curve. As an application, we we show that all smooth index-one Fano hypersurfaces in quadrics are K-stable, when the dimension is higher than one. To prove it, we apply the Abban-Zhuang method to reduce the stability problem to lower dimensions. Specifically, we construct suitable admissible flags such that the last term of the Abban-Zhuang estimate is a local stability threshold along multiple points on a smooth curve, and give a convex geometric interpretation of the estimate.

We compute the singular cohomology groups of odd biisotropic symplectic Grassmannians $\operatorname{S_2Gr}(k, 2n+1)$ by presenting an explicit dimension-preserving combinatorial bijection between the Bialynicki-Birula cells in $\operatorname{S_2Gr}(k, 2n+1)$ and the Schubert cells in $\operatorname{Gr}(k, 2n+2-k)$.

We show that ample stability suffices to apply Teleman quantisation to the endomorphism bundle of the universal representation on a quiver moduli space, and thus in particular to deduce its infinitesimal rigidity. This answers a question left open in our earlier work with Brecan, Franzen, and Reineke on the sheaf cohomology of universal bundles. This sufficiency makes several recent results applicable in the widest possible range, subject only to ample stability, which for coprime dimension vectors guarantees that the Picard rank of the moduli space is maximal.

Building on work of Turner, we propose a proof in characteristic two of Quillen's conjecture on finiteness properties of the cotangent complex for homomorphisms between noetherian rings of possibly infinite flat dimension.

We study the actions of hyperoctahedral groups on the cohomology of the minimal De Concini-Procesi wonderful models associated with reflection arrangements of type $B$. We express the generating series of their equivariant Poincaré polynomials in terms of the corresponding type $A$ series and the equivariant reduced characteristic series of type $B$. The key ingredient is a general inversion formula in equivariant incidence algebras relating equivariant Chow functions and super-reduced characteristic functions. Applying this formula to partition and signed partition lattices, we recover Getzler's compositional identity in type $A$ and establish its type $B$ analogue. We also obtain explicit plethystic and infinite-product formulas for the equivariant characteristic series of type $B$.

Let \(A\) be the coefficient ring of a smooth projective curve over \(\mathbb{F}_q\) with a closed point at infinity of arbitrary degree \(\mathrm{N}=\deg(\infty)\ge 1\), and let \(\varphi\) be a rank-one Drinfeld \(A\)-module. In this paper, we prove three main results concerning the module of special functions and its relation to Pellarin's \(\mathcal L(1)\)-series. First, the module \(\mathrm{sf}(\varphi)\) of special functions contains a Tate-algebra unit exactly when it is free of rank one, equivalently when its period lattice is isomorphic to the module of regular differentials. This settles the Gazda–Maurischat conjecture. The proof evaluates Cauchy kernels; on the period lattice all such evaluations agree, and their common characteristic residue recovers the period. Second, for suitable isogeny data to a principal-period target \(\psi\), a strictly normalized shtuka product, together with one characteristic residue, gives an explicit formula for the fundamental period \(\tilde{\pi}\) (the generator of the free lattice \(\Lambda_\psi\)). The same defect equation also yields the residue formula for the Drinfeld logarithm. Third, the first Frobenius twist of the normalized shtuka differential is identified with a pairing built from finite torsion traces, whose coefficients are finite étale traces over the field of definition. As an application, Pellarin's \(\mathcal L(1)\)-series is expressed exactly as the first Frobenius twist of the generating differential at a fundamental period. A notable feature of these results is that they hold for arbitrary $\mathrm{N} \geq 1$, and the source period lattice need not be free.

We construct degree-three isogenies of Hesse elliptic curves over an arbitrary field of characteristic different from three, which correspond to a step of the cubic arithmetic-geometric mean. As an application, we describe the Serre-Tate canonical lift of an ordinary elliptic curve in characteristic three by the 3-adic cubic arithmetic-geometric mean. We also discuss the relations of the Hesse curves with complex and finite hypergeometric functions.

We prove that the perverse filtration of the canonical $\mathrm{SL}_n$ Hitchin system is multiplicative when the rank and degree are coprime. Over the elliptic locus of an even-degree twisted Hitchin system, we compare the Fourier projectors for compactified Pryms with the endoscopic correspondence. We then use the square-zero property of the variant cohomology and a parabolic Higgs moduli reduction to pass to the full canonical moduli space.

