Random systems of holomorphic sections of a sequence of line bundles on compact Kähler manifolds

Abstract Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$, and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|\omega _{X} \otimes A^{\otimes m}|$ are algebraically hyperbolic. We prove the conjecture for spherical varieties with smooth orbit closures. As a corollary, we conclude that the conjecture holds for horospherical varieties, and for toroidal spherical varieties. Furthermore, for any spherical variety, we show that the conjecture holds modulo the complement of an open dense orbit.

We examine étale covers of genus 2 curves that occur in the linear system of a polarizing line bundle of type (1,d) on a complex abelian surface. We give results counting fixed points of involutions on such curves as well as decomposing their Jacobians into isogenous products.

Abstract We generalise the notions of scalar-valued holomorphic p -contact and s -symplectic structures introduced recently on compact complex manifolds by the second-named author jointly with H. Kasuya and L. Ugarte to their analogues with values in a holomorphic line bundle. We then study the resulting holomorphic p -contact and s -symplectic manifolds which, unlike their scalar counterparts that are never Kähler, can even be projective. In particular, we investigate the (lack of) positivity properties of the canonical bundle of these manifolds when it is given a possibly singular Hermitian fibre metric. One of the tools used is a very recent regularisation result for m -psh functions obtained jointly by S. Dinew and the second-named author.

Abstract We explore the impact of “slow-light” radiative transfer—i.e., general relativistic radiative transfer calculations in which the simulated fluid evolves while light rays are propagating through it—in general relativistic magnetohydrodynamic models of the M87 jet. Because the plasma in the jet-launching region is accelerated to relativistic velocities, and because the jet in M87 is nearly aligned with the line of sight (offset by ∼17°), a slow-light treatment is important for accurate modeling of the observable structure. While fast-light images exhibit prominent helical or loop-shaped features in the jet—which we associate with narrow bundles of magnetic field lines—these features become stretched and smoothed-out in slow-light images. Our slow-light images instead exhibit a double-edged, cone-like morphology that is more consistent with observations of M87 than conventional fast-light images. We find that the radius at which the plasma transitions from subrelativistic to relativistic velocities is imprinted on slow-light images via a transition from loop-dominated at small distances from the black hole (BH) to edge-dominated at a larger distance, with the loop-edge transition occurring at larger distances for lower BH spins. The jet image dynamics also vary with BH spin, with low-spin models producing jets that exhibit substantial “wobbling,” while high-spin models produce jets that are straighter and more stable in time. The spin-dependent jet morphology and variability are revealed by slow-light imaging both because slow-light effects are enhanced as the plasma velocity becomes more relativistic and because the plasma acceleration is itself a strong function of the spin.

We prove that the surface S ( X ) S(X) of bitangent lines of a general smooth quartic surface X X in P 3 \mathbb {P}^3 has unobstructed deformations of dimension 20 = h 1 ( S ( X ) , T S ( X ) ) 20=h^1(S(X), T_{S(X)}) . In addition, we show that the space of infinitesimal embedded deformations of X X injects into the one of S ( X ) S(X) . Finally, we prove that there is a natural birational map from the 20-dimensional moduli space of (polarised) double coverings of EPW-sextics to the moduli space of regular surfaces S S with p g = 45 p_g=45 and K S 2 = 360 K_S^2=360 polarised with a very ample line bundle H H such that H 2 = 40 H^2=40 , h 0 ( S , H ) = 6 h^0(S, H)=6 : the map sends a double covering of a EPW-sextic in P 5 \mathbb {P}^5 to the surface of double points of the EPW-sextic.

We study complex Lagrangians in Hitchin systems that factor through a proper subvariety of the Hitchin base non-trivially intersecting the regular locus. This gives a general framework for several examples in the literature. We compute the fiber-wise Fourier-Mukai transform of flat line bundles on visible Lagrangians. This proposes a construction of mirror dual branes to visible Lagrangians. Finally, we study a new example of visible Lagrangians in detail. Such visible Lagrangian exists whenever the underlying Riemann surface is a pillowcase cover. The proposed mirror dual brane turns out to be closely related to Hausel's toy model.

We give two new constructions of the harmonic algebra of a lattice polytope P P , a bigraded algebra whose character is the q q -Ehrhart series of P P defined by Reiner and Rhoades [Harmonics and graded Ehrhart theory, arXiv: 2407.06511 , 2024]. First, we show that the harmonic algebra is the associated graded algebra of the semigroup algebra of P P with respect to a certain natural filtration, clarifying it’s relationship with the more classical semigroup algebra. We then give a geometric interpretation of the harmonic algebra as a quotient of the ring of global sections of a certain family of line bundles on the blowup of the toric variety associated to P P at a generic point. Using this connection to toric geometry we resolve one the main conjectures of Reiner and Rhoades by showing that the harmonic algebra is not finitely generated in general.

