arXiv subjects

Carlos Florentino

Publications and source records attributed to Carlos Florentino.

29 records · Page 2Linked to original sources

arXiv2012

Unipotent Schottky bundles on Riemann surfaces and complex tori

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky.

math.AG↗

arXiv2012

Singularities of free group character varieties

Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating algebraic singularities to topological singularities, we show the moduli spaces X generally are not topological manifolds, except for a few examples we explicitly describe.

math.AG↗

arXiv2012

Polygons in Minkowski three space and parabolic Higgs bundles of rank two on CP^1

Consider the moduli space of parabolic Higgs bundles (E,Φ) of rank two on CP^1 such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by (E,Φ)\mapsto (E,-Φ). We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.

math.AG↗

arXiv2012

Stability of Affine G-varieties and Irreducibility in Reductive Groups

Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-Röhrle criteria, in the case $X=G^{N}$. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions.

math.RT↗

arXiv2011

Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas

We consider the metric space of all toric Kähler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of metrics on the toric variety, its quantization, and degeneration of generic divisors. The limits of the corresponding Kähler polarizations become degenerate along the Lagrangian fibration defined by the moment map. This allows us to interpolate continuously between geometric quantizations in the holomorphic and real polarizations and show that the monomial holomorphic sections of the prequantum bundle converge to Dirac delta distributions supported on Bohr-Sommerfeld fibers. In the second part, we use these families of toric metric degenerations to study the limit of compact hypersurface amoebas and show that in Legendre transformed variables they are described by tropical amoebas. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry.

math.DG↗

arXiv2011

Quantization of some moduli spaces of parabolic vector bundles on CP^1

We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over $\mathbf{CP}^1$ with four parabolic points and parabolic weights in {0,1/2}. Identifying such parabolic bundles as vector bundles on an elliptic curve, we obtain explicit expressions for the corresponding non-abelian theta functions. These non-abelian theta functions are described in terms of certain naturally defined distributions on the compact group SU(2).

math.AG↗

arXiv2011

A remark on "Connections and Higgs fields on a principal bundle"

We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.

math.DG↗

arXiv2009

The topology of moduli spaces of free group representations

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the SL(3,C)-character variety of a rank 2 free group is homotopic to an 8 sphere and the SL(2,C)-character variety of a rank 3 free group is homotopic to a 6 sphere.

math.GN↗

arXiv2006

On the BKS pairing for Kahler quantizations of the cotangent bundle of a Lie group

A natural one-parameter family of Kähler quantizations of the cotangent bundle $T^*K$ of a compact Lie group $K$, taking into account the half-form correction, was studied in \cite{FMMN}. In the present paper, it is shown that the associated Blattner-Kostant-Sternberg (BKS) pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work, from the point of view of \cite{AdPW}. The BKS pairing map is a composition of (unitary) coherent state transforms of $K$, introduced in \cite{Ha1}. Continuity of the Hermitian structure on the quantum bundle, in the limit when one of the Kähler polarizations degenerates to the vertical real polarization, leads to the unitarity of the corresponding BKS pairing map. This is in agreement with the unitarity up to scaling (with respect to a rescaled inner product) of this pairing map, established by Hall.

math.DG↗

arXiv2004

Geometric quantization, complex structures and the coherent state transform

It is shown that the heat operator in the Hall coherent state transform for a compact Lie group $K$ is related with a Hermitian connection associated to a natural one-parameter family of complex structures on $T^*K$. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizations of $T^*K$ for different choices of complex structures within the given family. In particular, these results establish a link between coherent state transforms for Lie groups and results of Hitchin and Axelrod, Della Pietra and Witten.

math.DG↗

arXiv2001

Schottky uniformization and vector bundles over Riemann surfaces

We study a natural map from representations of a free group of rank g in GL(n,C), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient condition for this map to be a submersion, when restricted to representations producing stable bundles. Using a generalized version of Riemann's bilinear relations, this condition is shown to be true on the subspace of unitary Schottky representations.

math.DG↗