Papers

Let $\boldsymbol{n}=(n_1,n_2,n_3):(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ be a holomorphic map germ admitting an injective representative. We prove that if $(d\boldsymbol{n})_0=0$, then $\mathrm{ord}_0(\boldsymbol{n})\in\{2,3,4\}$. This reduces Lê's conjecture to excluding potential counterexamples $\boldsymbol{n}$ of orders $2$, $3$, and $4$. The proof combines techniques from the theory of plane curve singularities, explicit cobordism constructions, and genus bounds obtained by applying properties of the $\Upsilon$ invariant coming from knot Floer homology.

We prove that every compact connected positive quaternionic–Kähler manifold of real dimension $4n$ with $n\ge2$ is isometric to a Wolf space after rescaling the metric by a positive constant. This proves the LeBrun–Salamon conjecture. The proof uses transverse rational curves and the contact structure of the twistor space.

Let $X_\sigma$ be the affine normal toric variety associated with a full-dimensional strongly convex rational polyhedral cone $\sigma$, and let $m$ be a primitive degree with slice $P_m=\sigma\cap[m=1]$. We construct an explicit flat algebraic family whose completion is universal for equivariant deformations in all degrees $-jm$, $j\ge1$, simultaneously. We prove that the irreducible components of the reduced base correspond bijectively to maximal lattice-friendly Minkowski decompositions of $P_m$, and describe the induced family on each component. When $m\in\sigma^\vee$, we also obtain a formally universal equivariant family for the pair $(X_\sigma,V(\chi^m))$. These results extend earlier miniversality and component theorems to arbitrary affine normal toric varieties and arbitrary primitive degrees.

Levin asked what can be said about the locus of points lying on a given curve in $\mathbb{G}_m^n$ that have a non-zero integer multiple on another given curve in $\mathbb{G}_m^n$ for $n \geq 3$. We give a definite answer to the abelian analogue of Levin's question, proving what is predicted by the Zilber–Pink conjecture in this case. An important ingredient in our proof is a strengthening of a height inequality by Vojta and Rémond.

Let $G$ be a simple abelian variety of dimension $g \in \mathbb{N}$ defined over $\mathbb{Q}^\mathrm{alg}$ and let $C_1, C_2 \subseteq G^N(\mathbb{C})$ be irreducible closed algebraic curves with $N \geq 3$. Further assume that at least one of $C_1$ and $C_2$ is not defined over $\mathbb{Q}^\mathrm{alg}$. Suppose that there does not exist an algebraic subgroup $H_1 \subseteq G^N(\mathbb{C})$ of dimension $g$ such that $C_1 \subseteq H_1$ and that there does not exist an algebraic subgroup $H_2 \subseteq G^N(\mathbb{C})$ of dimension $2g$ such that $C_1 \cup C_2 \subseteq H_2$. Denoting $\mathcal{N} = \{n \in \mathbb{N} \ | \ [n]C_1 \subseteq C_2\}$, we prove that $\bigcup_{n \in \mathbb{N} \setminus \mathcal{N}}\{x \in C_1 \ | \ nx \in C_2\}$ is finite.

We construct effective finite group actions on smooth projective complex curves for which Conjecture A of Polishchuk and Van den Bergh fails. For every $h\geq1$, an explicit $S_3$ action on the smooth projective model of $y^2=t^{12h-5}+t$ has quotient of genus $h$ and exactly one ramification orbit, with inertia of order three. The fixed quotient for the three-cycle class consists of two points, but its derived category admits no $\C$-linear exact fully faithful embedding into the equivariant derived category. More generally, when the coarse quotient has positive genus, we prove that the maximum number of pairwise completely orthogonal exceptional objects is $\sum_i\lfloor e_i/2\rfloor$, where $e_i$ are the inertia orders. We also characterize conjugacy-class decompositions with embeddings linear over the coarse quotient: such a decomposition exists if and only if $|[g]\cap H_i|\leq\lfloor e_i/2\rfloor$ for every nonidentity conjugacy class and every branch inertia group $H_i$. The sufficiency proof uses coloured paths of type $A$ and derived reflection functors, and produces kernels on the natural fibre products. The known ordered curve decomposition still yields a decomposition indexed by the connected components of the fixed quotients.

