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Indranil Biswas

Publications and source records attributed to Indranil Biswas.

At least 145 records · Page 8Linked to original sources
arXiv2023

Hodge rank of ACM bundles and Franchetta's conjecture

We prove that on a general hypersurface in $\mathbb{P}^N$ of degree $d$ and dimension at least $2$, if an arithmetically Cohen-Macaulay (ACM) bundle $E$ and its dual have small regularity, then any non-trivial Hodge class in $H^{n}(X, E\otimesΩ^n_X)$, $n = \lfloor\frac{N-1}{2}\rfloor$, produces a trivial direct summand of $E$. As a consequence, we prove that there is no universal Ulrich bundle on the family of smooth hypersurfaces of degree $d\geq 3$ and dimension at least $4$. This last statement may be viewed as a Franchetta-type conjecture for Ulrich bundles on smooth hypersurfaces.

math.AG↗
arXiv2023

On the geometry of the anti-canonical bundle of the Bott-Samelson-Demazure-Hansen varieties

Let $G$ be a semi-simple simply connected algebraic group over the field $\mathbb{C}$ of complex numbers. Let $T$ be a maximal torus of $G,$ and let $W$ be the Weyl group of $G$ with respect to $T$. Let $Z(w,\, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety corresponding to a tuple $\underline{i}$ associated to a reduced expression of an element $w \,\in\, W.$ We prove that for the tuple $\underline{i}$ associated to any reduced expression of a minuscule Weyl group element $w,$ the anti-canonical line bundle on $Z(w,\,\underline{i})$ is globally generated. As consequence, we prove that $Z(w,\,\underline{i})$ is weak Fano. Assume that $G$ is a simple algebraic group whose type is different from $A_2.$ Let $S\,=\,\{α_{1},\,\cdots,\,α_{n}\}$ be the set of simple roots. Let $w$ be such that support of $w$ is equal to $S.$ We prove that $Z(w,\,\underline{i})$ is Fano for the tuple $\underline{i}$ associated to any reduced expression of $w$ if and only if $w$ is a Coxeter element and $w^{-1}(\sum_{t=1}^{n}α_{t})\,\in\, -S$.

math.AG↗
arXiv2023

Genuinely ramified maps and pseudo-stable vector bundles

Let $X$ and $Y$ be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let $f : Y \rightarrow X$ be a finite generically smooth morphism such that the corresponding homomorphism between the étale fundamental groups $f_*:π^{\rm et}_{1}(Y) \rightarrowπ^{\rm et}_{1}(X)$ is surjective. Fix a polarization on $X$ and equip $Y$ with the pulled back polarization. For a point $y_0\in Y$, let $\varpi(Y, y_0)$ (respectively, $\varpi(X, f(y_0))$) be the affine group scheme given by the neutral Tannakian category defined by the strongly pseudo-stable vector bundles of degree zero on $Y$ (respectively, $X$). We prove that the homomorphism $\varpi(Y, y_0) \rightarrow \varpi(X, f(y_0))$ induced by $f$ is surjective. Let $E$ be a pseudo-stable vector bundle on $X$ and $F \subset f^*E$ a pseudo-stable subbundle with $μ(F)= μ(f^*E)$. We prove that $f^*E$ is pseudo-stable and there is a pseudo-stable subbundle $W \subset E$ such that $f^*W = F$ as subbundles of $f^*E$.

math.AG↗
arXiv2023

On some natural torsors over moduli spaces of parabolic bundles

A moduli space ${\mathcal N}$ of stable parabolic vector bundles, of rank $r$ and parabolic degree zero, on a $n$-pointed curve has two naturally occurring holomorphic $T^*{\mathcal N}$--torsors over it. One of them is given by the moduli space of pairs of the form $(E_*, D)$, where $E_* \in {\mathcal N}$ and $D$ is a connection on $E_*$. This $T^*{\mathcal N}$--torsor has a $C^\infty$ section that sends any $E_* \in {\mathcal N}$ to the unique connection on it with unitary monodromy. The other $T^*{\mathcal N}$--torsor is given by the sheaf of holomorphic connections on a theta line bundle over ${\mathcal N}$. This $T^*{\mathcal N}$--torsor also has a $C^\infty$ section given by the Hermitian structure on the theta bundle constructed by Quillen. We prove that these two $T^*{\mathcal N}$--torsors are isomorphic by a canonical holomorphic map. This holomorphic isomorphism interchanges the above two $C^\infty$ sections.

math.AG↗
arXiv2023

Holomorphic projective connections on compact complex threefolds

We prove that a holomorphic projective connection on a complex projective threefold is either flat, or it is a translation invariant holomorphic projective connection on an abelian threefold. In the second case, a generic translation invariant holomorphic affine connection on the abelian variety is not projectively flat. We also prove that a simply connected compact complex threefold with trivial canonical line bundle does not admit any holomorphic projective connection.

