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Indranil Biswas

Publications and source records attributed to Indranil Biswas.

At least 361 records · Page 20Linked to original sources
arXiv2014

On the essential dimension of coherent sheaves

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curve C, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Thélène, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne-Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curve C is elliptic.

math.AG↗
arXiv2014

Geometric prequantization on the path space of a prequantized manifold

Given a compact symplectic manifold $M$, with integral symplectic form, we prequantize a certain class of functions on the path space for $M$. The functions in question are induced by functions on $M$. We apply our construction to study the symplectic structure on the solution space of Klein-Gordon equation.

math.DG↗
arXiv2014

Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles

We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise, Integrable systems and algebraic surfaces, Duke Math. Jour. 83 (1996), 19--49.

math.AG↗
arXiv2014

Yang-Mills connections on compact complex tori

Let $G$ be a connected reductive complex affine algebraic group and $K\subset G$ a maximal compact subgroup. Let $M$ be a compact complex torus equipped with a flat Kähler structure and $(E_G ,θ)$ a polystable Higgs $G$-bundle on $M$. Take any $C^\infty$ reduction of structure group $E_K \subset E_G$ to the subgroup $K$ that solves the Yang--Mills equation for $(E_G ,θ)$. We prove that the principal $G$-bundle $E_G$ is polystable and the above reduction $E_K$ solves the Einstein--Hermitian equation for $E_G$. We also prove that for a semistable (respectively, polystable) Higgs $G$-bundle $(E_G , θ)$ on a compact connected Calabi--Yau manifold, the underlying principal $G$-bundle $E_G$ is semistable (respectively, polystable).

math.DG↗
arXiv2014

On the Gieseker Harder-Narasimhan filtration for principal bundles

We give an example of an orthogonal bundle where the Harder-Narasimhan filtration, with respect to Gieseker semistability, of its underlying vector bundle does not correspond to any parabolic reduction of the orthogonal bundle. A similar example is given for the symplectic case.

math.AG↗
arXiv2014

Anti-holomorphic involutive isometry of hyper-Kähler manifolds and branes

We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is complex Lagrangian with respect to one of the other complex structures, and (b) there exist Lagrangian submanifolds which are complex with respect to one complex structure and are not the fixed point locus of any involution which is anti-holomorphic with respect to one of the other complex structures.

math.DG↗
arXiv2014

Vassiliev Invariants from Symmetric Spaces

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

math.GT↗
arXiv2014

Fundamental group of a geometric invariant theoretic quotient

Let $M$ be an irreducible smooth projective variety, defined over an algebraically closed field, equipped with an action of a connected reductive affine algebraic group $G$, and let ${\mathcal L}$ be a $G$--equivariant very ample line bundle on $M$. Assume that the GIT quotient $M/\!\!/G$ is a nonempty set. We prove that the homomorphism of algebraic fundamental groups $π_1(M)\, \longrightarrow\, π_1(M/\!\!/G)$, induced by the rational map $M\, \longrightarrow\, M/\!\!/G$, is an isomorphism. If $k\,=\, \mathbb C$, then we show that the above rational map $M\, \longrightarrow \, M/\!\!/G$ induces an isomorphism between the topological fundamental groups.

math.AG↗
arXiv2014

A generalized Quot scheme and meromorphic vortices

Let X be a compact connected Riemann surface. Fix a positive integer r and two nonnegative integers d_p and d_z. Consider all pairs of the form (F, f), where F is a holomorphic vector bundle on X of rank r and degree d_z-d_p, and f : {\mathcal O}^{\oplus r}_X \rightarrow F is a meromorphic homomorphism which an isomorphism outside a finite subset of X and has pole (respectively, zero) of total degree d_p (respectively, d_z). Two such pairs $(F_1, f_1) and $(F_2, f_2) are called isomorphic if there is a holomorphic isomorphism of F_1 with F_2 over X that takes f_1 to f_2. We construct a natural compactification of the moduli space equivalence classes pairs of the above type. The Poincaré polynomial of this compactification is computed.

math.AG↗
arXiv2014

Poisson surfaces and algebraically completely integrable systems

One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there is one case of a ruled surface that does not seem to be covered. This is the case of projective bundle associated to the first jet bundle of a topologically nontrivial line bundle. We give the integrable system corresponding to this surface; it turns out to be a deformation of the Hitchin system.

math.AG↗
arXiv2014

Realization of an equivariant holomorphic Hermitian line bundle as a Quillen determinant bundle

Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$--equivariant holomorphic Hermitian line bundle $({\mathcal L}\,,h)$. This proves a conjecture of Dey and Mathai in \cite{DM}.

math.DG↗
arXiv2014

On the Zariski-Lipman conjecture for normal algebraic surfaces

We consider the Zariski-Lipman Conjecture on free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several classes of affine and projective surfaces.

math.AG↗
arXiv2014

H_1-semistability for projective groups

We initiate the study of the asymptotic topology of groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the $H_1$-semistability conjecture of Geoghegan for holomorphically convex groups. In view of a theorem of Eyssidieux, Katzarkov, Pantev and Ramachandran \cite{ekpr}, this implies that linear projective groups satisfy the $H_1$-semistability conjecture.

math.GT↗
arXiv2014

Low dimensional projective groups

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of cohomological dimension two, we show that $G$ is isomorphic to the fundamental group of a compact Riemann surface. As a consequence, we show that if a linear group $G$ has (rational) cohomological dimension two and is the fundamental group of a smooth complex projective variety, then $G$ is a (virtual) surface group.

math.GT↗
arXiv2014

Semistability of Principal Bundles on a Kähler Manifold with a Non-Connected Structure Group

We investigate principal $G$-bundles on a compact Kähler manifold, where $G$ is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal $G$-bundle $E_G$ admits an Einstein-Hermitian connection if and only if $E_G$ is polystable. We give an equivalent formulation of the (semi)stability condition. A question is to compare this definition with that of math.AG/0506511.

math.DG↗
arXiv2014

On the Kähler structures over Quot schemes

Let $S^n(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^n(X)$ admits a Kähler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n < d$. Let ${\mathcal Q}_X(r,n)$ be the Quot scheme parametrizing the torsion quotients of ${\mathcal O}^{\oplus r}_X$ of degree $n$. If $g \geq 2$ and $n \leq 2g-2$, we prove that ${\mathcal Q}_X(r,n)$ does not admit a Kähler structure such that all the holomorphic bisectional curvatures are nonnegative.

math.DG↗
arXiv2013

Rigidity of holomorphic maps between fiber spaces

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds $X$ and $Y$, a degree-one holomorphic map $f: Y\to X$ is a biholomorphism. Given that the real manifolds underlying $X$ and $Y$ are diffeomorphic, we provide a condition under which $f$ is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products $X=X_1\times X_2$ and $Y=Y_1\times Y_2$ of compact connected complex manifolds. When $X_1$ is a Riemann surface of genus $\geq 2$, we show that any non-constant holomorphic map $F:Y\to X$ is of a special form.

math.CV↗