The integral model of a G U ( n − 1 , 1 ) \mathrm {GU}(n-1,1) Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet L L -functions. We also determine the arithmetic volumes of Kudla-Rapoport divisors and relate them to coefficients of Eisenstein series.

In this paper, we study the positivity of an Ulrich vector bundle defined with respect to a globally generated ample line bundle. First, we prove a generalization of a Lopez theorem on the first Chern class and the bigness of an Ulrich bundle. Then, under some additional assumptions on the polarization, we give a description of its augmented base locus, which consequently leads to a characterization of the V-bigness and of the ampleness of an Ulrich bundle in this setting.

We study the holomorphic symplectic geometry of (the smooth locus of) the space of holomorphic sections of a twistor space with rotating circle action. The twistor space carries a line bundle with meromorphic connection constructed by Hitchin. We give an interpretation of Hitchin's meromorphic connection in the context of the Atiyah-Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue of the meromorphic connection serves as a moment map for the induced circle action, and furthermore the critical points of this moment map are studied. Particular emphasis is given to the example of Deligne-Hitchin moduli spaces.

A deformed Donaldson–Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a G 2 -manifold X satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. It can also be considered as an analogue of a G 2 -instanton. In this paper, we see that some important observations that appear in other geometric problems are also found in the dDT case as follows. (1) A dDT connection exists if a 7-manifold has full holonomy G 2 and the G 2 -structure is “sufficiently large”. (2) The dDT equation is described as the zero of a certain multi-moment map. (3) The gradient flow equation of a Chern–Simons type functional of Karigiannis and Leung, whose critical points are dDT connections, agrees with the Spin ( 7 ) version of the dDT equation on a cylinder with respect to a certain metric on a certain space. This can be considered as an analogue of the observation in instanton Floer homology for 3-manifolds.

Abstract Using representations of affine Lie algebras, we describe line bundles on a broad class of contractions of $$\overline{\textrm{M}}_{0,n}$$ M ¯ 0 , n , the moduli space of stable n -pointed rational curves, and show a variant of the cone and contraction theorem for these morphisms. These include the celebrated constructions of Kapranov, Keel, and Knudsen. Our main result suggests that while many so-called F-curves are not $$K_X$$ K X -negative, they exhibit behavior similar to $$K_X$$ K X -negative curves. This reveals a distinguished property of Knudsen’s construction $$f_{\textrm{Knu}}:\overline{\textrm{M}}_{0,n}\rightarrow \overline{\textrm{M}}_{0,n-1}\hspace{1.111pt}{\times }_{\overline{\textrm{M}}_{0,n-2}}\hspace{1.111pt}\overline{\textrm{M}}_{0,n-1}$$ f Knu : M ¯ 0 , n → M ¯ 0 , n - 1 × M ¯ 0 , n - 2 M ¯ 0 , n - 1 , allowing for the classification of all possible factorizations of $$f_{\textrm{Knu}}$$ f Knu , as well as further applications.

Deriving the Yukawa couplings and the resulting fermion masses and mixing angles of the Standard Model (SM) from a more fundamental theory remains one of the central outstanding problems in theoretical high-energy physics. It has long been recognized that string theory provides a framework within which this question can, at least in principle, be addressed. While substantial progress has been made in studying flavor physics in string compactifications over the past few decades, a concrete string construction that reproduces the full set of observed SM flavor parameters remains unknown. Here, we take a significant step in this direction by identifying two explicit <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>E</a:mi> <a:mn>8</a:mn> </a:msub> <a:mo>×</a:mo> <a:msub> <a:mi>E</a:mi> <a:mn>8</a:mn> </a:msub> </a:math> heterotic string models compactified on a Calabi-Yau threefold with Abelian, holomorphic, and polystable vector bundles with minimal supersymmetric (MS) SM spectrum. Subject to reasonable assumptions about the moduli, we show that these models reproduce the correct values of the quark and charged lepton masses, as well as the quark mixing parameters, at some point in their moduli spaces. The resulting four-dimensional theories are <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi mathvariant="script">N</c:mi> <c:mo>=</c:mo> <c:mn>1</c:mn> </c:math> supersymmetric, contain no exotic fields, and realize a <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>μ</f:mi> </f:math> -term suppressed to the electroweak scale. While the issues of moduli stabilization and supersymmetry breaking are not addressed here; our main result constitutes a proof of principle: There exist choices of topology and moduli within heterotic string compactifications which allow for an MSSM spectrum with the correct flavor parameters.