math.DG↗
arXiv2023

More assistance of entanglement, less rounds of classical communication

Classical communication plays a crucial role to distinguish locally a class of quantum states. Despite considerable advances, we have very little knowledge about the number of measurement and communication rounds needed to implement a discrimination task by local quantum operations and classical communications (in short, LOCC). In this letter, we are able to show the relation between round numbers with the local discrimination of a set of pure bipartite orthogonal quantum states. To demonstrate the possible strong dependence on the round numbers, we consider a class of orthogonal product states in $d\otimes d$, which require at least $2d-2$ round of classical communications. Curiously the round number can be reduced to $d$ by the assistance of one-ebit of entanglement as resource and can be reduced further by assistance of more entanglement. We are also able to show that the number of LOCC rounds needed for a discrimination task may depend on the amount of entanglement assistances.

quant-ph↗
arXiv2023

On the Seshadri constants of equivariant bundles over Bott-Samelson varieties and wonderful compactifications

We study torus-equivariant vector bundles $E$ on a complex projective variety $X$ which is either a Bott-Samelson-Demazure-Hansen variety or a wonderful compactification of a complex symmetric variety of minimal rank. We show that $E$ is nef (respectively, ample) if and only if its restriction to every torus--invariant curve in $X$ is nef (respectively, ample). We also compute the Seshadri constants $\varepsilon(E,x)$, where $x\, \in\, X$ is any point fixed by the action of a maximal torus.

math.AG↗
arXiv2023

Moduli spaces of vector bundles on a curve and opers

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\, \geq\,2$, and let $ξ$ be a holomorphic line bundle on $X$ with $ξ^{\otimes 2}\,=\, {\mathcal O}_X$. Fix a theta characteristic $\mathbb L$ on $X$. Let ${\mathcal M}_X(r,ξ)$ be the moduli space of stable vector bundles $E$ on $X$ of rank $r$ such that $\bigwedge^r E\,=\, ξ$ and $H^0(X,\, E\otimes{\mathbb L})\,=\, 0$. Consider the quotient of ${\mathcal M}_X(r,ξ)$ by the involution given by $E\, \longmapsto\, E^*$. We construct an algebraic morphism from this quotient to the moduli space of ${\rm SL}(r,{\mathbb C})$ opers on $X$. Since $\dim {\mathcal M}_X(r,ξ)$ coincides with the dimension of the moduli space of ${\rm SL}(r,{\mathbb C})$ opers, it is natural to ask about the injectivity and surjectivity of this map.

math.AG↗
arXiv2023

Pullback and direct image of parabolic connections and parabolic Higgs bundles

We provide an explicit algebraic construction for the pullback and direct image of parabolic bundles, parabolic Higgs bundles and parabolic connections through maps between Riemann surfaces. We show that these constructions preserve semistability and polystability, and we prove that they are compatible with the nonabelian Hodge correspondence.

math.AG↗
arXiv2023

Parabolic opers and differential operators

Parabolic SL(r,C)-opers were defined and investigated in [BDP] in the set-up of vector bundles on curves with a parabolic structure over a divisor. Here we introduce and study holomorphic differential operators between parabolic vector bundles over curves. We consider the parabolic SL(r,C)-opers on a Riemann surface X with given singular divisor S and with fixed parabolic weights satisfying the condition that all parabolic weights at any point $x_i$ in S are integral multiples of $\frac{1}{2N_i+1}$, where $N_i > 1$ are fixed integers. We prove that this space of opers is canonically identified with the affine space of holomorphic differential operators of order r between two natural parabolic line bundles on X (depending only on the divisor S and the weights $N_i$) satisfying the conditions that the principal symbol of the differential operators is the constant function 1 and the sub-principal symbol vanishes identically. The vanishing of the sub-principal symbol ensures that the logarithmic connection on the rank r bundle is actually a logarithmic SL(r, C)-connection.

math.AG↗
arXiv2023

Framed Parabolic sheaves on a trinion

We consider for structure groups ${\rm SU}(n)\,\subset\, {\rm SL}(n,\mathbb C)$ a densely defined toric structure on the moduli of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In combination with previous degeneration results, these extend to similar moduli for arbitrary Riemann surfaces.