Let $p$ be an odd prime, $\psi$ a nontrivial additive character of $\mathbb F_p$, $\psi_q=\psi\circ\mathrm{Tr}_{\mathbb F_q/\mathbb F_p}$ and $\lambda\in\mathbb F_p^\times$. For $h\in(\mathbb F_q^\times)^n$ let $\Sigma_n(h)$ be the sum of $\psi_q(u_1^{-1}+\cdots+u_n^{-1})$ over $u\in(\mathbb F_q^\times)^n$ with $h_1u_1+\cdots+h_nu_n=-\lambda$, an additively twisted average of products of Kloosterman sums. For every $n\ge2$, the cohomology computing $\Sigma_n(h)$ is concentrated in one degree and has dimension $2^n-1$. It has $r(n)=2^n-\binom n{\lfloor n/2\rfloor}$ eigenvalues of weight $n-1$, and its other eigenvalues are the integers $q^{k-1}$, $1\le k\le\lfloor n/2\rfloor$, with ballot numbers as multiplicities. The part of top weight forms a lisse sheaf $\mathcal W_n$ of rank $r(n)$ on the torus $\mathbb G_m^n$. We determine its local monodromy at the coordinate boundary and its determinant: $\det\mathcal W_n$ is the Artin–Schreier sheaf of the sum $-2^{n-1}\lambda^{-1}(h_1+\cdots+h_n)$ of the critical values of the phase, twisted by $p^{(n+1)r(n)/2}$. The geometric monodromy group of $\mathcal W_n$, and the arithmetic one after normalization to weight $0$, are equal to $\{g\in\mathrm{GL}_{r(n)}:\det(g)^p=1\}$. Consequently the normalized sums become equidistributed like traces of Haar-random elements of $\{g\in\mathrm U(r(n)):\det(g)^p=1\}$. The proofs use Laumon's Fourier transform, middle convolution, the Cayley cubic surface and Larsen's alternative, but no classification of finite groups.

Let $L$ be a line bundle of negative degree on an elliptic curve $E$. While the ruled surface $M := \mathbb P(\mathcal O_E \oplus L)$ is one of the simplest varieties in algebraic geometry, its tangent bundle $T_M$ is surprisingly complex: we first show that the ring of symmetric tensors is not finitely generated. We then construct a holomorphic function on the canonical extension $Z_M$ which allows us to show that $Z_M$ is not Stein.

We prove that any counter-example to the two dimensional Jacobian Conjecture with integral coefficients and determinant of its Jacobian matrix equals one, has only paths of asymptotic values that can contain only finitely many points of the planar integral lattice. The second part of our paper gives some nontrivial applications of the theory of certain Theta functions to the two dimensional Jacobian Conjecture over the real field, and over the complex field.

We study the effect of deleting a horizontal divisor from an arithmetic scheme on its étale fundamental group. Assume that the scheme is regular along the divisor, that each prime component contains a normal crossings point of residue characteristic $2$, and that the generic fibers of the boundary components remain geometrically integral after finite étale base change. We prove that the kernel of the induced surjection on fundamental groups is topologically perfect. In particular, the inclusion induces an isomorphism on maximal prosolvable quotients. We apply this result to projective hyperplane complements and to the extension of representations over complete local coefficient rings.

We establish several general results on the categorical dynamics of abelian varieties. (1) We prove the exponential and polynomial Gromov–Yomdin equalities for every autoequivalence of the derived category of an arbitrary abelian variety. The exponential equality extends earlier results of Kikuta for elliptic curves and of Yoshioka for abelian surfaces and simple abelian varieties. (2) We prove the categorical polynomial entropy of every autoequivalence is bounded above by $g^2$, where $g$ is the dimension of the abelian variety. This gives a categorical generalization of the polynomial log-volume growth of Lin–Oguiso–Zhang and recovers their upper bound as a direct consequence. (3) We identify the shifting number of every autoequivalence of an abelian variety with the pullback of the normalized Barge–Ghys symplectic translation number on the universal cover of $\text{Sp}(4g,\mathbb{R})$. (4) We prove that the set of reduced shifting numbers is finite if and only if the nef cone is rational polyhedral. Moreover, when this set is infinite, there exists an autoequivalence with transcendental shifting number. In particular, this gives a negative answer to a question of Dimitrov–Haiden–Katzarkov–Kontsevich concerning the algebraicity of categorical entropy functions.