We show that no meaningful version of the Cone Conjecture holds for non-generic Sakai surfaces.

Let $f:\mathbf P^n_{\mathbb C}\to\mathbf P^n_{\mathbb C}$ be an endomorphism of degree greater than one. We prove that every irreducible Zariski closed subset $V$ with $f^{-1}(V)=V$ is a linear subspace.

We give a combinatorial criterion for detecting roots of Bernstein-Sato polynomials of ideals. As applications, we prove the strong monodromy conjecture for determinantal and Pfaffian ideals.

Using the recent proof of the Cassels–Swinnerton-Dyer conjecture for cubic surfaces, we finish the classification of $G$-unirational complex cubic threefolds. In particular, we prove that the Klein cubic threefold is $\mathsf{PSL}_2(\mathbb{F}_{11})$-unirational, establishing that the essential dimension of $\mathsf{PSL}_2(\mathbb{F}_{11})$ is $3$. This disproves a conjecture of Dolgachev that the essential dimension of a group is at least its Cremona dimension.

We prove invariance of the geometric Arveson-Douglas conjecture under Veronese embeddings of projective varieties. This provides new examples of singular varieties satisfying the conjecture. Furthermore, it reduces the conjecture to projective varieties whose defining ideals admit quadratic Gröbner bases, by keeping the same Schatten regularity thresholds. Our results lead to canonical examples of subproduct-systems whose Toeplitz algebras are not $KK$-equivalent to the complex numbers. Finally, we show that Douglas' fundamental class conjecture, asserting that the associated Toeplitz extension implements $\mathbb{T}$-equivariant duality in $KK$-theory, is invariant under Veronese embeddings as well. While the latter result bootstraps the fundamental class conjecture, it also leads to infinitely many counterexamples.

We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if $f\in (x_1,\ldots,x_n)^2\subsetneq k[[x_1,\ldots,x_n]]$ has an isolated critical point and $\operatorname{char}k=0$, then its Euler conductor $J_f:f$ is not contained in the Tjurina ideal $(J_f,f)$. More generally, if $A=R/I$ is a nonzero Noetherian $k$-algebra and $a=[f]\in A$ is nilpotent with $d_{A/k}a=0$, then $I:f\nsubseteq(I,f)$. For a local ring $(R,\mathfrak n)$ we obtain the stronger noncontainment $I:f\nsubseteq(I,f)+\mathfrak n(I:f)$. The proof reduces a hypothetical containment to a self-exact square-zero element and detects its differential by the trace of a regular representation over the dual numbers. Finally, an explicit five-variable isolated singularity satisfies $J_f:f\subseteq\overline{J_f}$, disproving the integral-closure strengthening proposed by Ma and Zuo.

A collection of $m \geq 2$ lines through the origin in $\mathbb{R}^2$ is balanced if the sum of the orthogonal projections onto these lines equals $m/2$ times the identity. A line arrangement in $\mathbb{RP}^2$, endowed with the round metric of curvature $1$, is point-balanced if the tangent lines at every vertex form a balanced collection. A line arrangement $\mathcal{A}$ in $\mathbb{RP}^2$ has the Hirzebruch property if it consists of $3k$ lines and every line contains exactly $k+1$ vertices. Using a Kempf–Ness convexity argument, we show that an irreducible arrangement $\mathcal{A}$ has the Hirzebruch property if and only if its projective equivalence class contains a point-balanced representative, unique up to orthogonal transformations. We then combine elementary properties of balanced collections of lines in $\mathbb{R}^2$ with spherical geometry to prove that every irreducible point-balanced arrangement is a reflection arrangement. This gives a new proof of Panov's classification of real Hirzebruch arrangements.

We prove an Ax-Lindemann theorem for the rank one Riemann-Hilbert correspondence on a smooth projective complex algebraic curve and recover a result of Simpson.

