We study mirror symmetry of families of elliptic K3 surfaces with elliptic fibers of type $E_6$, $E_7$ and $E_8$. We consider a moduli space $\mathsf{T}$ of the mirror K3 surfaces enhanced with the choice of differential forms. We show that coordinates on $\mathsf{T}$ are given by the ring of quasi modular forms in two variables, with modular groups adapted to the fiber type. We furthermore introduce

We present an example of a $\mathcal{C}^1$-robustly transitive skew-product with non-trivial, non-hyperbolic action on homology. The example is conservative, ergodic, non-uniformly hyperbolic and its fiber directions cannot be decomposed into two dominated subbundles.

We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of inhomogeneous quadratic polynomials. It is shown that the boundedness of such operators is implied by the boundedness of the corresponding Weyl symbols.

We prove Burns–Krantz type boundary rigidity theorems for holomorphic self-maps of some fibered domains.

We give examples of tight high dimensional contact manifolds admitting a contactomorphism whose powers are all smoothly isotopic but not contact-isotopic to the identity. This is a generalization of an observation in dimension $3$ by Gompf, also reused by Ding and Geiges.

In this paper, we study germs of smooth CR mappings sending a closed orbit of $SU(\ell, m)$ into a closed orbit of $SU(\ell^\prime , m^\prime)$ in Grassmannian manifolds. We show that if the signature difference of the Levi forms of two orbits is not too large, then the mapping can be factored into a simple form and one of the factors extends to a totally geodesic embedding of the ambient Grassmannian

For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that its group of automorphisms faithfully acts on its cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and groups of automorphisms for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and

A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $\mu$ of the random walk. The drift is the “translation distance” of the random walk. In this paper, we prove that the drift varies continuously with the transition probability measures, under the assumption that the distance and the horofunctions

We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective. One application is that the Seifert genus is monotonic under strongly homotopy-ribbon concordance.

Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\mathcal{M}_{lc} (n,d)$ denote the moduli space of pairs $(E, D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $\mathfrak{gl} (n, \mathbb{C})$, where $n$ and $d$ are mutually corpime. Let $L$ denote a

We obtain conservation laws at negative regularity for the Benjamin–Ono equation on the line and on the circle. These conserved quantities control the $H^s$ norm of the solution for $-\frac{1}{2} \lt s \lt 0$. The conservation laws are obtained from a study of the perturbation determinant associated to the Lax pair of the equation.

We prove that the enveloping algebra $\mathcal{U}(\mathfrak{q})$ of a finite-dimensional Lie algebra $\mathfrak{q}$ contains a commutative subalgebra of the maximal possible transcendence degree ($\operatorname{dim} \mathfrak{q} + \operatorname{ind} \mathfrak{q}) / 2$.

We find new examples of the $P = W$ identities of de Cataldo–Hausel–Migliorini by studying cluster varieties. We prove that the weight filtration of 2D cluster varieties correspond to the perverse filtration of elliptic fibrations which are deformation equivalent to the elliptic fibrations of types $I_b$ or $2I_b$. These examples do not arise from character varieties or moduli of Higgs bundles.

Let $k$ be a finitely generated field of characteristic $p \gt 0$ and $X$ a smooth and proper scheme over $k$. Recent works of Cadoret, Hui and Tamagawa show that, if $X$ satisfies the $\ell$-adic Tate conjecture for divisors for every prime $\ell \neq p$, the Galois invariant subgroup $Br(X_{\overline{k}}) {[p^\prime]}^{\pi_1(k)}$ of the prime-to-$p$ torsion of the geometric Brauer group of $X$ is

For every integer $g$, we construct a $2$-solvable and $2$-bipolar knot whose topological $4$-genus is greater than $g$. Note that $2$-solvable knots are in particular algebraically slice and have vanishing Casson–Gordon obstructions. Similarly all known smooth $4$-genus bounds from gauge theory and Floer homology vanish for $2$-bipolar knots. Moreover, our knots bound smoothly embedded height four

This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups. We prove that the Birman exact sequence does not admit a section over any subgroup $\Gamma$ contained in the Torelli group with finite index. A fortiori this implies that there is no multi-section for the universal surface bundle with Torelli monodromy. This theorem was announced in a 1990 preprint

