We study persistence properties of the solution of the Benjamin-Ono equation in weighted Sobolev spaces. Roughly, we show that for beta < 7/2, the solution u(x, t) of the BO remains in the space L-2(|x|(2 beta )dx) if and only if its data u(x, 0) belongs to this space and it is regular enough, i.e. u(0) is an element of H-beta(R).
We show that every projective Oka manifold is elliptic in the sense of Gromov. This gives an affirmative answer to a long-standing open question.
We provide new examples of anti-symplectic involutions on moduli spaces of stable sheaves on K3 surfaces. These involutions are constructed through (anti) autoequivalences of the bounded derived category of coherent sheaves on K3 surfaces arising from spherical bundles. We analyze these induced maps in the moduli space, imposing restrictions on the Mukai vector and considering the preservation of stability conditions. Our construction extends and unifies classical examples, such as the Beauville involutions, Markman-O'Grady reflections and a more recent construction by Beri-Manivel.
In this short note, we will explain that the good moduli space morphisms behave as if they are proper when we consider sheaf operations, though they are not separated. For example, the decomposition theorem and the base change theorem hold for these morphisms, which have applications to the cohomological study of moduli spaces.
We construct exceptional Fano varieties with the smallest known minimal log discrepancies in all dimensions. These varieties are well-formed hypersurfaces in weighted projective space. Their minimal log discrepancies decay doubly exponentially with dimension, and achieve the optimal value in dimension 2.
In this note we record a comparison theorem on the B-model variation of semi-infinite Hodge structures. This result is considered a folklore theorem by experts in the field. We only take this opportunity to write it down. Our motivation is to apply it in the study of B-model categorical enumerative invariants.
In this paper, we provide formulas calculating the partition functions of two types of plane partitions using the crystal melting model method introduced by Okounkov, Reshetikhin and Vafa. As applications, we obtain a product formula for the partition function of the plane partitions with a limit shape boundary. A corollary of this formula is the demonstration of the equivalence between this partition function and the open-closed string amplitude of the double$-\mathbb{P}^1$ model. We also derive a product formula for the partition function of symmetric plane partitions with a limit shape boundary.
The m-thick part of the modular surface X is the smallest compact subsurface of X with horocycle boundary containing all the closed geodesics which wind around the cusp at most m times. The m-thick parts form a compact exhaustion of X. We are interested in the geodesics that lie in the m-thick part (so called m low-lying geodesics). We produce a complete asymptotic expansion for the number of m low-lying geodesics of length equal to 2n in the modular surface. In particular, we obtain the asymptotic growth rate of the m low-lying geodesics in terms of their word length using the natural generators of the modular group. After establishing a correspondence between this counting problem and the problem of counting necklaces with n beads, we perform a careful singularity analysis on the associated generating function of the sequence.
We prove that 3-dimensional ellipsoids invariant under a 2-torus action contain infinitely many distinct immersed minimal tori, with at most one exception. These minimal tori bifurcate from the 2-torus orbit of largest volume at a dense set of eccentricities, and remain invariant under a circle.
This article studies a class of Dirac operators of the form $D_\varepsilon= D+\varepsilon^{-1}\mathcal A$, where $\mathcal A$ is a zeroth order perturbation vanishing on a subbundle. When $\mathcal A$ satisfies certain additional assumptions, solutions of the Dirac equation have a concentration property in the limit $\varepsilon\to 0$: components of the solution orthogonal to $\ker(\mathcal A)$ decay exponentially away from the locus $\mathcal Z$ where the rank of $\ker(\mathcal A)$ jumps up. These results are extended to a class of non-linear Dirac equations. This framework is then applied to study the compactness properties of moduli spaces of solutions to generalized Seiberg-Witten equations. In particular, it is shown that for sequences of solutions which converge weakly to a $\mathbb Z_2$-harmonic spinor, certain components of the solutions concentrate exponentially around the singular set of the $\mathbb Z_2$-harmonic spinor. Using these results, the weak convergence to $\mathbb Z_2$-harmonic spinors proved in existing convergence theorems is improved to $C^\infty_{loc}$.
We study the vanishing of $L(f,\chi,j)$ as $f$ runs through all classical forms in a $p$-adic Hida family (including forms with arbitrarily high nebentype at $p$), $\chi$ runs through all characters of $p$-power conductor, and $j$ is a critical value. We show that if infinitely many of these $L$-values vanish (apart from the ones forced to vanish by the sign of their functional equation) then this infinitude of vanishing must be exceptionally regular, so regular in fact that one can typically rule out this possibility in any given example. Indeed, we systematically verified that such regular vanishing does not occur in multiple Hida families twisted by a wide range of quadratic characters by computing the corresponding two-variable $p$-adic $L$-functions via overconvergent modular symbols.
In this paper, we establish an & ell;(2) decoupling inequality for the convex hypersurface {(xi 1, . . . , xi(n-1), xi(m)(1) + + xi(m)(n-1) ) : (xi 1, . . . , xi(n-1)) is an element of [0, 1](n-1)} associated with the decomposition adapted to hypersurfaces of finite type, where n >= 2 and m >= 4 is an even number. The key ingredients of the proof include & ell;(2) decoupling inequalities for convex hypersurfaces of the form {(xi(1), . . . , xi(n-1), phi(1)(xi(1)) + + phi(s)(xi(s)) + xi(m)(s+1) + + xi(m)(n-1) ) : (xi(1), . . . , xi(n-1)) is an element of [0, 1](n-1)} , 0 <= s <= n - 1, with phi(1), . . . , phi(s) being m-nondegenerate.
Phong and Stein [22] first studied the composition of operators associated with different kinds of homogeneities which arise naturally in the partial derivative-Neumann problem. To characterize kernels of the composition operators with different homogeneities, Nagel, Ricci, Stein and Wainger [20] introduced a class of two-flag kernels, a class of kernels P(E) and a restricted class of kernels P-0(E). They studied the regularity properties of these classes of kernels in L-p, 1 < p < infinity. In this article, we develop the Hardy space theory for these kernels. Via decomposition characterizations, we introduce the two-flag Hardy and local Hardy spaces associated with different homogeneities in the two-step case for these classes of kernels and study the relations between different Hardy spaces. As applications, we prove the boundedness of the convolution operators with these kernels on corresponding Hardy (or local Hardy) spaces.
We determine the g.c.d.of degrees of all finite base field extensions, performing a prescribed partial splitting of a generic central simple algebra with a quadratic pair, or, equivalently, of a generic torsor under a split projective orthogonal group. We also compute this invariant for the generic central simple algebras with quadratic pairs of trivial discriminant, or, equivalently, for the generic torsors under the split projective special orthogonal groups.
We prove sharp lower bounds for the charged Hawking mass of stable surfaces in electrostatic space-times in various contexts. An upper bound for the genus of stable surfaces in the electrostatic system is provided. We also study the positivity for the charged Hawking mass of a minimal surface with index one in the electrostatic space-times. A criterion for a CMC surface in the Reissner-Nordstrom deSitter space to be stable is presented.