A combination of recent results of Canning–Larson–Payne–Willwacher and Payne–Willwacher is that the motivic Euler characteristic of the moduli space $M_{g,n}$ of nonsingular curves with $n$ markings is polynomial in the Tate motive if and only if $g=0$ or $3g+2n<25$. In the cases when $g\geq 1$ and $3g+2n<25$, there are $32$ pairs $(g,n)$ where the motivic Euler characteristic has been computed. We compute the motivic Euler characteristic for $9$ more pairs, leaving only $6$ pairs where the motivic Euler characteristic is polynomial but unknown. In genus $4$, we moreover compute the answer when $n=7$, i.e.~the first non-polynomial case.

In [6], we introduced the notion of a Hermitian-relative curve, which is a plane curve defined by $(x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A (x,y,z)^t =0$ with $A \in GL(3, \mathbb{F}_q)$. After investigating their basic properties, we classified those curves with two or more rational inflexions. In this paper, we first complete the classification of curves whose rational points are all inflexions by demonstrating the existence of curves with exactly one rational point. As a continuation, we then classify Hermitian-relative curves that possess at least one rational point which is not an inflexion. We categorize these curves according to the ordered pair consisting of the number of rational points and the number of inflexions. Finally, we enumerate all possible pairs and prove that each pair is realized by a suitable curve.

For any regular projective surface $X$ over an arbitrary field $k$ of characteristic $p>0$ with $K_X\equiv0$, we prove that \[ 60K_X\sim0 . \] This uniform bound is optimal. To show that the factor $5$ is necessary, we construct an example in characteristic $5$ for which $K_X$ has exact order five. If $X$ is geometrically normal, then $12K_X\sim0$. Our main focus is the geometrically non-normal case, where we obtain more precise bounds according to the Albanese morphism. If it is nontrivial, then $p=2$ or $3$, and we prove $12K_X\sim0$ and $6K_X\sim0$, respectively; both bounds are optimal. If it is trivial, then $K_X\sim0$ for $p\ge7$, while $5K_X\sim0$, $3K_X\sim0$, and $4K_X\sim0$ for $p=5,3,2$, respectively.

We construct regular integral models for a class of Shimura varieties of symplectic and orthogonal type with maximal parahoric level at an odd prime $p$. These models are defined over the ring of integers of the reflex field and have special fiber with normal crossings and irreducible components of multiplicity one or two. Our construction uses explicit equivariant modifications of the corresponding canonical local models for symplectic and split even orthogonal similitude groups, with the minuscule cocharacters corresponding to maximal isotropic Grassmannians. These modifications are obtained by successively blowing up Schubert varieties in the special fiber. This proves the equivariant modification conjecture of Pappas at maximal parahoric level in the above cases.

In this paper, we prove that $\mathbb{Q}$-Gorenstein projective complex varieties admitting int-amplified endomorphisms are of Calabi–Yau type.

Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold of Picard number one. We prove that, if $X$ has terminal singularities, then $(-K_X)^4\le648$, with equality precisely for $\mathbb{P}(1,1,1,1,2)$. If the singularities are canonical, we obtain $(-K_X)^4\le7332$. Under the additional assumption that a primitive Weil polarization is Cartier outside finitely many points, the latter bound improves to the sharp bound $1024$; if its non-Cartier locus has dimension at most one, we obtain $6084$.

We investigate birational automorphisms of hypersurfaces that preserve the projection from a point. Let $V \subset \mathbb{P}^{n+1}$ be an irreducible hypersurface of degree $d \ge 3$ with $\dim \mathrm{Sing}(V) \le n-2$, and let $\pi_P : V \dashrightarrow \mathbb{P}^n$ denote the projection from a point $P \in \mathbb{P}^{n+1} \setminus \mathrm{Sing}(V)$. Assume moreover that if $d = 3$ then $P \notin V$. We prove that any $\sigma \in \mathrm{Bir}(V)$ satisfying $\pi_P \circ \sigma = \pi_P$ extends uniquely to a projective transformation of $\mathbb{P}^{n+1}$. By combining the above projective lifting theorem with the theory of complex reflection groups, we investigate Galois and quasi-Galois points for the Fermat variety $F_d^n \subset \mathbb P^{n+1}$ of dimension $n$ and degree $d$. We show that, for $d \ge 4$, $F_d^n$ has exactly $n+2$ Galois points and, in addition, $d(n+2)(n+1)/2$ quasi-Galois points. When $d=3$, $F_3^n$ has exactly $n+2$ outer Galois points.

