Papers

Let $p>3$ be a prime number. If $A$ and $B$ are two principally polarized superspecial abelian surfaces over $\mathbb{F}_{p^{2}}$ with $p^{2}$-Frobenius $[-p]$, we prove that there exists a separable polarized isogeny between them with multiplier at most $p/\sqrt{2}$. The multiplier bound is uniform in the pair and asymptotically optimal up to a small multiplicative constant. Given quaternionic coordinates, we provide a deterministic algorithm that satisfies this upper bound in polynomial time. For unrestricted multipliers, $\mathrm{KLPT}^{2}$ provides a heuristic algorithm with a multiplier upper bound of order $p^{6+o(1)}$ for arbitrary pairs and $p^{3+o(1)}$ when one polarization matrix is the identity matrix. Our algorithm's improvement is an explicit upper bound below $p$ with an unconditional deterministic guarantee.

The locus $S_d(X)$ of $d$-secant lines to a projective variety $X$ plays an important role in the study of projective varieties via projections (Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010). For a projective scheme $X\subseteq\mathbb{P}^r$ satisfying $\textbf{N}_{d,2}$, we show that $S_d(X)\cup X$ is cut out set-theoretically by the $r$-minors of a matrix $M$ constructed from $d$-forms scheme-theoretically defining $X$ and their linear syzygies. In particular, $S_d(X)=\mathbb{P}^r$ if and only if every $r$-minor of $M$ vanishes. Using this, we find a singular threefold $X\subseteq\mathbb{P}^5$ with $S_4(X)\ne \mathbb{P}^5$ and $(I_X)_3=0$, and a normal fourfold $X\subseteq\mathbb{P}^7$ with $S_3(X)\ne \mathbb{P}^7$ and $(I_X)_2=0$. The nonexistence of such varieties in the smooth case is an open question raised by the second author and by Gruson-Peskine. Next, we consider the locus $S_{k,d}(X)$ of $k$-planes $L$ such that $L\cap X$ contains a hypersurface of degree $\ge d$ in $L$ so that $S_{1,d}(X)=S_d(X)$. The locus $S_{k,d}(X)\cup X$ is set-theoretically defined by the $(r+1-k)$-minors of $M$. As a result, we obtain determinantal equations vanishing on the $(q+1)$-secant variety $\sigma_{q+1}X$ from the equations of $\sigma_qX$ and their linear syzygies when $\sigma_qX$ satisfies $\textbf{N}_{q+1,2}$. Finally, we extend the $d$-secant lemma for lines to the locus $S_{k,d}(X)$ of $k$-planes. This yields a lower bound on the number of linear syzygies.

It was known that the canonical bundle of a smooth toroidal compactification of a noncompact finite-volume ball quotient is nef in dimension $n\geq3$ and ample in dimension $n\geq6$. We prove that the canonical bundle is ample in dimension $n\geq4$ and show that this dimension bound is sharp. We construct infinitely many pairwise noncommensurable noncompact Picard modular threefolds whose smooth projective toroidal compactifications have non-ample canonical bundle. Each compactification contains a smooth rational curve meeting the interior and having canonical degree zero. More generally, in every dimension $n\geq2$, we construct infinitely many pairwise noncommensurable noncompact Picard modular varieties whose smooth toroidal compactifications contain smooth rational curves meeting the interior. In particular, such curves persist in arbitrarily high dimension, even when the canonical bundle is ample.

