Line bundle
View article: On real bisectional curvature for Hermitian manifolds
On real bisectional curvature for Hermitian manifolds Open
Motivated by the recent work of Wu and Yau on the ampleness of a canonical line bundle for projective manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called real bisectional curvature for Hermit…
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Machine learning line bundle cohomology Open
We investigate different approaches to machine learning of line bundle cohomology on complex surfaces as well as on Calabi-Yau three-folds. Standard function learning based on simple fully connected networks with logistic sigmoids is revie…
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Positivity of Line Bundles and Newton-Okounkov Bodies Open
The purpose of this paper is to describe asymptotic base loci of line bundles on projective varieties in terms of Newton-Okounkov bodies. As a result, we obtain equivalent characterizations of ampleness and nefness via convex geometry.
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Notes on the universal elliptic KZB connection Open
The universal elliptic KZB equation is the integrable connection on the\npro-vector bundle over M_{1,2} whose fiber over the point corresponding to the\nelliptic curve E and a non-zero point x of E is the unipotent completion of\n\\pi_1(E-…
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A tropical approach to nonarchimedean Arakelov geometry Open
Chambert-Loir and Ducros have recently introduced a theory of real valued\ndifferential forms and currents on Berkovich spaces. In analogy to the theory\nof forms with logarithmic singularities, we enlarge the space of differential\nforms …
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Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds Open
We introduce neural networks (NNs) to compute numerical Ricci-flat Calabi–Yau (CY) metrics for complete intersection and Kreuzer–Skarke (KS) CY manifolds at any point in Kähler and complex structure moduli space, and introduce the package …
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Remarks on Contact and Jacobi Geometry Open
We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., …
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The Verlinde formula for Higgs bundles Open
We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde formula for the case of $SU(n)$ proposed previ…
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Universal polynomials for tautological integrals on Hilbert schemes Open
We show that tautological integrals on Hilbert schemes of points can be written in terms of universal polynomials in Chern numbers. The results hold in all dimensions, though they strengthen known results even for surfaces by allowing inte…
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Machine learning and algebraic approaches towards complete matter spectra in 4d F-theory Open
Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d F-theory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To qu…
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Bundle geometry of the connection space, covariant Hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method Open
A bstract We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satis…
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On the cohomology of the classifying spaces of projective unitary groups Open
Let [Formula: see text] be the classifying space of [Formula: see text], the projective unitary group of order [Formula: see text], for [Formula: see text]. We use a Serre spectral sequence to determine the ring structure of [Formula: see …
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Injectivity theorem for pseudo-effective line bundles and its applications Open
We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness, Kollár’s vanishing theorem, and a gener…
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Formality conjecture for K3 surfaces Open
We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra RHom*(F; F) is formal for any sheaf F polystable with respect to an ample line bundle. Our main tool …
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Higgs bundles for real groups and the Hitchin–Kostant–Rallis section Open
We consider the moduli space of polystable -twisted -Higgs bundles over a compact Riemann surface , where is a real reductive Lie group and is a holomorphic line bundle over . Evaluating the Higgs field on a basis of the ring of polynomial…
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An $\varepsilon$-regularity theorem for line bundle mean curvature flow Open
In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given K\"ahler manifold. The goal o…
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TANGENT BUNDLE ENDOWED WITH QUARTER-SYMMETRIC NON-METRIC CONNECTION ON AN ALMOST HERMITIAN MANIFOLD Open
In this paper, we have studied the tangent bundle endowed with quarter-symmetric non-metric connection obtained by vertical and complete lifts of a quarter-symmetric non-metric connection on the base manifold and, also, proposed the study …
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Effectivity in the Hyperbolicity-related problems Open
The aim of this work is to deal with effective questions related to the Kobayashi and Debarre conjectures, and based on the work of Damian Brotbek and Lionel Darondeau. We first show that if a line bundle $L$ generates $k$-jets, the $k$-th…
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Bochner Laplacian and Bergman kernel expansion of semipositive line bundles on a Riemann surface Open
We generalize the results of Montgomery (Commun Math Phys 168:651–675, 1995) for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion for semipositiv…
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Nef Line Bundles on Calabi–Yau Threefolds, I Open
We prove that a nef line bundle ${\mathcal{L}}$ with $c_1({\mathcal{L}})^2 \ne 0$ on a Calabi–Yau threefold $X$ with Picard number $2$ and with $c_3(X) \ne 0$ is semiample, that is, some multiple of $\mathcal L$ is generated by global sect…
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Geometry and stability of tautological bundles on Hilbert schemes of points Open
We explore the geometry and establish the slope-stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general, we complete a series of results of Schlickewei and Wandel, who …
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Quantisation of derived Lagrangians Open
We investigate quantisations of line bundles $\mathcal{L}$ on derived Lagrangians $X$ over $0$-shifted symplectic derived Artin $N$-stacks $Y$. In our derived setting, a deformation quantisation consists of a curved $A_{\infty}$ deformatio…
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Szegő kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds Open
Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in\set{0,1,\ldots,n-1}$, let $\Box^{(q)}_…
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Pluripotential theory and convex bodies Open
In their seminal paper, Berman and Boucksom exploited ideas from complex geometry to analyze asymptotics of spaces of holomorphic sections of tensor powers of certain line bundles $L$ over compact, complex manifolds as the power grows. Thi…
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Liberté et accumulation Open
The principle of Batyrev and Manin and its variants gives a precise conjectural interpretation for the dominant term for the number of points of bounded height on an algebraic variety for which the opposite of the canonical line bundle is …
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Remarks on motives of moduli spaces of rank 2 vector bundles on curves Open
Let $C$ be an algebraic curve of genus $g \geq 2$ and $M_L$ be the moduli space of rank 2 stable vector bundles on $C$ whose determinants are isomorphic to a fixed line bundle $L$ of degree 1 on $C.$ S. del Bano studied motives of moduli s…
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Calabi-Yau manifolds with isolated conical singularities Open
Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau …
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Variation of canonical height and equidistribution Open
Let $π: E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that th…
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Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions Open
Let M be a connected d-dimensional complex projective manifold, and let A be a holomorphic positive Hermitian line bundle on M, with normalized curvature. Let G be a compact and connected Lie group of dimension d(G), and let T be a compact…
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mpleness of the CM line bundle on the moduli space of canonically polarized varieties Open
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