Abstract We consider multiple radial Schramm–Loewner evolution (SLE) curves with various time parameterizations and possible spiraling behavior. We construct them by tilting independent radial SLEs with a suitable local martingale, generalizing the earlier construction by Healey and Lawler. We prove that the curves are almost surely transient (i.e., they emanate from boundary points and terminate at a common interior target point). We show that they enjoy the resampling property: conditional on all of the curves but one, the remaining curve is distributed as chordal SLE in the remaining domain. We also verify that the multiradial SLE measure satisfies a natural boundary perturbation property analogous to that of the known SLE variants, involving its partition function (which is finite). Interestingly, in the parlance of Coulomb gas formalism in conformal field theory, partition functions of multiradial SLE processes with spiral involve both electric and magnetic charges.
Abstract We study the behavior of volumes of divisors in a family. We show that the volume of a divisor on the generic fiber equals the infimum of its volumes on fibers over any dense subset of the base. As an application, we show that the volume function is upper semicontinuous in flat families with reduced and irreducible fibers. Given $\epsilon>0$, we also prove that for a $\mathbb{Q}$-Cartier divisor $B$ on a family of varieties $X\rightarrow T$, if $(X_{t},|B_{t}|_{\mathbb{Q}})$ is $\epsilon $-lc and the volume of $B_{t}$ is constant for densely many closed points $t\in T$, then the generic fiber $(X_\eta ,|B_\eta |_{\mathbb{Q}})$ is also $\epsilon $-lc. Throughout this paper, the ground field is $\mathbb{C}$. By a variety, we mean an integral, separated scheme of finite type over a field (possibly non-algebraically closed).
Abstract Our primary objective is to construct closed subschemes of the character variety of the two-holed torus and the twisted character variety of a genus-$2$ surface that satisfy the following properties: (1) They are invariant under the action of the corresponding pure Mapping Class Groups (MCGs). (2) They contain characters arising from dense representations into $\operatorname{SU}_{2}(\mathbb{C})$ and $\operatorname{SL}_{2}(\mathbb{Z}_{p})$ for primes $p> 5$. (3) For any characteristic zero field $F$, and every absolutely irreducible character $x$ in the $F$-points of these subschemes, the Zariski closure of the pure MCG orbit of $x$ in the modified relative character variety has strictly smaller dimension. Our construction shows that [9, Theorem 1.4] is incorrect. In [5], we prove that these subschemes precisely characterize the locus of exceptional infinite orbits under the action of pure MCGs.
Abstract The main subject of study of this paper is general properties of Harish-Chandra algebras and modules, with special focus on the transfer of properties to a “spherical subalgebra”. We discuss in some detail the ring theoretical property of being a quasicommutative algebra, showing it to be Morita invariant, among other things. Then we obtain the main result of our paper: if an algebra $U$ and its spherical subalgebra $eUe$, where $e$ is a suitable idempotent, are Morita equivalent, and their Harish-Chandra subalgebras are compatible in a suitable way, we obtain an equivalence for their categories of Harish-Chandra modules. Along the way we realize an important category of modules for rational Cherednik algebras, the category $\widehat{\mathcal O}_{\mathfrak{c}}$, as a Harish-Chandra category, and as an application we obtain the equivalence of the categories of Harish-Chandra modules for two important algebras in Coulomb branch theory. Then we prove some general theorems about Galois rings and orders and later specialize our discussion to invariants of rings of differential operators on the torus and the Weyl algebra. In the last section we show that category $\mathcal{O}$ for fixed rings of the Weyl algebras in the case of complex reflection groups belongs to Harish-Chandra module categories with respect to two very different Harish-Chandra subalgebras. To Pan for all the smiles that she gives me.
Abstract For manifolds equipped with group actions, we have the following natural question: To what extent does the equivariant cohomology determine the equivariant diffeotype? We resolve this question for Hamiltonian circle actions on compact, connected symplectic four-manifolds. They are equivariantly diffeomorphic if and only if their equivariant cohomology rings are isomorphic as algebras over the equivariant cohomology of a point. In fact, we prove a stronger claim: each isomorphism between their equivariant cohomology rings is induced by an equivariant diffeomorphism.
In this paper, we study the connectedness of the fibers of integrable systems that extend complexity one $T$-spaces with proper moment maps, assuming that every tall singular point is non-degenerate. Our main result states that if there are no tall singular points with a hyperbolic block and connected $T$-stabilizer, then each fiber is connected. Moreover, we prove that the above condition is necessary if either some reduced space is simply connected or the moment map for the integrable system is generic in a natural sense.