Let $X/\mathbb{Q}$ be a smooth, projective, geometrically integral curve of genus $g \geq 2$ with a point $ b \in X(\mathbb{Q})$ and geometrically CM Jacobian. Fix an odd prime $p$ of good reduction, and let $U$ be the étale pro-unipotent $\mathbb{Q}_p$-fundamental group of $X$. We prove, for all $n \gg 0,$ the inequality $\dim_{\mathbb{Q}_p} H^1_f(G_T, U_n) < \dim_{\mathbb{Q}_p}H^1_f(G_p, U_n)$ of Bloch–Kato Selmer schemes, with their ratio having limit superior at most $1/2$. Here $U_n$ is the lower central series quotient, and $T$ a finite set containing $p$ and the primes of bad reduction. This answers a particular case of a conjecture by Kim. Following the Iwasawa-theoretic method of Coates–Kim, we reduce the problem to bounding certain character multiplicities at degree $n$. We then define a twisted character lattice $B$ and its $p$-adic completion $B_p.$ Our main idea is to regard normalized multiplicities as $\mu_n(\lbrace x \in B_p: f(x,n)=0\rbrace)$ for a $p$-adic analytic function $f$ coming from Iwasawa theory and a sequence of measures $(\mu_n)_n$ weakly converging to the Haar probability measure $\mu$ on $B_p,$ which is the $p$-adic analogue of the uniform measure. We prove convergence by considering characters $\eta$ of $B_p$ and comparing $\int_{B_p} \eta \ d\mu_n$ to traces of operators induced by symplectic operators $h_\eta$, followed by a character formula. A compactness argument for sequences of $p$-adic loci gives our main Iwasawa estimate. Poitou-Tate duality, the global Euler characteristic formula, and Hodge filtration estimates then give the dimension inequality.

In this paper we consider the $n-3$ dimensional linear section $X_\Pi$ of the Grassmannian $\mathbb G(1,n)$ of lines in $\mathbb P^n$ with a general linear space $\Pi$ of codimension $n+1$, with $n=2k+1$ odd and $n\geq 5$, that is a Calabi-Yau variety. Related to $X_\Pi$ there is a degree $k+1$ Pfaffian hypersurface $\mathcal C_\Pi$ that is the intersection of the dual of $\mathbb G(1,n)$ with $\Pi^\perp$. We prove that $\mathcal C_\Pi$ is rational and we provide a birational parametrization of it. Moreover we study in some detail the congruence of lines of $\mathbb P^n$ given by the centers of the linear complexes parametrised by the points of $\mathcal C_\Pi$.

We prove uniform height gaps for subvarieties of abelian varieties in arbitrary characteristic, extending the new gap principle of Gao-Ge-Kühne. Our proof goes through the theory of adelic curves and globally valued fields, and proves the Bogomolov conjecture for an arbitrary globally valued field. This gives a different way to obtain uniform Bogomolov-type results, differing from the approaches of Dimitrov-Gao-Habegger-Kühne and Yuan. First, we prove the Bogomolov conjecture over globally valued fields for divisors. We then reduce the case of general subvarieties to the case of divisors by an induction argument. Specializing to the case of global function fields, we obtain a new proof of the geometric Bogomolov conjecture following the strategy of Gubler and Yamaki.

We show that the wild locus of an étale morphism of smooth adic spaces over an affinoid field is either empty or contains a divisorial point. As a consequence, we establish the equivalence of curve tameness and valuation tameness without using resolution of singularities. Moreover, given a finite étale map of regular schemes $f \colon Y \to X$ that is curve tame over a base $S$, we construct a compactification $\bar{f} \colon \overline{Y} \to \overline{X}$ over $S$ that is numerically tame.

We show that integers act invertibly on a derived scheme $S$ if and only if they act invertibly on the reduced motivic suspension spectrum of the affine line in $\mathrm{MS}_S$. As a consequence, over regular locally noetherian schemes and derived schemes on which some nonzero integer acts as zero, the motivic sphere spectrum is $\mathbf{A}^1$-invariant after inverting primes that are not invertible in the base scheme. In particular, over regular locally noetherian $\mathbf{Q}$-schemes, the motivic sphere spectrum is $\mathbf{A}^1$-invariant.

Let $S$ be a Cohen-Macaulay normal domain containing a field $k$ with an action of a finite group $G$ with $|G|$ invertible in $k$. We prove that $G$-equivariant noncommutative crepant resolutions of $S$ induce noncommutative crepant resolutions of $S^G$ if the $G$-action is small. We also discuss similar results for equivariant maximal modification algebras. We introduce the notion of $G$-maximal modifying ($G$-MM) module, and we show that $G$-MM module over $S$ induces MM $S^G$-module in some special cases. Furthermore, we prove that Gorenstein MMAs of three-dimensional $\mathbb{Q}$-Gorenstein Cohen-Macaulay rings are derived equivalent. As an application of these results, we prove the existence of an MM module and derived equivalences of MMAs for three-dimensional terminal singularities of index two.

