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Register a new userTotally real theta characteristics
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Authors
Mario Kummer
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No Arabic abstract
A totally real theta characteristic of a real curve is a theta characteristic which is linearly equivalent to a sum of only real points. These are closely related to the facets of the convex hull of the canonical embedding of the curve.
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Using the theory of totally real number fields we construct a new class of compact complex non-K{a}hler manifolds in every even complex dimension and study their analytic and geometric properties.
A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincare showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.
We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequalities involving the Laplacian of the warping function and the squared mean curvature of $C$-totally real warped product submanifolds of $(kappa ,mu ) $-space forms, Sasakian space forms and non-Sasakian $(kappa ,mu) $-manifolds. Then we obtain obstructions to the existence of minimal isometric immersions of $C$-totally real warped product submanifolds in $(kappa ,mu) $-space forms, non-Sasakian $(kappa ,mu) $-manifolds and Sasakian space forms. In the last, we obtain an example of a warped product $C$-totally real submanifold of a non-Sasakian $(kappa ,mu) $-manifold, which satisfies the equality case of the basic inequality.
We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed orientable (4k+1)-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4k-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.
In this article, we construct a $theta$-density for the global sections of ample Hermitian line bundles on a projective arithmetic variety. We show that this density has similar behaviour to the usual density in the Arakelov geometric setting, where only global sections of norm smaller than $1$ are considered. In particular, we prove the analogue by $theta$-density of two Bertini kind theorems, on irreducibility and regularity respectively.
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