V. Balaji

Bio: V. Balaji is an academic researcher from Chennai Mathematical Institute. The author has contributed to research in topics: Moduli space & Vector bundle. The author has an hindex of 15, co-authored 36 publications receiving 491 citations. Previous affiliations of V. Balaji include University of Liverpool.

Principal bundles over projective manifolds with parabolic structure over a divisor

Abstract: Principal $G$-bundles with parabolic structure over a normal crossing divisor are defined along the line of the interpretation of the usual principal $G$-bundles as functors from the category of representations, of the structure group $G$, into the category of vector bundles, satisfying certain axioms. Various results on principal bundles are extended to the more general context of principal bundles with parabolic structures, and also to parabolic $G$-bundles with Higgs structure. A simple construction of the moduli space of parabolic semistable $G$-bundles over a curve is given, where $G$ is a semisimple linear algebraic group over $C$.

Holonomy Groups of Stable Vector Bundles

Abstract: We define the notion of holonomy group for a stable vector bundle F on a variety in terms of the Narasimhan‐Seshadri unitary representation of its restriction to curves. Next we relate the holonomy group to the minimal structure group and to the decomposition of tensor powers of F. Finally we illustrate the principle that either the holonomy is large or there is a clear geometric reason why it should be small.

Holonomy groups of stable vector bundles

Abstract: We define the notion of holonomy group for a stable vector bundle F on a variety in terms of the Narasimhan--Seshadri unitary representation of its restriction to curves. Next we relate the holonomy group to the minimal structure group and to the decomposition of tensor powers of F. Finally we illustrate the principle that either the holonomy is large or there is a clear geometric reason why it should be small.

Stability of the Poincare Bundle

Abstract: Let C be a nonsingular projective curve of genus g ≥ 2 defined over the complex numbers, and let M ξ denote the moduli space of stable bundles of rank n and determinant ξ on C, where ξ is a line bundle of degree don C and n and d are coprime. It is shown that a universal bundle U ξ on C × M ξ is stable with respect to any polarisation on C × M ξ . Similar results are obtained for the case where the determinant is not fixed and for the bundles associated to the universal bundles by irreducible representations of GL(n, C). It is shown further that the connected component of the moduli space of bundles with the same Hilbert polynomial as U ξ on C × M ξ containing U ξ is isomorphic to the Jacobian of C.

Moduli of representations of the fundamental group of a smooth projective variety, II . Finite group actions and étale cohomology . A rank theorem for analytic maps between power series spaces . Some groups whose reduced C[*]-algebra is simple . Existence de 1-formes fermées non singulières dans une classe de cohomologie de de Rahm

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Wall-Crossing in Coupled 2d-4d Systems

Abstract: We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N = 2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactied on a circle, we get a 3d theory with a su- persymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of classS, that is, for those theories ob- tained by compactifying the six-dimensional (0; 2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A1 theories of classS. Finally, we indi- cate how our results can be used to produce solutions to the A1 Hitchin equations on the Riemann surface C.

Algebraic integrability of foliations with numerically trivial canonical bundle

Abstract: Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.

On the classification of rank-two representations of quasiprojective fundamental groups

Abstract: Suppose $X$ is a smooth quasiprojective variety over $\cc$ and $\rho : \pi _1(X,x) \rightarrow SL(2,\cc )$ is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then $\rho$ factors through a map $X\rightarrow Y$ with $Y$ either a DM-curve or a Shimura modular stack.