Specialization maps are natural maps from equivariant vanishing cycle cohomology to the equivariant cohomology of the ambient space. We study their injectivity problem for moduli stacks of representations of quivers with potentials, including those associated with generalized conifolds and finite subgroups of $SO(3,\mathbb{C})$. Applications to the irreducibility of modules of Reshetikhin type shifted Yangians are also given.

Let $k$ be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over $k$ and over $\mathbb{C}$ equipped with the hybrid norm. In both settings, we show an analogue of Mané–Sad–Sullivan's theorem about the continuity of the Julia set (in the Hausdorff topology). The non-archimedean case relies on the maximum modulus principle, which we show for analytic functions over analytic spaces without boundary.

We study the quantized $K$-theoretic Coulomb branches of framed star-shaped quivers with central gauge dimension $n\geq2$, $r-1$ one-dimensional outer gauge vertices, and a one-dimensional framing at the central vertex. For $n=2$ and arbitrary $r\geq1$, and for every $n\geq2$ with $1\leq r\leq5$, we prove that the loop-graded Koszul-perverse heart is a monoidal categorification of the associated quantum cluster algebra over $\mathbb{Z}[v^{\pm1}]$. This quantum cluster algebra coincides with its upper quantum cluster algebra. For all $n\geq2$ and $r\geq1$, we construct an explicit initial seed and identify the associated upper quantum cluster algebra with the Grothendieck ring of this heart. Every mutation is realized by a short exact sequence, and every quantum cluster monomial is represented by a simple object.

For an almost simple complex algebraic group $G$ of semisimple rank $m$, we prove that a finite family of parabolic subgroups contains a common maximal torus whenever every subfamily of at most $m+2$ members does. We determine the resulting Helly number $(h(G))$ for almost simple groups of types $A$, $B$, $C$, $G_2$, $F_4$, $E_7$, and $E_8$, and obtain bounds differing by one in types $D$ and $E_6$. For general reductive groups of positive semisimple rank, the Helly number is the maximum of those of the almost simple factors. We apply these results to establish a fan-theoretic criterion characterizing smooth toric varieties on which every toric principal $G$-bundle splits equivariantly. In particular, for $d \ge 2$, this splitting property holds on $\mathbb{P}^d$ if and only if $d \ge h(G)$

We establish some basic properties of t-flat fine log schemes, i.e., fine log schemes whose underlying schemes are flat over their log structure (in a suitable sense). Notably, we show that the depth of a t-flat fs log scheme is simply determined by its logarithmic stratification. We also show that there is a good theory of log cotangent complexes for t-flat fine log schemes, and use the log cotangent complex to give a homological criterion for t-flatness.

Let a finite group act faithfully by symplectic automorphisms on a manifold of $K3^{[n]}$ type for $n\geq2$. We prove Mongardi's conjecture that the strict inequality $\mathrm{rk}(S)+\ell(A_S)<24$ holds for its associated Leech coinvariant lattice $S$. The converse was already proved independently by Huybrechts and Mongardi: any Leech coinvariant lattice $S$ satisfying the strict inequality and of rank at most $20$ is realized by such an action for some $n\geq2$. The key idea is to consider the minimum norms of vectors representing discriminant classes of $S$. The absence of numerical walls gives lower bounds for certain classes, while the structure of Leech lattice implies upper bounds for all discriminant classes. Comparing these bounds reduces the proof to finitely many values of $n$. We exclude the remaining cases using further lattice arguments, together with Höhn–Mason's classification of fixed-point sublattices of Leech lattice, thereby proving Mongardi's conjecture.