math.AG↗
arXiv2023

Connections and genuinely ramified maps of curves

Given a singular connection $D$ on a vector bundle $E$ over an irreducible smooth projective curve $X$, defined over an algebraically closed field, we show that there is a unique maximal subsheaf of $E$ on which $D$ induces a nonsingular connection. Given a generically smooth map $ϕ: Y \rightarrow\ X$ between irreducible smooth projective curves, and a singular connection $(V, D)$ on $Y$, the direct image $ϕ_*V$ has a singular connection. Let $\textbf{R}(ϕ_*{\mathcal O}_Y)$ be the unique maximal subsheaf on which the singular connection on $ϕ_*{\mathcal O}_Y$ -- corresponding to the trivial connection on ${\mathcal O}_Y$ -- induces a nonsingular connection. We prove that the homomorphism of étale fundamental groups $ϕ_*: π_1^{\rm et}(Y, y_0) \rightarrow π_1^{\rm et}(X, ϕ(y_0))$ induced by $ϕ$ is surjective if and only if ${\mathcal O}_X \subset \textbf{R}(ϕ_*{\mathcal O}_Y)$ is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism $ϕ_*$ is surjective if and only if ${\mathcal O}_X = \textbf{R}(ϕ_*{\mathcal O}_Y)$. For any nonsingular connection $D$ on a vector bundle $V$ over $X$, there is a natural map $V\hookrightarrow {\bf R}(ϕ_*ϕ^*V)$. When the characteristic of the base field is zero, we prove that the map $ϕ$ is genuinely ramified if and only if $V ={\bf R}(ϕ_*ϕ^*V)$.

math.AG↗
arXiv2023

Degenerations of Bundle Moduli

Over a family $\mathbb X$ of genus $g$ complete curves, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of ${\rm SL}(n, \mathbb C)$ bundles over the generic smooth curve $X_t$ in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by $(\mathbb C^*)^s$ of a moduli space on the desingularisation. Taking a "maximal" degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we obtain a degeneration of the moduli space of bundles into a $(\mathbb C^*)^{(3g-3)(n-1)}$-quotient of a $(2g-2)$-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem, the moduli of bundles on the smooth curve is a space of representations of the fundamental group into ${\rm SU}(n)$ (the "symplectic picture"). We obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that our limit space is also a $(S^1)^{(3g-3)(n-1)}$ symplectic quotient of a $(2g-2)$-th power of a space associated to the three-pointed sphere.

math.AG↗
arXiv2022

Entangled state distillation from single copy mixed states beyond LOCC

No pure entangled state can be distilled from a $2\otimes 2$ or $2\otimes 3$ mixed state by separable operations. In $3\otimes 3$, pure entanglement can be distilled by separable operation but not by LOCC. In this letter, we proved the conjecture [PRL. 103, 110502 (2009)] that it is possible to distill pure entanglement for $2\otimes 4$ system by LOCC and further improve these in higher dimensions to distill a pure entangled state of Schmidt rank $d$ from a $m\otimes n$ mixed state by separable operation when $m+n \geqslant 3d$. We found results for tripartite systems with target state $d$-level GHZ-type state. These results provide a class of systems where separable operation is strictly stronger than LOCC.

quant-ph↗
arXiv2022

Classification, reduction and stability of toric principal bundles

Let $X$ be a complex toric variety equipped with the action of an algebraic torus $T$, and let $G$ be a complex linear algebraic group. We classify all $T$-equivariant principal $G$-bundles $\mathcal{E}$ over $X$ and the morphisms between them. When $G$ is connected and reductive, we characterize the equivariant automorphism group $\text{Aut}_T(\mathcal{E} )$ of $\mathcal{E}$ as the intersection of certain parabolic subgroups of $G$ that arise naturally from the $T$-action on $\mathcal{E}$. We then give a criterion for the equivariant reduction of the structure group of $\mathcal{E}$ to a Levi subgroup of $G$ in terms of $\text{Aut}_T(\mathcal{E} )$. We use it to prove a principal bundle analogue of Kaneyama's theorem on equivariant splitting of torus equivariant vector bundles of small rank over a projective space. When $X$ is projective and $G$ is connected and reductive, we show that the notions of stability and equivariant stability are equivalent for any $T$-equivariant principal $G$-bundle over $X$.

math.AG↗
arXiv2022

On the stability of pulled back parabolic vector bundles

Take an irreducible smooth projective curve $X$ defined over an algebraically closed field of characteristic zero, and fix finitely many distinct point $D\, =\, \{x_1,\, \cdots,\, x_n\}$ of it; for each point $x\, \in\, D$ fix a positive integer $N_x$. Take a nonconstant map $f\, :\, Y\, \longrightarrow \, X$ from an irreducible smooth projective curve. We construct a natural subbundle $\mathcal{F}\, \subset\, f_*{\mathcal O}_Y$ using $(D,\, \{N_x\}_{x\in D})$. Let $E_*$ be a stable parabolic vector bundle whose parabolic weights at each $x\, \in\, D$ are integral multiples of $\frac{1}{N_x}$. We prove that the pullback $f^*E_*$ is also parabolic stable, if ${\rm rank}(\mathcal{F})\,=\, 1$.

math.AG↗