Normal functions and the height of Gross-Schoen cycles

AbstractWe prove a variant of a formula due to Zhang relating the Beilinson– Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

Download Full-text

Socle pairings on tautological rings

Épijournal de Géométrie Algébrique ◽

10.46298/epiga.2019.volume3.3784 ◽

2019 ◽

Vol Volume 3 ◽

Author(s):

Felix Janda ◽

Aaron Pixton

Keyword(s):

Explicit Formula ◽

Moduli Space ◽

Compact Type ◽

Stable Curves ◽

Tautological Ring ◽

Tautological Rings

We study some aspects of the $\lambda_g$ pairing on the tautological ring of $M_g^c$, the moduli space of genus $g$ stable curves of compact type. We consider pairing kappa classes with pure boundary strata, all tautological classes supported on the boundary, or the full tautological ring. We prove that the rank of this restricted pairing is equal in the first two cases and has an explicit formula in terms of partitions, while in the last case the rank increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on the tautological ring of $M_g$. Comment: 18 pages, 1 figure; v3: journal version; v2: minor revisions to sections 1.1 and 4.1, results unchanged

Download Full-text

Explicit soft supersymmetry breaking in the heterotic M-theory B − L MSSM

Journal of High Energy Physics ◽

10.1007/jhep08(2021)033 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Anthony Ashmore ◽

Sebastian Dumitru ◽

Burt A. Ovrut

Keyword(s):

Line Bundle ◽

Supersymmetry Breaking ◽

Moduli Space ◽

Initial Conditions ◽

Low Energy ◽

Strongly Coupled ◽

Hidden Sector ◽

Effective Lagrangian ◽

Soft Supersymmetry Breaking ◽

M Theory

Abstract The strongly coupled heterotic M-theory vacuum for both the observable and hidden sectors of the B − L MSSM theory is reviewed, including a discussion of the “bundle” constraints that both the observable sector SU(4) vector bundle and the hidden sector bundle induced from a single line bundle must satisfy. Gaugino condensation is then introduced within this context, and the hidden sector bundles that exhibit gaugino condensation are presented. The condensation scale is computed, singling out one line bundle whose associated condensation scale is low enough to be compatible with the energy scales available at the LHC. The corresponding region of Kähler moduli space where all bundle constraints are satisfied is presented. The generic form of the moduli dependent F-terms due to a gaugino superpotential — which spontaneously break N = 1 supersymmetry in this sector — is presented and then given explicitly for the unique line bundle associated with the low condensation scale. The moduli-dependent coefficients for each of the gaugino and scalar field soft supersymmetry breaking terms are computed leading to a low-energy effective Lagrangian for the observable sector matter fields. We then show that at a large number of points in Kähler moduli space that satisfy all “bundle” constraints, these coefficients are initial conditions for the renormalization group equations which, at low energy, lead to completely realistic physics satisfying all phenomenological constraints. Finally, we show that a substantial number of these initial points also satisfy a final constraint arising from the quadratic Higgs-Higgs conjugate soft supersymmetry breaking term.

Download Full-text

A pathological case of the C1 conjecture in mixed characteristic

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004118000178 ◽

2018 ◽

Vol 167 (01) ◽

pp. 61-64 ◽

Cited By ~ 1

Author(s):

INDER KAUR

Keyword(s):

Line Bundle ◽

Moduli Space ◽

Projective Curve ◽

Free Sheaf ◽

Smooth Projective Curve ◽

Mixed Characteristic ◽

Locally Free Sheaf ◽

Fixed Rank ◽

Pathological Case

AbstractLet K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

Download Full-text

Bounded properties of curvatures of perturbed Ricci metric on curve moduli

International Journal of Mathematics ◽

10.1142/s0129167x14501134 ◽

2014 ◽

Vol 25 (12) ◽

pp. 1450113

Author(s):

Xiaorui Zhu

Keyword(s):

Finite Volume ◽

Ricci Curvature ◽

Moduli Space ◽

Point Of View ◽

Bisectional Curvature ◽

Bounded Geometry ◽

Holomorphic Bisectional Curvature ◽

Chern Form ◽

Thick Part ◽

Kahler Metric

As is well-known, the Weil–Petersson metric ωWP on the moduli space ℳg has negative Ricci curvature. Hence, its negative first Chern form defines the so-called Ricci metric ωτ. Their combination [Formula: see text], C > 0, introduced by Liu–Sun–Yau, is called the perturbed Ricci metric. It is a complete Kähler metric with finite volume. Furthermore, it has bounded geometry. In this paper, we investigate the finiteness of this new metric from another point of view. More precisely, we will prove in the thick part of ℳg, the holomorphic bisectional curvature of [Formula: see text] is bounded by a constant depending only on the thick constant and C0 when C ≥ (3g - 3)C0, but not on the genus g.

Download Full-text