Let $K$ be a function field over an $F$-finite field $k$ of characteristic $p > 0$ and let $\nu$ be a valuation of $K/k$. We show $\nu$ is an Abhyankar valuation of $K/k$ if and only if the corresponding valuation ring is Frobenius split. The proof proceeds by analyzing when ideals of a valuation ring are uniformly $F$-compatible and establishing a general ramification-theoretic characterization of Frobenius split valuation rings of $F$-finite fields. In addition, when the ground field $k$ is not $F$-finite, we show that the equivalence between Frobenius splitting and the Abhyankar property can fail by constructing an example of an excellent Frobenius split DVR of a function field whose corresponding valuation is not divisorial. Our methods do not rely on local uniformization results.

Let $\delta=xz-yt$. We give an explicit tame factorization of the Anick automorphism $(x+\delta t,y+\delta z,z,t)$ of $F[x,y,z,t]$. After a linear change of coordinates, the factorization consists of sixteen elementary automorphisms, all fixing $z$. The proof uses a commutator construction with variables already present in the algebra. A complete list of the elementary factors is included in Section 5. The explicit polynomial factorization identities are defined over $\ZZ[1/2]$; no conclusion in characteristic $2$ is asserted.

In quantum mechanics of genus-1 potentials, perturbation theory around the topologically trivial saddle and non-perturbative instanton saddle are constructively related via the P/NP relation. Whether such a relation persists for higher-genus potentials has remained an open problem. In this work, we resolve it using tools from complex algebraic geometry combined with exact WKB. The classical energy conservation relation $p^2 = 2(E-V(q))$ defines a genus-g Riemann surface $X$, whose periods are naturally organized by its Jacobian variety $J(X) = \mathbb{C}^{\text{g}}/\Lambda$ via the Abel-Jacobi map, and independently by the solution basis of the associated Picard-Fuchs equations. We show that the physically relevant, WKB-active cycles are generally related to the Picard-Fuchs basis by a linear transformation that lies in $\text{SL}(2\text{g},\mathbb{Z})$ but not in the symplectic group $\text{Sp}(2\text{g},\mathbb{Z})$ once $\text{g}\geq2$, in contrast to the genus-1 case, where the two groups coincide. This mismatch is the obstruction that has made an explicit higher-genus P/NP relation elusive. We show that it can be resolved by passing to a modified Riemann bilinear identity, built from a suitably transformed intersection matrix, which the WKB-active periods do satisfy exactly. The result is an all-orders quantum P/NP relation valid for potentials of arbitrary genus, which we derive explicitly for genus-2 (quintic and sextic) and genus-3 (septic and octic) anharmonic oscillators.

We provide effective bounds on adjoint line bundles on $X$ for which the secant variety $\Sigma(X)$ satisfies property $N_{3,p}$. The given bounds only depend on $\operatorname{dim}(X)$ and $p$.

We prove an exact identity, as convergent logarithmic germs at large Euclidean momentum, between the equal-mass two-dimensional $(n-1)$-loop sunset Feynman integral, with $n\geq4$, and the genus-zero one-point descendant invariants of the local Calabi–Yau $n$-fold $X_n=\operatorname{Tot}(K_{F_n})$, where $F_n\subset(\mathbb P^1)^{n}$ is a smooth $(1,\ldots,1)$ hypersurface. Writing $s$ for the momentum variable and $y=-1/s$, the identity reads \[ I^{(n-1)}_{\circleddash}(s)=(-1)^{n-1}yA\,\Bigl[nR^{n-1}-\theta_Q{\mathcal D}_0^\Delta\,R^{n-2} +\sum_{j=0}^{n-3}c_jR^j\Bigr], \] where $R=\log y+O(y)$ is the local mirror coordinate, $A=\theta_yR$, $Q=e^R$, ${\mathcal D}_0^\Delta(Q)$ is the primary one-point potential, and each $c_j$ has the form \[ c_j=c_j^{(0)}+\sum_{a=0}^{n-2}\bigl(c_{j,a}\,{\mathcal D}_a^\Delta(Q) +c'_{j,a}\,\theta_Q{\mathcal D}_a^\Delta(Q)\bigr) \] with the coefficients in $\mathbb Q[\zeta(3),\zeta(5),\ldots]$ the odd zeta values coming from the Gamma class. The descendant series ${\mathcal D}_a^\Delta(Q)$ are given explicitly by the ambient hypergeometric series and the inverse mirror map. We also prove global univalence of the principal mirror map, with explicit upper and lower bounds on the $Q$-Taylor radii, extract the full primary evaluation divisor and its integral cover transform, and reduce the ambient endpoint constants to a finite universal Gamma-class part and $\lfloor(n-1)/2\rfloor$ real parameters.

