arXiv subjects

Sébastien Picard

Publications and source records attributed to Sébastien Picard.

7 recordsLinked to original sources

arXiv2026

Balanced and Aeppli Parameters for the Heterotic Moduli

In this paper, we fix the complex structure and explore the moduli space of the heterotic system by considering two different yet "dual" deformation paths starting from a Kähler solution. They correspond to deformation along the Bott-Chern cohomology class and the Aeppli cohomology class respectively. Using the implicit function theorem, we prove the local existence of heterotic solutions along these two paths and hence establish an initial step to construct local moduli coordinates around a Kähler solution.

math.DG↗

arXiv2026

Special Lagrangian submanifolds and circle collapse on K3

We consider $K3$ surfaces collapsing to a three-dimensional affine base. We show that certain affine lines on the base lift to degenerating sequences of special Lagrangian two-spheres and tori in the collapsing $K3$ surface. In particular, we construct special Lagrangian two-spheres connecting pairs of Taub-NUT bubbles. These examples fit into the broader program of reconstructing special submanifolds from graphs and combinatorial data on a collapsed affine limit.

math.DG↗

arXiv2026

Flows of conformally coclosed $G_2$-structures with dilaton

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the Kähler--Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures. We compare the $G_2$-anomaly flow to the $G_2$-Laplacian coflow, and investigate short-time existence and fixed points.

math.DG↗

arXiv2025

Gromov-Hausdorff continuity of non-Kähler Calabi-Yau conifold transitions

We study the geometry of Calabi-Yau conifold transitions. This deformation process is known to possibly connect a Kähler threefold to a non-Kähler threefold. We use balanced and Hermitian-Yang-Mills metrics to geometrize the conifold transition and show that the whole operation is continuous in the Gromov-Hausdorff topology.

math.DG↗

arXiv2025

The Strominger System and Flows by the Ricci Tensor

This is a survey on the Strominger system and a geometric flow known as the anomaly flow. We will discuss various aspects of non-Kähler geometry on Calabi-Yau threefolds. Along the way, we discuss balanced metrics and balanced classes, the Aeppli cohomology class associated to a solution to the Strominger system, the equations of motion of heterotic supergravity, and a version of Ricci flow in this special geometry.

math.DG↗

arXiv2025

Calabi-Yau threefolds across quadratic singularities

These are lecture notes on non-Kähler complex threefolds presented at the MATRIX program ``The geometry of moduli spaces in string theory''. We review some basics of Calabi-Yau geometry in Section 1, describe topological features of the conifold transition in Section 2, and survey recent developments on the geometrization of conifold transitions in Section 3.

math.DG↗

arXiv2023

Flows of $G_2$-Structures associated to Calabi-Yau Manifolds

We establish a correspondence between a parabolic complex Monge-Ampère equation and the $G_2$-Laplacian flow for initial data produced from a Kähler metric on a complex $2$- or $3$-fold. By applying estimate for the complex Monge-Ampère equation, we show that for this class of initial data the $G_2$-Laplacian flow exists for all time and converges to a torsion-free $G_2$-structure induced by a Kähler Ricci-flat metric. Similar results are obtained for the $G_2$-Laplacian coflow, and in this case the coflow is related to the Kähler-Ricci flow.

math.DG↗