Mathematische Annalen最新文献

Let (overline{{mathcal {H}}}_{g,n}) denote the Hodge bundle over (overline{{mathfrak {M}}}_{g,n}), and ({mathbb {P}}overline{{mathcal {H}}}_{g,n}) its associated projective bundle. Let ({mathcal {H}}_{g,n}) and ({mathbb {P}}{mathcal {H}}_{g,n}) be respectively the restriction of (overline{{mathcal {H}}}_{g,n}) and ({mathbb {P}}overline{{mathcal {H}}}_{g,n}) to the smooth part ({mathfrak {M}}_{g,n}) of (overline{{mathfrak {M}}}_{g,n}). The Hodge norm provides us with a Hermtian metric on ({mathscr {O}}(-1)_{{mathbb {P}}{mathcal {H}}_{g,n}}). Let (Theta ) denote the curvature form of this metric. In this paper, we show that if (overline{{mathcal {N}}}) is a subvariety of ({mathbb {P}}overline{{mathcal {H}}}_{g,n}) that intersects ({mathcal {H}}_{g,n}), then the integral of the top power of (Theta ) over the smooth part of (overline{{mathcal {N}}}cap {mathbb {P}}{mathcal {H}}_{g,n}) equals the self-intersection number of the tautological divisor (c_1({mathscr {O}}(-1)_{{mathbb {P}}overline{{mathcal {H}}}_{g,n}})cap overline{{mathcal {N}}}) in (overline{{mathcal {N}}}). This implies that the volume of a linear subvariety of ({mathbb {P}}{mathcal {H}}_{g,n}) whose local coordinates do not involve relative periods can be computed by the intersection number of its closure in ({mathbb {P}}overline{{mathcal {H}}}_{g,n}) with some power of any divisor representing the tautological line bundle. We also genralize this statement to the bundles ({mathbb {P}}overline{{mathcal {H}}}^{(k)}_{g,n}), (k in {mathbb {Z}}_{ge 2}), of k-differentials with poles of order at most ((k-1)) over (overline{{mathfrak {M}}}_{g,n}). To obtain these results, we use the existence of an appropriate desingularization of (overline{{mathcal {N}}}) and a deep result of Kollár (Subadditivity of the Kodaira Dimension: Fiber of General Type, Algebraic Geometry, Sendai, 1985, Advanced studies in Pure Math. (1987)) on variation of Hodge structure.

Given an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of tubes. We also develop a general framework on tube formulas for valuations in Riemannian manifolds.

For the non-rotating gaseous stars modeled by the compressible Euler–Poisson system with general pressure law, Lin and Zeng (Comm Pure Appl Math 75: 2511–2572, 2022) proved a turning point principle, which gives the sharp linear stability/instability criteria for the non-rotating gaseous stars. In this paper, we prove that the sharp linear stability criterion for the non-rotating stars also implies nonlinear orbital stability against general perturbations provided the global weak solutions exist. If the perturbations are further restricted to be spherically symmetric, then nonlinear stability holds true unconditionally in the sense that the existence of global weak solutions near the non-rotating star can be proved.

We study the continuity set (the set of all continuous points) of pullback random attractors from a parametric space into the space of all compact subsets of the state space with Hausdorff metric. We find a general theorem that the continuity set is an IOD-type (a countable intersection of open dense sets) with the local similarity under appropriate conditions of random dynamical systems, and we further show that any IOD-type set in the parametric space has the continuous cardinality, which affirmatively answers the unsolved question about the cardinality of the continuity set of attractors in the literature. Applying to the random nonautonomous nonlocal parabolic equations on an unbounded domain driven by colored noise, we establish the existence and IOD-type continuity of pullback random attractors in time, sample-translation and noise-size, moreover, we prove that the continuity set of the pullback random attractor on the plane of time and sample-translation is composed of diagonal rays whose number of bars is the continuous cardinality.

In this paper we investigate the Fourier coefficients of harmonic Maass forms of negative half-integral weight. We relate the algebraicity of these coefficients to the algebraicity of the coefficients of certain canonical meromorphic modular forms of positive even weight with poles at Heegner divisors. Moreover, we give an explicit formula for the coefficients of harmonic Maass forms in terms of periods of certain meromorphic modular forms with algebraic coefficients.

Recently Pawłucki showed that compact sets that are definable in some o-minimal structure admit triangulations of class ({{mathcal {C}}}^p) for each integer (pge 1). In this work, we make use of these new techniques of triangulation to show that all continuous definable maps between compact definable sets can be approximated by differentiable maps without changing their image. The argument is an interplay between o-minimal geometry and PL geometry and makes use of a ‘surjective definable version’ of the finite simplicial approximation theorem that we prove here.

It is known that the optimal Noether inequality ({text {vol} }(X) ge frac{4}{3}p_g(X) - frac{10}{3}) holds for every 3-fold X of general type with (p_g(X) ge 11). In this paper, we give a complete classification of 3-folds X of general type with (p_g(X) ge 11) satisfying the above equality by giving the explicit structure of a relative canonical model of X. This model coincides with the canonical model of X when (p_g(X) ge 23). We also establish the second and third optimal Noether inequalities for 3-folds X of general type with (p_g(X) ge 11). A novel phenomenon shows that there is a one-to-one correspondence between the three Noether inequalities and three possible residues of (p_g(X)) modulo 3. These results answer two open questions by Chen et al. (Duke Math J 169(9):1603–1164, 2020), and in dimension three an open question by Chen and Lai (Int J Math 31(1):2050005, 2020).

We obtain piecewise regularity results for linear elliptic systems with piecewise regular coefficients arising from composite materials. We find Hölder continuous functions related to the higher-order derivatives by compensating the possible discontinuity due to the geometry and the discontinuity of the coefficients. This leads to the piecewise regularity of the higher-order derivatives and solves an open problem introduced by Li and Vogelius, and Li and Nirenberg for general composite geometry.

Given (h,g in {mathbb {N}}), we write a set (X subset {mathbb {Z}}) to be a (B_{h}^{+}[g]) set if for any (n in {mathbb {Z}}), the number of solutions to the additive equation (n = x_1 + dots + x_h) with (x_1, dots , x_h in X) is at most g, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative (B_{h}^{times }[g]) set analogously. In this paper, we prove, amongst other results, that there exist absolute constants (g in {mathbb {N}}) and (delta >0) such that for any (h in {mathbb {N}}) and for any finite set A of integers, the largest (B_{h}^{+}[g]) set B inside A and the largest (B_{h}^{times }[g]) set C inside A satisfy

In fact, when (h=2), we may set (g = 31), and when h is sufficiently large, we may set (g = 1) and (delta gg (log log h)^{1/2 - o(1)}). The former makes progress towards a recent conjecture of Klurman–Pohoata and quantitatively strengthens previous work of Shkredov.

Bounds for the volume of sections of convex bodies which are in the (L_p) John ellipsoid positions are established. Specifically, when the convex bodies are in the LYZ ellipsoid position, we construct a set of Hanner polytopes attaining the sharp bounds.