We study the codegree Tur & aacute;n density of $\mathcal{C}_\ell <^>{r}$, the $r$-uniform hypergraph tight cycle of length $\ell $. A result of Han, Lo, and Sanhueza-Matamala states that if $\ell $ is sufficiently large and $r/\gcd (r,\ell )$ is even, then the codegree Tur & aacute;n density of $\mathcal{C}_\ell <^>{r}$ is $1/2$. We prove that whenever the latter assumption is not satisfied, there is a significant drop in the codegree Tur & aacute;n density. That is, if $\ell $ is sufficiently large and $r/\gcd (r,\ell )$ is odd, then the codegree Tur & aacute;n density of $\mathcal{C}_\ell <^>{r}$ can be at most $1/3$. Moreover, this bound is tight for infinitely many uniformities $r$ and all sufficiently large $\ell $ in the corresponding residue classes modulo $r$. Our proof makes use of a group-theoretic connection between Tur & aacute;n-type theorems for tight cycles and "oriented colorings" of the edge set of a hypergraph.
Abstract We give criteria for determining the positivity of line bundles coming from vertex operator algebras (VOAs) on the moduli space $\overline{\textrm{M}}_{0,n}$ of rational curves with $n$ marked points. The criteria use the multiplicative structure of VOA representations encoded in the fusion ring. Using them, we construct positive line bundles on $\overline{\textrm{M}}_{0,n}$ from certain parafermion VOAs. These give the first examples of commutant VOAs producing positive line bundles.
In this paper, we prove the mass equidistribution theorem restricted to vertical geodesic segments for holomorphic Hecke newforms of large square-free level. We utilize the effective proof of Quantum Unique Ergodicity in the level aspect. Moreover, we study this $L<^>{2}$ mass in the full geodesics and relate it to second moments of twisted $L$-functions.
Let $\Psi ( extbf{z}, extbf{a},q)$ be a fundamental solution matrix of the quantum difference equation of a Nakajima variety $X$. In this work, we prove that the operator $$ \begin{align*} & \Psi( extbf{z}, extbf{a},q) \Psi( extbf{z}<^>p, extbf{a}<^>p,q<^>{p<^>2})<^>{-1} \end{align*} $$ has no poles at the primitive complex $p$-th roots of unity $q=\zeta _{p}$. As a byproduct, we show that the iterated product of the operators $ extbf{M}_{\mathcal{L}}( extbf{z}, extbf{a},q )$ from the $q$-difference equation on $X$ $$ \begin{align*} & extbf{M}_{\mathcal{L}} ( extbf{z} q<^>{(p-1)\mathcal{L}}, extbf{a},q) \cdots extbf{M}_{\mathcal{L}} ( extbf{z} q<^>{\mathcal{L}}, extbf{a},q) extbf{M}_{\mathcal{L}} ( extbf{z}, extbf{a},q) \end{align*} $$ evaluated at $q=\zeta _{p}$ has the same eigenvalues as $ extbf{M}_{\mathcal{L}} ( extbf{z}<^>{p}, extbf{a}<^>{p},q<^>{p})$.Upon a reduction of the quantum difference equation of $X$ to the quantum differential equation over the field of finite characteristic, the above iterated product transforms into a Grothendiek-Katz $p$-curvature of the corresponding quantum connection whereas $ extbf{M}_{\mathcal{L}} ( extbf{z}<^>{p}, extbf{a}<^>{p},q<^>{p})$ becomes a certain Frobenius twist of that connection. In this way, we give an explicit description of the spectrum of the $p$-curvature of quantum connection for Nakajima varieties.
Viewing $\mathbb{S}<^>{n-1}=\partial \mathbb{B}<^>{n}\subset \mathbb{R}<^>{n}$, the boundary Laplacian $\Delta _{\mathbb{S}<^>{n-1}}$ can be explicitly expressed in terms of $\Lambda $, the Dirichlet-to-Neumann map, as $\Delta _{\mathbb{S}<^>{n-1}}=\Lambda <^>{2}+(n-2)\Lambda $. In this paper, we seek to characterize those manifolds for which such an exact relationship holds, and more generally to measure the failure of such a relationship in terms of geometric data. To this end, we obtain a stability estimate which shows that a smoothly bounded domain $\Omega $ in $\mathbb{R}<^>{3}$ must be close to the ball if the commutator $[\Lambda ,\Delta _{\partial \Omega }]$ is small. We then study the case of manifolds conformal to the ball, show that such a relationship implies a radial metric structure, and discuss stability in this setting. Finally, we provide a modern exposition of Gohberg's lemma, a foundational result in microlocal analysis which we employ as a starting step for our reasoning.