We determine the equal-mass two-dimensional three- and four-loop sunset Feynman integrals $I^{(3)}_\circleddash$ and $I^{(4)}_\circleddash$ at large Euclidean momentum in terms of the genus-zero Gromov–Witten theory of the local Calabi–Yau fourfold and fivefold $X_L=\operatorname{Tot}(K_{F_L})$, where $F_L\subset(\mathbb P^1)^{L+1}$, $L=3,4$, is a smooth hypersurface of multidegree $(1,\ldots,1)$. At two loops the sunset integral is governed by the prepotential of a local Calabi–Yau threefold. For $L\geq3$ the moduli spaces of unmarked stable maps have positive virtual dimension and there is no prepotential, therefore the Feynman integral is governed instead by the one-point potential $\mathcal F_{\gamma_L}$ with insertion $\gamma_L=H^{L-1}/(L-1)!$, by its gravitational descendants, and by $\zeta(3)$ boundary terms coming from the Gamma class. At both loop orders the coefficient of $R^{L-1}$ in the Frobenius expansion in the mirror coordinate $R$ is $-\theta_Q\mathcal F_{\gamma_L}$ with the Novikov variable $Q=\exp(R)$. At three loops the whole Feynman integral is an explicit expression in $\mathcal F_{\gamma_3}$ alone, because genus-zero recursion determines all descendant series from it; at four loops two diagonal series suffice, and the $\zeta(3)$ term is itself corrected by $\theta_Q\mathcal F_{\gamma_4}$. At three loops we also derive the analytic normalization independently, from the Picard–Fuchs equation and the polar part of a Mellin transform, prove integrality of the multiple-cover transforms for arbitrary integral insertions together with an explicit quantum-$K$ formula, and compute the exact period connection polynomials. Finally we give endpoint values along the Euclidean axis, bounds on the Taylor radii in the Novikov variable $Q$, a computer-assisted proof of the four-loop endpoint nonanalyticity, and a comparison of the normalizations at two, three and four loops.

We develop a modular wall crossing theory for quasimaps to a projective variety, allowing independent variation of the boundary coefficients and the quasimap weight. Building on the K-stability of quasimaps introduced by Hashizume and the first author, we construct projective moduli spaces in the stable, Calabi–Yau, and log Fano regimes, together with wall crossing morphisms. A central construction is the moduli theory of boundary polarized Calabi–Yau quasimaps, which retains an ample polarization at the numerically trivial locus and allows comparison with suitable perturbations toward the stable and log Fano regions. The stable theory applies in arbitrary genus, while the comparisons through the Calabi–Yau locus concern genus zero. For degree-one boundary divisors, the resulting framework relates weighted stable maps and quasimaps to Hassett spaces and GIT quotients of weighted points on $\mathbb P^1$. In a companion paper, we apply this framework to give a modular interpolation between Miranda's GIT compactification of rational elliptic surfaces and the Baily–Borel compactification of an eight-dimensional ball quotient.