Tautological systems developed in [8, 9] are Picard–Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to zero loci of global sections of vector bundles. In particular, we obtain similar criterion as in [8, 9] about holonomicity and regularity of the systems. We also prove solution rank formulas and geometric realizations of

We show that normalized quantum $K$-theoretic vertex functions for cotangent bundles of partial flag varieties are the eigenfunctions of quantum trigonometric Ruijsenaars–Schneider (tRS) Hamiltonians. Using recently observed relations between quantum Knizhnik–Zamolodchikov (qKZ) equations and tRS integrable system we derive a nontrivial identity for vertex functions with relative insertions.

We propose a method to construct $\mathrm{G}_2$–instantons over a compact twisted connected sum $\mathrm{G}_2$–manifold, applying a gluing result of Sá Earp and Walpuski to instantons over a pair of $7$-manifolds with a tubular end. In our example, the moduli spaces of the ingredient instantons are non-trivial, and their images in the moduli space over the asymptotic cross-section K3 surface intersect

The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler–Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions

This paper is based on a formulation of the Navier–Stokes equations developed by Iyer and Constantin, where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. Our contribution is to establish this probabilistic representation formula for mild solutions of the Navier–Stokes equations on $\mathbb{R}^d$.

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture

The purpose of this note is to give an account of a well-known folklore result: the Hodge structure on the second cohomology of a compact hyperkähler manifold uniquely determines Hodge structures on all higher cohomology groups.We discuss the precise statement and its proof, which are somewhat difficult to locate in the literature.

We study the miniversal deformations of minimally elliptic two-dimensional singularities of multiplicities of $5$, $6$ and $7$. By restricting the miniversal deformations on the line transverse to the discriminant locus, we construct many new three-dimensional isolated rational Gorenstein singularities with one-dimensional equisingular deformation and nontrivial $T^2$. In fact the three-dimensional

We refine and prove the central conjecture of our first paper for annuli with at least two marked intervals on each boundary component by computing the derived Hall algebras of their Fukaya categories.

We prove that a twisted Enriques (respectively, untwisted bielliptic) surface over an algebraically closed field of positive characteristic at least $3$ (respectively, at least $5$) has no non-trivial Fourier–Mukai partners.

We prove a basic inequality for the $d$-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map between two aspherical Seifert homology spheres is homotopic to a homeomorphism if and

Let $k$ be a perfect field of positive characteristic and let $X$ be a smooth irreducible quasi-compact scheme over $k$. The Drinfeld–Kedlaya theorem states that for an irreducible $F$-isocrystal on $X$, the gap between consecutive generic slopes is bounded by one. In this note we provide a new proof of this theorem. Our proof utilizes the theory of $F$‑isocrystals with $r$‑log decay. We first show

Peng Wu [P. Wu, “Einstein four-manifolds with self-dual Weyl tensor of nonnegative determinant”, Int. Math. Res. Not. (2021), no. 2, 1043–1054] recently announced a beautiful characterization of conformally Kähler, Einstein metrics of positive scalar curvature on compact oriented $4$-manifolds via the condition $\operatorname{det}(W^{+}) \gt 0$. In this note, we buttress his claim by providing an entirely

We compute the archimedean doubling zeta integrals which appear in the interpolation formulas for the $p$‑adic $L$-functions of Siegel modular forms constructed in [Z. Liu, “$p$‑adic $L$-functions for ordinary families of symplectic groups”, J. Inst. Math. Jussieu 19 (2020), no. 4, 1287–1347].