Let $f:S\to C$ be a relatively minimal semistable fibration of genus $g\ge2$ over an algebraically closed field of characteristic $p>0$ with smooth and geometrically connected generic fiber. Let $\Sigma\subset C$ denote the set of points over which the fibers are singular, with $s=|\Sigma|$. If the total surface $S$ is Hodge Witt and $s\ge2$, we prove the classical Arakelov inequality $$ \text{deg}\, f_*\omega_{S/C}\le \frac g2 \text{deg}\, \Omega_C^1(\log\Sigma), $$ where $\omega_{S/C}$ denotes the relative dualizing sheaf. For $C=\mathbb P^1$, we obtain the stronger estimate $$ \text{deg}\, f_*\omega_{S/\mathbb P^1}\le \frac g2(s-2)-\frac12 b_1(S), $$ where $b_1(S)$ is the first Betti number of $S$. We also construct a genus-$2$ semistable fibration over $\mathbb P^1$ with exactly $4$ singular fibers in characteristic $5$. Its total surface is Hodge Witt, and $$ \text{deg}\, f_*\omega_{S/\mathbb P^1}=2=\frac g2(s-2). $$ Thus Nguyen's lower bound $s\ge4$ and the Hodge–Witt Arakelov inequality are both sharp. As applications, we derive a canonical-class inequality and a Szpiro-type inequality with coefficients linear in $g$.

Assuming a weak form of resolution of singularities in positive characteristic, we show several at-the-characteristic numerical invariants, including Hodge numbers, are given by homomorphisms out of the Grothendieck ring of varieties. Under the same assumption, we obtain two consequences: (i) $\widehat K$-equivalent varieties have the same Hodge numbers, answering a question of de Fernex–Mere; and (ii) universally homeomorphic varieties need not have the same class in the Grothendieck ring of varieties, answering a question of Nicaise–Sebag. In the appendix, we provide unconditional evidence for (i) by proving that birational Calabi–Yau varieties of dimension at most four have the same Hodge numbers.

For a smooth projective $3$-fold $V$ of intermediate Kodaira dimension, we establish Noether-type inequalities relating the Iitaka volume $\operatorname{Ivol}(V)$ and the geometric genus $p_g(V)$. Specifically, for a smooth projective $3$-fold $V$ of Kodaira dimension $2$, we show that $\operatorname{Ivol}(V)\geq \frac{3}{4}p_g(V)-\frac{5}{4};$ while if the Kodaira dimension of $V$ is $1$, then $\operatorname{Ivol}(V)\geq p_g(V)-1.$ Both inequalities are sharp. In the Kodaira dimension $2$ case, we prove that the equality holds only if the bicanonical system of $V$ induces the Iitaka fibration and the canonical system of $V$ is composed with a pencil of surfaces whose general member has Iitaka volume $\frac12$ provided that $p_g(V)\geq 4$.

We prove that the wedge local model for a ramified unitary group of signature $(r,s)$ ($r>s$) at self-dual level is flat when $p\geq 2s-1$, confirming Pappas's conjecture in this case. The proof first reduces the problem to determining the defining equations of a nilpotent orbit closure of self-adjoint matrices, and then to computing the global sections of the corresponding Kempf resolution. The computation is carried out by reducing to the type $A$ case using the Koszul complex.

The off-diagonal Gram map for h vectors in an M-dimensional quadratic space is flat with reduced fibers of rational singularities (FRS) in characteristic zero if and only if M > max_{2 <= r <= h} (2h - r + 1 - 2h/r). Equivalently, this gives the exact FRS convolution threshold of x |-> (x_i x_j){i < j}. For its zero-fiber ideal I{M,h}, we also prove lct(I_{M,h}) = min{ binom(h,2), min_{0 <= k <= h-2} ( binom(k+1,2) + M(h-k)/2 ) }. In the strict range, all truncated coefficient maps have geometrically integral complete-intersection fibers and are faithfully flat over Z[1/2], uniformly over graphs on h vertices. The proof uses a sharp weighted estimate for hollow symmetric matrices over finite local principal ideal rings, paired with isotropic lower bounds. Within the established point-counting approach to rational singularities, this estimate also gives an explicit modulus of continuity for Gram-map densities over every local field of odd residue characteristic.