Using K-moduli spaces for log quasimaps $q_t\colon\left(\mathbb P^1,\frac{1-t}{12}D\right)\to[\mathbb A^2/\mathbb G_m]$ of degree twelve with twelve points and weight $t/12$, we construct a modular interpolation $\{\mathcal M_t\}_{0\le t\le1}$ between the Baily–Borel compactification of the Heckman–Looijenga ball quotient $X_o$ and Miranda's GIT compactification of the moduli space of rational elliptic surfaces. We completely determine the wall-crossing and, for every rational $t\in[0,1]$, identify \[ \mathcal M_t\cong\operatorname{Proj}R\!\left(X_o,\mathcal L+\frac t2\Delta(6)+\frac t3\Delta(9)\right), \] where $\mathcal L$ is the automorphic $\mathbb Q$-line bundle and $\Delta(6),\Delta(9)$ are distinguished Heegner divisors. On the automorphic side, we construct a new automorphic form on $X_o$ via a Borcherds product, whose divisor gives an independent relation among the Heegner divisors. This relation provides a key input for determining the birational transformations in the K-moduli wall-crossing. As part of this analysis, we show that the first positive chamber $\mathcal M_t$ for $t\in (0,1/7)$ is Looijenga's semi-toroidal compactification.

Let $\mathop{Mod}_{g,n}$ be the mapping class group of a genus-$g$ surface with $n$ punctures and let $\mathcal{C}_{g,n}$ be its curve complex. For a finite-index subgroup $G < \mathop{Mod}_{g,n}$, Boggi proved that $\widetilde{H}_k(\mathcal{C}_{g,n}/G;\mathbb{Q}) = 0$ for $g+n \gg k$. This plays an important role in the work of Putman-Wieland on the virtual first Betti number of $\mathop{Mod}_{g,n}$. The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.

We give a criterion for the box product of a Fourier–Mukai kernel to induce a fully faithful functor to the derived category of a symmetric quotient stack. This leads to many new examples of embeddings of derived categories of moduli spaces. We study in more detail the case of the moduli space of stable bundles on a curve. Concretely, we identify pieces of the left-orthogonal complement of our embedding, and extend the known semi-orthogonal decomposition of the moduli space of bundles with rank 2 and fixed determinant to the moduli space of bundles with varying determinant.

We characterize Fourier quasicrystal measures with positive integer masses as real sections of effective Lee-Yang cycles, with weighted local intersection degrees. The section matrix has positive maximal minors, and the mass at each root is its weighted local intersection degree. For Delone support the reduced support of the cycle can be chosen strict. The support of every integer-mass Fourier quasicrystal measure is the common complex zero set of finitely many real-valued trigonometric polynomials. For unit masses the representing cycle can be chosen reduced, with local degree one at every physical root. These results answer Questions (1), (2), and (4) of Alon, Kummer, Kurasov, and Vinzant. The proof recovers a finite-dimensional torus from the spectrum, proves that the resulting analytic set is algebraic, and reconstructs its complex branches and their integer coefficients from their moments. A support-height theorem for holonomic sequences gives the required linear spectral bound. Singular Lee-Yang sections are Fourier quasicrystal measures with their natural local intersection degrees; discarding those degrees can destroy the Fourier quasicrystal property.

For every positive integer $n$, we prove that there is a constant $C_n$, depending only on $n$, such that $Vol(-K_X) \leq C_n \epsilon^{-(2^n-n-1)}$ whenever $X$ admits an $\epsilon$-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one. In particular, the optimal exponent for fourfolds is eleven. We also prove an anticanonical interpolation theorem with optimal exponent $2^n-1$. The proof combines signed discrepancy estimates under projection, finite morphisms to projective space, and the canonical bundle formula. A reduction to a base bounded independently of $\epsilon$, followed by a volume estimate along a flag, yields the sharper volume exponent.

We prove that a very general hypersurface of degree at least five and any positive dimension does not have a decomposition of the diagonal, over an uncountable algebraically closed field of characteristic not $2$. In particular, such hypersurfaces are not stably or retract rational. For the proof, we construct (in each dimension $\geq 8$) a certain quintic hypersurface with a nonzero unramified cohomology class, and which is rationally fibered by quadrics over a lower-dimensional projective space. From our construction, the main result follows using methods developed by Schreieder.

We construct complex structures on the six-sphere $S^6$ with arbitrarily large local Kuranishi dimension. As a corollary, we find the same phenomenon holds for the smooth manifold underlying complex projective 3-space $\mathbb{P}^3$.

We prove the abundance conjecture for threefolds in characteristic $>42$.