In this paper, we extend our result on a depth preserving property of the local Langlands correspondence for quasi-split unitary groups ([16]) to non-quasi-split unitary groups by using the local theta correspondence. The key ingredients are a depth preserving property of the local theta correspondence proved by Pan and a description of the local theta correspondence via the local Langlands correspondence

We exhibit explicit examples of very general special cubic fourfolds with discriminant $d$ admitting an associated (twisted) K3 surface, which have non-isomorphic Fourier–Mukai partners. In particular, in the untwisted setting, we show that the number of Fourier–Mukai partners for a very general special cubic fourfold with discriminant $d$ and having an associated K3 surface, is equal to the number

We prove sharp bounds on the discriminants of stable rank two sheaves on surfaces in three-dimensional projective space. The key technical ingredient is to study them as torsion sheaves in projective space via tilt stability in the derived category. We then proceed to describe the surface itself as a moduli space of rank two vector bundles on it. Lastly, we give a proof of the Bogomolov inequality

In this paper we prove that, over a Hilbertian ground field, surfaces with two conic fibrations whose fibres have non-zero intersection product have the Hilbert property. We then give an application of this result, namely the verification of the Hilbert property for certain del Pezzo varieties.

Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable cases, the observable is cohomologous to its ergodic minimizing value on the Mather set, and the discrete weak KAM solution plays the role of the transfer function.

We prove a generalization of the Monge–Cayley–Salmon theorem on osculation and ruled submanifolds using geometric measure theory.

We introduce a process, which we call Levi–Kähler reduction, for constructing Kähler manifolds and orbifolds from CR manifolds (of arbitrary codimension) with a transverse torus action. Most of the paper is devoted to the study of Levi–Kähler reductions of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of the orbifolds

We study the radius of analyticity $R(t)$ in space, of strong solutions to systems of scale-invariant semi-linear parabolic equations. It is well-known that near the initial time, $R(t)t^{-\frac{1}{2}}$ is bounded from below by a positive constant. In this paper we prove that $\displaystyle\liminf_{t\to 0} R(t)t^{-\frac{1}{2}}=\infty$, and assuming higher regularity for the initial data, we obtain

We study the Fulton–Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$‑theory defined by Anderson and Payne.

In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\operatorname{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_\ell)$, with $F$ a totally real number field and $G$ a reductive group, to geometric $\ell$‑adic representations. In this note we take $G$ to be a classical group and construct many examples of $G$-irreducible representations to which

We construct holomorphically varying families of Fatou–Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $\mathbb{C}^n$ for $n \gt 1$. This provides a simple proof of the recent result of $Y$. Kusakabe to the effect that the complement $\mathbb{C}^n \setminus K$ of any polynomially convex subset $K$ of $\mathbb{C}^n$ is an Oka manifold. The analogous

We develop a general framework for studying relative weight representations for certain pairs consisting of an associative algebra and a commutative subalgebra. Using these tools we describe projective and simple weight modules for a common localisation of two quantum versions of Weyl algebras, one by Maltsiniotis and the other one by Akhavizadegan and Jordan, for generic values of the deformation

We study the conditions under which the highest nonvanishing local cohomology module of a domain $R$ with support in an ideal $I$ is faithful over $R$, i.e., which guarantee that $H^c_I (R)$ is faithful, where $c$ is the cohomological dimension of $I$. In particular, we prove that this is true for the case of positive prime characteristic when $c$ is the number of generators of $I$.

Ruberman gave the first examples of self-diffeomorphisms of four-manifolds that are isotopic to the identity in the topological category but not smoothly so. We give another example of this phenomenon, using the Dehn twist along a $3$-sphere in the connected sum of two $K3$ surfaces.

We prove that, if $C$ is a smooth projective curve over the complex numbers, and $E$ is a stable vector bundle on $C$ whose slope does not lie in the interval $[-1, n-1]$, then the associated tautological bundle $E^{[n]}$ on the symmetric product $C^{(n)}$ is again stable. Also, if $E$ is semi-stable and its slope does not lie in $[-1, n-1]$, then $E^{[n]}$ is semi-stable.

We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1 (\Omega)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.

We consider the following nonlinear Schrödinger equation on exterior domain:\[\begin{cases}iu_t+ \Delta_g u + ia(x)u-|u|^{p-1}u=0 & (x,t) \in \Omega \times (0,+ \infty), \\u {\big \vert}_{\Gamma}=0 & t \in (0,+ \infty), \\u (x,0) = u_0(x) & x \in \Omega,\end{cases}\] where $1\lt p \lt \frac{n+2}{n-2}$, $\Omega \subset \mathbb{R}^n (n \geq 3)$ is an exterior domain and $(\mathbb{R}^n , g)$ is a complete

Three different generalizations will be given for Nishino’s rigidity theorem asserting the triviality of Stein families of $\mathbb{C}$ over the polydisc, in connection to generalizations of Hartogs’s theorem on the analyticity criterion for continuous functions.