In the present article, we define a notion of good height functions on quasi-projective varieties V defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions are defined as limits of height functions associated with semi-positive adelic metrization on big and nef ℚ -line bundles on projective models of V satisfying mild assumptions. Building on a recent work of the author and Vigny as well as on a classical estimate of Call and Silverman, and inspiring from recent works of Kühne and Yuan and Zhang, we deduce the equidistribution of generic sequence of preperiodic parameters for families of polarized endomorphisms with marked points.

Abstract Let ℰ be a rank-2 vector bundle over an elliptic curve E , decomposable as a sum of line bundles of degrees d ' &gt; d ≥ 2, and ℒ the determinant of ℰ. The subspace L (ℰ) ⊂ ℙ n −1 ≅ ℙExt 1 (ℒ, 𝒪 E ) consisting of classes of extensions with middle term isomorphic to ℰ is one of the symplectic leaves of a remarkable Poisson structure on ℙ n −1 defined by Feigin–Odesskii and Polishchuk, and all symplectic leaves arise in this manner, as shown in earlier work that realizes L (ℰ) as the base space of a principal Aut(ℰ)-fibration. Here, we embed L (ℰ) into a larger, projective base space L ˜ ( E ) $\widetilde{L}(\mathcal{E})$ of a principal Aut(ℰ)-fibration whose total space parametrizes sections of ℰ. The embedding realizes L ( E ) ⊂ L ˜ ( E ) $L(\mathcal{E})\subset \widetilde{L}(\mathcal{E})$ as a complement of an anticanonical divisor Y (one of the main results), and we give an explicit description of the normalization of Y as a projective-space bundle over a projective space. For d = 2 , L ˜ ( E ) $d=2,\, \widetilde{L}(\mathcal{E})$ is one of the three Hirzebruch surfaces Σ i , i = 0, 1, 2; we determine which occurs when and hence also the cases when L (ℰ) is affine. Separately, we prove that for d &lt; n /2 the singular locus of the secant slice Sec d , z ( E ) ⊂ ℙ n −1 , the portion of the d th secant variety of E consisting of points lying on spans of d -tuples with sum z ∈ E , is precisely Sec d −2 . This strengthens the result that L (ℰ) is smooth, appearing in prior joint work with R. Kanda and S. P. Smith.

Abstract We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for families $$f : \mathcal {X} \rightarrow \mathcal {S}$$ f : X → S with big monodromy and large period image defined over the ring of N -integers $$\mathcal {O}_{L}[1/N]$$ O L [ 1 / N ] of a number field L , we produce a proper closed subscheme $$\mathcal {E} \subsetneq \mathcal {S}$$ E ⊊ S outside of which all line bundles appearing in positive characteristic fibres of f admit characteristic zero lifts. This in particular applies to elliptic surfaces over $$\mathbb {P}^1$$ P 1 and projective hypersurfaces in $$\mathbb {P}^3$$ P 3 of degree $$d \ge 5$$ d ≥ 5 . We also study the locus $$\mathcal {E}$$ E in more detail in the $$h^{0, 2} = 2$$ h 0 , 2 = 2 case.

We construct a 2-parameter family of new triaxial SU(2)-invariant complete negative Einstein metrics on the complex line bundle over . The metrics are conformally compact and neither Kähler nor self-dual. The proof involves using rigorous numerics to produce an approximate Einstein metric to high precision in a bounded region containing the singular orbit or "bolt", which is then perturbed to a genuine Einstein metric using fixed-point methods. At the boundary of this region, the latter metric is sufficiently close to hyperbolic space for us to show that it indeed extends to a complete, asymptotically hyperbolic Einstein metric.

We present a geometric interpretation of the viscous Burgers equation in terms of an abelian logarithmic connection on a complex line bundle. The velocity field is shown to define the local component of a gauge potential, while the Cole-Hopf transformation corresponds to a logarithmic trivialization of the associated flat connection. In this framework, the nonlinear Burgers dynamics is reinterpreted as an evolution on the space of connections, whereas the Cole-Hopf transform induces a linear Schrödinger-type (heat) equation on sections of the bundle. We analyze the resulting gauge structure and show that the construction can be organized using concepts from prequantum geometry, without implying a genuine quantization scheme. The flatness of the connection and the absence of a non-degenerate symplectic form are discussed as intrinsic obstructions to full quantization. This viewpoint clarifies both the linearization mechanism of the Cole-Hopf transformation and its geometric limitations. In this paper, no new analytical results are claimed.

A line bundle is immaculate if its cohomology vanishes in every dimension. We give a criterion for when a smooth toric Deligne–Mumford stack has infinitely many immaculate line bundles. This answers positively a question of Borisov and Wang. As a byproduct, we describe the asymptotic behavior of the collection of immaculate line bundles.