Abstract Farre, Pozzetti, and Viaggi proved that any $(d-k)$-hyperconvex subgroup of $\textsf{PSL}(d,\mathbb C)$ is virtually isomorphic to a convex cocompact Kleinian group and that its $\alpha _{k}$-critical exponent is at most 2. We show that a $(d-k)$-hyperconvex subgroup is isomorphic to a uniform lattice in $\textsf{PSL}(2,\mathbb C)$ if and only if its $\alpha _{k}$-critical exponent is exactly 2. Furthermore, we show that if a strongly irreducible $(d-k)$-hyperconvex subgroup has $\alpha _{k}$-critical exponent $2$, then it is the image of a uniform lattice in $\textsf{PSL}(2,\mathbb C)$ by an irreducible representation of $\textsf{PSL}(2,\mathbb C)$ into $\textsf{PSL}(d,\mathbb C)$.
Given a compact Riemann surface $C$ and its Jacobian $J_{C}$, the line in $H^{0}(J_{C},\, 2\Theta )$ orthogonal to the sections of $2\Theta $ vanishing at $0\, \in \, J_{C}$ produces a natural projective structure on $C$. We investigate the properties of this projective structure.
Abstract We develop the deformation-obstruction calculus for morphisms of complexes with a fixed lift of the codomain, in the setting of derived categories of deformations of abelian categories. This extends the existing deformation-obstruction calculi for abelian categories and objects therein. As an application, we show that semiorthogonal decompositions deform uniquely in smooth and proper families of schemes.
Let 1 <= k <= n and M be a random n & times; n matrix with independent uniformly random {+/- 1}-entries. We show that there exists an absolute constant c > 0 such that P[rank(M) <= n-k] <= exp(-cnk). This confirms a well-known prediction in the area, extending a result of Rudelson (who previously proved this same result under the restriction k <= i/n, via different methods).
Abstract By $p$-adically interpolating the branching law for the spherical pair $\left (U_{n}, U_{n+1} \times U_{n}\right )$ of definite unitary groups, we construct a $p$-adic $L$-function attached to cohomological automorphic representations of $U_{n+1} \times U_{n}$. Under a further multiplicity one assumption, we extend the construction to Coleman families. Our $p$-adic $L$-function interpolates the square root of the central critical $L$-value. It has weight and anticyclotomic variables and its construction relies on the proof of the unitary Gan–Gross–Prasad conjecture.
Let $\operatorname{\mathfrak h}$ be a Cartan subalgebra of a complex semisimple Lie algebra $\operatorname{\mathfrak g}.$ We define a compactification $\bar{\operatorname{\mathfrak h}}$ of $\operatorname{\mathfrak h}$, which is analogous to the closure $\bar{H}$ of the corresponding maximal torus $H$ in the adjoint group of $\operatorname{\mathfrak g}$ in its wonderful compactification, which was introduced and studied by De Concini and Procesi [11]. We observe that $\bar{\operatorname{\mathfrak h}}$ is a matroid Schubert variety and prove that the irreducible components of the boundary $\bar{\operatorname{\mathfrak h}} - \operatorname{\mathfrak h}$ of $\operatorname{\mathfrak h}$ are divisors indexed by root system data. We prove that $\bar{\operatorname{\mathfrak h}}$ is a normal variety and find an affine paving of $\bar{\operatorname{\mathfrak h}},$ where the strata are given by the orbits of $\operatorname{\mathfrak h}.$ We show that the strata of $\bar{\operatorname{\mathfrak h}}$ correspond bijectively to subspaces of the corresponding Coxeter hyperplane arrangement studied by Orlik and Solomon, and prove that the associated posets are isomorphic. As a consequence, we express the Betti numbers of $\bar{\operatorname{\mathfrak h}}$ in terms of well-known combinatorial invariants in the classical cases. We show that the Weyl group $W$ acts on $\bar{\operatorname{\mathfrak h}}$, and describe $H<^>{\bullet }(\bar{\operatorname{\mathfrak h}}, \operatorname{\mathbb C})$ as a representation of $W$, and compute the cup product for $H<^>{\bullet }(\bar{\operatorname{\mathfrak h}}, \operatorname{\mathbb Z})$.
In this note, we prove-in dimension at most $4$-a conjecture of Hao, which says that a morphism $f: X o A$ to a simple abelian variety $A$ from a smooth projective $X$ is smooth if and only if there is a global $1$-form pulled back from $A$ without any zeros. We also give a complete classification of four-folds with a $1$-form without zeros and not admitting a map to an elliptic curve.
We study the uncertainty principle & Vert;mu(xi)|xi|(beta)& Vert;(alpha)(infinity)(integral|x|(alpha)d mu)(beta)>= C-alpha,C-beta,(d)& Vert;mu & Vert;(alpha+beta)(TV) and that extremizers exist. Moreover, we obtain an abstract characterization of the extremizers, which allows us to describe their asymptotic behavior and, for certain parameter values, to determine them explicitly.