We compare five natural classes of birational automorphisms of an algebraic variety: projectively regularizable, regularizable, projectively pseudo-regularizable, pseudo-regularizable, and those admitting an algebraically stable model. These classes provide different geometric frameworks in which the dynamics of a birational automorphism, and in particular its first dynamical degree, can be studied. We determine the general inclusion relations among them and show that, for $\mathbb{P}^k$ with $k\geqslant 3$, all the resulting non-trivial inclusions are strict; we prove this by giving explicit examples.

For a primitive symplectic variety $X$ admitting a holomorphic Lagrangian fibration $f: X \to B$, we classify the possible Mumford–Tate groups of the very general fiber. We extract as a consequence that the discriminant of $f$ has codimension one in $B$. In the special case when $X$ is a hyper-Kähler manifold which is general among those admitting a Lagrangian fibration, and $b_2(X) \geq 7$, we further show that the very general fiber of $f$ is Hodge-generic. Finally, as an application of our classification, we confirm an expectation of Li–Tosatti that when $f$ is not isotrivial, the special Kähler metric on $B$ has positive holomorphic sectional curvature on the regular values of $f$.

We refine the real enumerative invariants insensitive to changes in the real structure introduced in our previous paper on del Pezzo surfaces, and extend these refined invariants to arbitrary rational surfaces. The construction involves a choice of an auxiliary conic bundle together with a Pin-structure. We prove that the resulting invariants are independent of these auxiliary choices and establish recursive relations for their computation.

We construct explicit non-malleable affine extractors for every constant entropy rate, with linear output length and exponentially small error, against any fixed number of affine tamperings without fixed points. For every fixed $0<\eta<1$ and $t$, we also obtain entropy threshold $C_{\eta,t}n/\log n$, output length $\lfloor n^{1-\eta}\rfloor$, and error $2^{-n^{1-\eta}}$ against $t$ tamperings. Our extractors, as well as the directional affine extractors of Li and Zhong (CCC 2024), yield explicit Boolean functions with correlation $2^{-\Omega(n)}$ against weakly read-once linear branching programs of size $2^{\Omega(n)}$. For non-oblivious decision trees, we prove linear depth lower bounds for queries of each fixed degree $r\ge2$. Applying Li's sumset extractor (FOCS 2023) gives depth $\Omega_\delta((n/\ell)\log\ell)$ for growing locality $\ell\le n^{1-\delta}$, where $0<\delta<1$ is fixed. In the same range, directional affine extractors give correlation $2^{-\Omega(n/\sqrt\ell)}$ against local trees of depth $c(n/\ell)\log\ell/\log\log\ell$, for a sufficiently small constant $c>0$. Our extractors derandomize the lossless lifting of Efremenko and Itsykson (STOC 2026). For every fixed $0<\xi<1$, this gives explicit polynomial-size unsatisfiable CNFs on $N$ variables whose $\mathrm{Res}(\oplus)$ refutations of resolution depth at most $N$ require size at least $2^{(1-\xi)N}$. Separately, parity substitutions give polynomial-size CNFs on $N$ variables with polynomial-size ordinary-resolution proofs for which every $\mathrm{Res}(\oplus)$ refutation of size $S$ and depth $d$ satisfies $d\log(2S)=\Omega(N^2)$. This removes the $\log^2 N$ loss in the tradeoff of Itsykson, Podolskii, and Shekhovtsov (CCC 2026).

We study deformations of foliated nonsingular varieties defined by locally free subsheaves of tangent sheaves and locally free subshseaves of cotangent sheaves in the language of functors of Artin rings. We identify first-order deformations and obstructions.

We give explicit positive factorizations for a genus-19 Lefschetz fibration over the torus and a family of genus-4 Lefschetz fibrations over a genus-2 surface. All these examples have only three nodes, and their total spaces are symplectic 4-manifolds of general type on the Bogomolov-Miyaoka-Yau line. In the course of deriving the monodromy relation in genus 19, we obtain several interesting identities, including a factorization of a Dehn twist into a product of three order-3 mapping classes. We show that the total space of this fibration is topologically s-cobordant to the Cartwright-Steger surface, and we determine its real quotient up to homeomorphism. The genus-4 family consists of variations of the Bauer-Catanese fibrations and contains 65 isomorphism classes of Lefschetz fibrations on 45 diffeomorphism types of compact complex ball quotients. We describe their topology and that of the real quotients of the three original surfaces. The handle monodromies in our main factorizations are pseudo-Anosov, and we show more generally that for holomorphic genus-4 Lefschetz fibrations on the BMY line, this holds for every choice of handle generators.