We prove that every cluster monomial of type $A_n$ is log-concave and unimodal with respect to its Laurent expansion in a fan initial cluster. We derive a coefficient formula for laminar interval products arising from compatible diagonals in the polygon model. In even rank, we reduce log-concavity in fan coordinates to log-concavity along integer lines. In odd rank, we use convolution to handle the coefficient sums arising from the fan substitution. This gives partial affirmative answers to the log-concavity conjecture of Chen-Huang-Sun and the unimodality conjecture of Chen. Our result also complements Shen's study on the supports of products of canonical basis elements for cluster X-varieties of type $A_n$.

Let $k$ be algebraically closed of characteristic $p$. Reichstein and Vistoli showed that every $\Z/p^n$-torsor becomes at most one-dimensional over a field extension of degree prime to $p$, and Ledet conjectured $\ed_k(\Z/p^n)=n$. Using the finite-degree profile $\edd d_k(\tau)$ and its jump degrees introduced in \cite{PA}, we study the generic $\Z/p^n$-torsor $\tgen$, whose profile records how many Artin–Schreier–Witt layers an extension of degree $\le d$ can absorb. An Abel–Jacobi rigidity argument, in which the purely transcendental total space of $\tgen$ forces divisor classes on twisted curves to be constant, shows that a $G$-stable linear system on a faithful $\Z/p^n$-curve has degree $\ge p^{n-1}$. Hence $\edd d_k(\tgen)\ge2$ for $d<p^{n-1}$ and $d_1(\tgen)=p^{n-1}$ for all $p$ and $n\ge2$; in particular the truncation tower is optimal in degree $p^{n-2}$. We prove a quantitative Reichstein–Vistoli theorem, $d_1^{(p')}(\tgen)\le l_n(p)=1+(p-1)\sum_{j=1}^{n-1}p^{2j-1}$, and show that it is attained: $d_1^{(p')}(\tgen)=l_n(p)$ for all $p$ and $n$. We determine the profile at $(p,n)=(2,2)$ and every slot but the first at $(2,3)$, where the remaining slot is $\ed_k(\Z/8)\in\{2,3\}$, that is, Ledet's conjecture at level $3$. We show that the subgroup formula $\edd{p}_k(\tau_n)=n-1$ is equivalent to Ledet's conjecture at level $n-1$ together with a statement about extensions of degree $\le p$ disjoint from $L$, so that the profile relocates, but does not decide, Ledet's conjecture.

Let $L$ be a smooth Hermitian holomorphic line bundle on a compact complex manifold $X$ of dimension $n$, and let $r$ be its maximal curvature rank. For every fixed holomorphic vector bundle $E$, we prove $h^q(X,L^p\otimes E)=O_\varepsilon(p^{r+\varepsilon})$ for every $\varepsilon>0$ and every Dolbeault degree, with the sharp bounds $h^0,h^n=O(p^r)$ at the endpoints. The curvature may have variable rank and arbitrary signature, and no global foliation or Kähler assumption is imposed. The all-degree estimate follows from local rescaling and holomorphic restriction estimates that remain uniform across rank changes. An independent maximum-principle argument locates the maximal rank of a section map at a semipositive curvature point; it yields $\kappa(L|_{X_0})\le r$ on each connected component $X_0$ and gives the endpoint bounds.

Let $k$ be an algebraically closed field of characteristic $0$ or $p\neq3$. We construct a terminal projective 4-fold $X$ and an ample Cartier divisor $D$ such that \[ 3K_X\sim0,\qquad D-K_X\ \text{is ample},\qquad H^0(X,D)=0. \] In char 0, this disproves Ambro–Kawamata effective non-vanishing conjecture. From this example we construct a smooth projective 4-fold $Y$ with a semiample and big Cartier divisor $D_Y$ such that $D_Y-K_Y$ is basepoint-free and big, and $H^0(Y,D_Y)=0$. We also construct a Gorenstein terminal 6-fold with $K\sim0$, other geometric avatars, and a terminal order-5 Jacobian quotient.

