Research
My research is in the areas of differential geometry and algebraic geometry and has strong links with geometric analysis and mathematical physics. The main projects I have carried out include special metrics and connections in complex geometry (Kähler and non-Kähler), spin geometry, and the theory of moduli spaces in algebraic geometry.
My current main research interests are complex non-Kähler geometry, special holonomy, and mirror symmetry. In my work I transfer techniques from Kähler geometry to address problems in complex non-Kähler geometry, using Courant algebroids and other objects reminiscent from higher gauge theory. I am also interested in analogue problems in dimensions 7 and 8, in relation to G2 and Spin(7) geometry. My research builds towards providing mathematical underpinnings for a conjectural generalization of mirror symmetry, known as (0,2)-mirror symmetry, using special holonomy with torsion and the theory of vertex algebras. In particular, in the last 8 years I have extensively studied the Hull-Strominger system of partial differential equations, which provides a common ground for these topics.
Publications
Book
M. Garcia-Fernandez, J. Streets, Generalized Ricci Flow, University Lecture Series, Volume 76, 2021, 248 pp, American Mathematical Society.
Papers
25) M. Garcia-Fernandez, R. Rubio, C. Tipler, Gauge theory for string algebroids, to appear in Journal of Differential Geometry, arXiv:2004.11399.
24) M. Garcia-Fernandez, R. Gonzalez Molina, Harmonic metrics for the Hull-Strominger system and stability, to appear in International Journal of Mathematics (special issue in honour of Oscar Garcia-Prada) , arXiv:2301.08236.
23) L. Álvarez-Cónsul, A. De Arriba, M. Garcia-Fernandez, (0,2) Mirror symmetry on homogeneous Hopf surfaces, International Mathematics Research Notices, Volume 2024, Issue 2 (2024) 1211–1298, https://doi.org/10.1093/imrn/rnad016.
22) M. Garcia-Fernandez, J. Jordan, J. Streets, Non-Kähler Calabi-Yau geometry and pluriclosed flow, J. Math. Pure Appl. 177 (2023) 329-367.
21) L. Álvarez-Cónsul, M. Garcia-Fernandez, O. García-Prada, V. Pingali, C. Yao, Obstructions to the existence of solutions of the self-dual Einstein-Maxwell-Higgs equations on a compact surface, Bulletin des Sciences Mathématiques 183 (3) (2023) 103233 .
20) M. Garcia-Fernandez, R. Rubio, C. Shahbazi, C. Tipler, Canonical metrics on holomorphic Courant algebroids, Proc. London Math. Soc. 125 (3) (2022) 700-758.
19) A. Clarke, M. Garcia-Fernandez, C. Tipler, T-Dual solutions and infinitesimal moduli of the G2-Strominger system, Advances in Theoretical and Mathematical Physics 26 (6) (2022) 1669-1704.
18) M. Garcia-Fernandez, V. Pingali, C. Yao, Gravitating vortices with positive curvature, Advances in Mathematics 388 (2021) 107851.
17) L. Álvarez-Cónsul, M. Garcia-Fernandez, O. García-Prada, V. Pingali, Gravitating vortices and the Einstein-Bogomol'nyi equations, Mathematische Annalen 379 (2021) 1651-1684.
16) M. Garcia-Fernandez, R. Rubio, C. Tipler, Holomorphic string algebroids, Trans. Amer. Math. Soc. 373 (2020) 7347-7382.
15) M. Garcia-Fernandez, T-dual solutions of the Hull-Strominger system on non-Kähler threefolds, J. reine angew. Math. 766 (2020) 137--150.
14) L. Álvarez-Cónsul, M. Garcia-Fernandez, O. Garcia-Prada, On the Kähler-Yang-Mills-Higgs equations, Pure and Applied Mathematics Quarterly 15 (2019) 1181–1217, special issue in honor of Prof. Donaldson's 60th birthday, Part 2 of 2.
13) M. Garcia-Fernandez, Ricci flow, Killing spinors, and T-duality in generalized geometry, Adv. Math. 350 (2019) 1059-1108.
12) M. Garcia-Fernandez, R, Rubio, C. Tipler, Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry, Mathematische Annalen 369 (2017) 539-595.
11) S. A. Filippini, M. Garcia-Fernandez, J. Stoppa, Stability data, irregular connections and tropical curves, Selecta Mathematica New Ser. 23 (2) (2017) 1355-1418.
10) L. Álvarez-Cónsul, M. Garcia-Fernandez, O. Garcia-Prada, Gravitating vortices, cosmic strings and the Kähler-Yang-Mills equations, Comm. Math. Phys. 351 (1) (2017) 361-385.
9) M. Garcia-Fernandez, J. Ross, Balanced metrics on twisted Higgs bundles, Mathematische Annalen 367 (3) (2017) 1429-1471.
8) M. Garcia-Fernandez, Lectures on the Strominger system, Travaux Mathématiques, Special Issue: School GEOQUANT at the ICMAT, Vol. XXIV (2016) 7-61, arXiv:1609.02615.
7) B. Andreas, M. Garcia-Fernandez, Note on Solutions of the Strominger System from Unitary Representations of cocompact lattices of SL(2;C), Comm. Math. Phys. 332 (2014) 1381-1383.
6) M. Garcia-Fernandez, Torsion-free generalized connections and Heterotic Supergravity, Comm. Math. Phys. 332 (2014) 89-115.
5) M. Garcia-Fernandez, C. Tipler, Deformation of complex structures and the Coupled Kähler–Yang–Mills equations, J. London Math. Soc. 89 (2014) 779-796.
4) M. Garcia-Fernandez, J. Ross, Balanced metrics on vector bundles and polarised manifolds, Proc. London Math. Soc. 106 (2013) 1143--1156.
3) L. Álvarez-Cónsul, M. Garcia-Fernandez, O. Garcia-Prada, Coupled equations for Kähler metrics and Yang–Mills connections, Geometry and Topology 17 (2013) 2731-2812.
2) B. Andreas, M. Garcia-Fernandez, Heterotic Non-Kähler Geometries via polystable bundles on Calabi-Yau threefolds, J. Geom. Phys. 62 (2012) 183-188.
1) B. Andreas, M. Garcia-Fernandez, Solutions of the Strominger System via stable bundles on Calabi-Yau threefolds, Comm. Math. Phys. (1) 315 (2012) 153-168.
Preprints
• A. A. da Silva Jr., M. Garcia-Fernandez, J. D. Lotay, H. N. Sá Earp, Coupled G_2 Instantons, arXiv:2404.12937 (2024).
• L. Álvarez-Cónsul, A. De Arriba, M. Garcia-Fernandez, Vertex algebras from the Hull-Strominger system, arXiv:2305.06836 (2023).
• M. Garcia-Fernandez, R. Gonzalez Molina, Futaki Invariants and Yau's Conjecture on the Hull-Strominger system, arXiv:2303.05274 (2023).
• M. Garcia-Fernandez, J. Keller, J. Ross, Quantization of Hitchin's equations for Higgs bundles I, arXiv:1601.04960 (2016).
• M. Garcia-Fernandez, C. Shahbazi, Self-dual generalized metrics for pure N=1 six-dimensional Supergravity, arXiv:1505.03088 (2015).