For every integer $g \gt 1$ and prime $p \gt 0$, we give an example of a standard graded domain $R$ (where Proj $R$ is a nonsingular projective curve of genus $g$ over an algebraically closed field of characteristic $p$), such that the set of $F$-thresholds of the irrelevant maximal ideal of $R$ is not discrete. This answers a question posed by Mustaţӑ–Takagi–Watanabe ([MTW], 2005). These examples

The classical four-vertex theorem describes a fundamental property of simple closed planar curves. It has been extended to space curves, namely a smooth, simple closed curve in $\mathbb{R}^3$ has at least four points with vanishing torsion if it lies on a convex surface. More recently, Ghomi [6] extended this property to curves lying on locally convex surfaces. In this paper we provide an alternative

In [CVX3] (T. H. Chen, K. Vilonen, and T. Xue, “Springer correspondence for the split symmetric pair in type A”, Compos. Math. 154 (2018), no. 11, 2403–2425), we have established a Springer theory for the symmetric pair $(\operatorname{SL}(N), \operatorname{SO}(N))$. In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These

In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex projective space; we also show that all Schur polynomials are numerically positive for

The Eisenbud–Green–Harris (EGH) conjecture states that a homogeneous ideal in a polynomial ring $K[x_1, \dotsc, x_n]$ over a field $K$ that contains a regular sequence $f_1, \dotsc , f_n$ with degrees $a_i, i =1, \dotsc , n$ has the same Hilbert function as a lex-plus-powers ideal containing the powers $x^{a_i}_i , i = 1, \dotsc , n$. In this paper, we discuss a case of the EGH conjecture for homogeneous

Let $f : X \to Y$ be a proper, dominant morphism of smooth varieties over a number field $k$. When is it true that for almost all places $v$ of $k$, the fibre $X_P$ over any point $P \in Y (k_v)$ contains a zero-cycle of degree $1$? We develop a necessary and sufficient condition to answer this question. The proof extends logarithmic geometry tools that have recently been developed by Denef and Lo

We prove that termination of lower dimensional flips for generalized $\operatorname{klt}$ pairs implies termination of flips for $\operatorname{log}$ canonical generalized pairs with a weak Zariski decomposition. Under the same hypothesis we prove that the existence of weak Zariski decompositions for pseudo-effective log canonical pairs implies the existence of weak Zariski decompositions for pseudo-effective

Given a smooth variety $X$ with an action of a finite group $G$, and a semiorthogonal decomposition of the derived category, $\mathcal{D}([X/G])$, of $G$-equivariant coherent sheaves on $X$ into subcategories equivalent to derived categories of smooth varieties, we construct a similar semiorthogonal decomposition for a smooth $G$-invariant divisor in $X$ (under certain technical assumptions). Combining

We show that any one-relator group $G = F / {\langle \negthinspace \langle} w {\rangle \negthinspace \rangle}$ with torsion is coherent — i.e., that every finitely generated subgroup of $G$ is finitely presented — answering a 1974 question of Baumslag in this case.

Let $M$ be a compact Riemannian manifold without boundary. We investigate the integrals of $L^2$-normalized Laplace eigenfunctions over closed submanifolds. General bounds for these quantities were obtained by Zelditch [23], and are sharp in the case where $M$ is the standard sphere. However, as with sup norms of eigenfunctions, there are many interesting settings where improvements can be made to

Let $S$ be a Burniat surface with $K^2_S = 6$. Then we show that $\operatorname{glct}(S, K_S) = \frac{1}{2}$ by showing that $\operatorname{glct}(S, 2K_S) = \operatorname{lct} (S, E) = \frac{1}{4}$ for some divisor $E \in \lvert 2K_S \rvert$. This implies that Tian’s conjecture (which fails in general) holds for the polarized pair $(S, 2K_S)$, since the corresponding graded algebra is generated by

We prove that the mapping class group of a closed connected orientable surface of genus at least eight is generated by three involutions.