We study the fundamental groups of a normal projective Calabi-Yau variety $X$ with klt singularities and of its regular locus $X_{reg}$. We prove that both groups are almost abelian, and that their ranks are determined by the irregularities of finite etale and quasi-etale covers, respectively. Our approach is differential geometric, and relies on recent advances in the geometric theory of complex Monge-Ampere equations.

We study the conjectural characteristic class analogue of the Hodge index theorem for singular complex algebraic varieties, formulated by Brasselet-Schürmann-Yokura which expresses the Goresky-MacPherson homology $L$-classes in terms of suitable Hodge-theoretic $L$-classes, for quotient spaces $X/G$, where $G$ is a finite group and $X$ is a pure-dimensional complex projective variety. Assuming that an equivariant $K$-theoretical version of the conjecture holds for $X$, we show that the characteristic class conjecture holds for $X/G$. Without assuming the equivariant $K$-theoretical version, we show that the characteristic class conjecture holds for $X/G$, provided that it holds for all fixed point sets $X^g$ and that a suitable normally nonsingular inclusion assumption is satisfied. Our approach is to work in equivariant analytic $K$-homology and equivariant algebraic $G$-theory and to identify the corresponding localized classes. On the analytic side, we identify the Banagl-Zagier equivariant $L$-classes with localized Chern character of the equivariant $K$-homology class of the signature operator defined by Banagl-Leichtnam-Piazza. On the algebraic side, we compute the localization of the equivariant motivic Hodge-Chern class transformation of the intersection Hodge module.

We prove that for a geometrically connected modular curve $X_H$ of prime level larger than $311$, all automorphisms are modular, i.e., they arise from the moduli interpretation of $X_H$. The proof relies on a general criterion involving the gonality of $X_H$ and the dimension of the CM part of the Jacobian of $X_H$. For arbitrary level, we give algorithms to compute the modular automorphisms and, in many cases, to prove that there are no other automorphisms. Finally, we apply these algorithms to the curves in the database LMFDB.

We construct a separated logarithmic compactification of a divisor in the moduli of polarised irreducible holomorphic symplectic (IHS) varieties of generalised Kummer type in dimension at least four. This compactification carries a universal family of logarithmic IHS varieties, and its tropicalisation carries a universal family of tropical generalised Kummer varieties. The compactification is built from a more general construction of explicit, projective, and logarithmically smooth models for type~II and type~III degenerations of generalised Kummer varieties of arbitrary dimension and over base logarithmic schemes of any dimension. Over a base of any logarithmic rank, these degenerations come with a tropicalisation map to a family of integral affine manifolds with singularities, and with boundary in the type II case. This tropical family is constructed through a tropical generalised Kummer construction. Our approach uses logarithmic Hilbert schemes to extend the conventional generalised Kummer construction to families of logarithmic abelian varieties. We describe fibres of our degenerations, and thus every point in the boundary of our compactification. Proper models correspond to polyhedral structures on the associated tropical generalised Kummer variety, with the resulting polyhedral complex serving as the dual intersection complex; each irreducible component is constructed by applying the generalised Kummer construction to a product of logarithmic group compactifications of semi-abelian varieties.

In this paper, we study joint deformations of pairs $(X,E)$, where $X$ is a compact complex manifold and $E\rightarrow X$ is a holomorphic vector bundle. We prove that $(X,E)$ has unobstructed deformations if $X$ is a Fujiki manifold with torsion canonical bundle and $H^2(X,\textrm{End}^0E)=0$, where $\textrm{End}^0 E$ denotes the trace-free endomorphism bundle. When the canonical bundle is trivial, the Fujiki assumption can be replaced by three weak $\partial\bar\partial$-conditions. We construct examples satisfying these conditions whose Frölicher spectral sequences do not degenerate at $E_1$. These results provide non-Kähler extensions of the theorems of Li–Pan and Iacono–Manetti. Without this vanishing assumption, we obtain both unobstructed and obstructed pairs. In Thomas's example, we prove unobstructedness of the pair, although the bundle has obstructed deformations with the manifold fixed. On the other hand, we prove that every strict projective Calabi–Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations. We also construct obstructed pairs with Hermitian flat bundles on complex tori. These results answer two questions raised by Felten.