Research

Picture taken in Laugarnes, Reykjavik, Iceland

My research area is in Several Complex variables and Complex Geometry. 

I am interested in L^2 estimates of the dbar-operator on pseudoconvex domains in complex manifolds. My interests are both in sufficient conditions for their existence and in their applications, related to the Diederich-Fornaess exponent, the tangential Cauchy-Riemann equation, as well as the study of Levi-flat hypersurfaces.

My research is connected to complex dynamics, CR geometry, complex geometry and pluripotential theory.

My publications:

• with J Brinkschulte and M. Adachi, A residue formal for meromorphic connections and applications to stable sets of foliations. Journal of Geometric Analysis, 338 (2023).

• with M. Adachi, On Levi-flat hypersurfaces with transversely affine foliation. Mathematische Zeitschrift. 301(2022) no.1. DOI:10.1007/s00209-021-02927-z

• S. Biard, Symmetry and Interpolation of the estimates for the complex Green operator, Proceedings of RIMS, Kyoto, Japan (2021).

• S. Biard and J.E. Fornaess and J. Wu, Weighted-L^2 version of Mergelyan and Carleman approximation, Journal of Geometric Analysis (2020).

• S. Biard and A. Iordan, Nonexistence of Levi flat hypersurfaces with positive normal bundle in compact Kahler manifolds of dimension ≥ 3, International Journal of Mathematics, Vol. 31 (2020), no.01.DOI: 10.1142/S0129167X20500044

• S. Biard and J.E Fornaess and J. Wu, Weighted-L2 polynomial approximation in C. Transactions of AMS, 373 (2), (2020), 919-938. 

• S. Biard and E. Straube, Estimates for the complex Green operator: Symmetry, percolation and Interpolation, Transactions of AMS, Vol. 371 (2019).

• S. Biard and E. Straube, L2-Sobolev Theory for the complex Green Operator, International Journal of Mathematics, Vol. 28, N◦9 (2017).

• S. Biard, On L2 estimates for ∂ ̄ on pseudoconvex domains in complete Kähler manifolds with positive holomorphic bisectional curvature, The Journal of Geometric Analysis, Volume 24, Issue 3, p.1583-1612 (2014).

•  S. Biard, On L2 estimates for ∂ ̄ on pseudoconvex domains in complete Kähler mani- folds with positive holomorphic bisectional curvature, Comptes Rendus de l’Académie des Sciences Mathématiques, vol. 350, fascicule 11-12, juin 2012

Submitted Papers:

• S.Biard, J Brinkschulte and M. Adachi, A residue formal for meromorphic connections and applications to stable sets of foliations. Submitted. https://arxiv.org/abs/2210.09273

Ongoing Collaborations with:

• John Erik Fornaess and Jujie Wu

• Masanori Adachi

• Ragnar Sigurdsson, Benedikt Magnusson and Tyson Ritter

Service:

• Reviewer for Mathematical Reviews.