## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Tuesday, July 21, 2015 — 1:00 PM EDT

**Alejandra Vicente Colmenares, Pure Mathematics, University of Waterloo**

*"Semistable rank 2 co-Higgs bundles over Hirzebruch surfaces"*

It has been observed by S. Rayan that the complex projective surfaces that potentially admit non-trivial examples of semistable co-Higgs bundles must be found at the lower end of the Enriques-Kodaira classification. Motivated by this remark, we study the geometry of these objects (in the rank 2 case) over Hirzebruch surfaces, giving special emphasis to $\P \times \P$. Two main topics can be identified throughout the dissertation: non-emptiness of the moduli spaces of rank 2 semistable co-Higgs bundles over Hirzebruch surfaces, and the description of these moduli spaces over $\P \times \P$.

The existence problem consists in determining for which pairs of Chern classes $(c_1,c_2)$ there exists a non-trivial semistable rank 2 co-Higgs bundle with Chern classes $c_1$ and $c_2$. We approach this problem from two different perspectives. On one hand, we restrict ourselves to certain natural choices of $c_1$ and give necessary and sufficient conditions on $c_2$ that guarantee the existence of non-trivial semistable co-Higgs bundles with these Chern classes; we do this for arbitrary polarizations when $c_2 \leq 2$. On the other hand, for arbitrary $c_1$, we also provide necessary and sufficient conditions on $c_2$ that ensure the existence of non-trivial semistable co-Higgs bundles; however, we only do this for the standard polarization.

As for the description of the moduli spaces $\MPP(c_1,c_2)$ of rank 2 semistable co-Higgs bundles over $\P \times \P$, we restrict ourselves to the standard polarization. We then discuss how to use the spectral construction and the Hitchin correspondence to understand generic rank 2 semistable co-Higgs bundles. Furthermore, we give an explicit description of the moduli spaces when $c_2=0, 1$ for certain choices of $c_1$. Finally, we explore the first order deformations of points in the moduli space $\MPP(c_1,c_2)$.

Location

MC 5479

,

Canada

,

Canada

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Indigenous Initiatives Office.