arXiv subjects

Achinta Kumar Nandi

Publications and source records attributed to Achinta Kumar Nandi.

6 recordsLinked to original sources

arXiv2026

Abundance of Bergman metrics with constant positive holomorphic sectional curvature

An outstanding open question, which has attracted renewed attention following the pioneering work of Huang--Li--Treuer, is whether, for a given positive integer $m$, there exists a complex manifold whose Bergman metric is locally isometric to $m$ times the Fubini--Study metric. Previously, this question had only been resolved in the case $m=1$. In this paper, we construct, for any pair of positive integers $(m,n)$ with $n \geq 2$, an $\mathbb{R}$-parameter (hence uncountable) family of Reinhardt domains in $\mathbb{C}^n$ whose Bergman metrics are all locally isometric to $m$ times the Fubini--Study metric. Moreover, we show that the domains in this family are mutually Bergman inequivalent. This not only answers the folklore question, but also suggests that a reasonable classification of the geometry of such complex manifolds is infeasible. We also note such examples cannot exist in dimension one. The results complete the remaining open case in the study of complex manifolds whose Bergman space separates points and whose Bergman metric has constant holomorphic sectional curvature. Our approach differs from existing methods in the literature. We reduce the construction to a mapping problem and apply a Brouwer fixed point argument to establish the existence of the desired domains.

math.CV↗

arXiv2025

Proper maps of annuli

We study proper holomorphic maps of annuli in complex Euclidean spaces, that is, domains with $U(n)$ as the automorphism group. By the Hartogs phenomenon and a result of Forstneri\v{c}, such maps are always rational and extend to proper maps of balls. We first prove that a proper map of annuli from $n$ dimensions to $N$ dimensions where $N < \binom{n+1}{2}$ is always an affine embedding. This inequality is sharp as the homogeneous map of degree 2 satisfies $N=\binom{n+1}{2}$. Next we find a necessary and sufficient condition for a map to be homogeneous: A proper map of annuli is homogeneous if and only if its general hyperplane rank, the affine dimension of the image of a general hyperplane, is exactly $N-1$. As a corollary, we obtain a classification of homogeneous proper maps of balls. A homogeneous proper ball map takes all spheres centered at the origin to spheres centered at the origin. We show that if a proper ball map has general hyperplane rank $N-1$ and takes one sphere centered at the origin to a sphere centered at the origin, then it is homogeneous. Another corollary of this result is a complete classification of proper maps of annuli from dimension 2 to dimension 3. Finally, we give a complete normal form of rational proper maps of annuli of degree 2.

math.CV↗

arXiv2025

On the dimension of the $p$-Bergman spaces

The investigation of the dimension of Bergman spaces has long been a central topic in several complex variables, uncovering profound connections with potential theory and function theory since the pioneering work of Carleson, Wiegerinck, and others in the 1960s. We investigate the dimension of $p$-Bergman spaces associated with pseudoconvex domains in $\mathbb{C}^n$. By constructing $L^p$-versions of the extension theorems of Ohsawa and Ohsawa-Takegoshi, we establish several geometric and potential-theoretic criteria that ensure the spaces are infinite-dimensional. Sufficient conditions for the infinite dimensionality of $p$-Bergman spaces of complete N-circled fibered Hartogs domains, balanced domains, and weighted $p$-Fock spaces are obtained by applying the mentioned $L^p$-analogs of extension theorems and generalizing a sufficient condition of Jucha.

math.CV↗

arXiv2025

Asymptotic behavior of geodesics in conformally compact manifolds

We study the asymptotic behavior of geodesics near the boundary of a conformally compact Riemannian manifold $(X,g)$. In the case where the sectional curvature at infinity is constant (the asymptotically hyperbolic case) it is known that non-trapped geodesics extend to the conformal infinity as smoothly immersed curves in $\overline{X}$ and that the conformal infinity can be locally smoothly parametrized by the initial directions of such non-trapped geodesics starting at a given point. We show that in the general case non-trapped geodesics typically only extend to the conformal infinity with $C^{1,α}$ regularity, due to the presence of a logarithmic singularity sourced by the gradient of the limiting sectional curvature. In spite of this singular behavior, we show that the endpoints of such geodesics depend smoothly on the initial conditions, so that the conformal infinity can still be locally smoothly parametrized using the initial velocities of geodesics. In particular, the smooth structure of the conformal infinity is canonically determined by $(X,g)$. We also show how to specify `initial data' for geodesics at the conformal infinity and obtain an asymptotic expansion that parametrizes the geodesics tending toward a given boundary point.

math.DG↗

arXiv2024

Proper maps of ball complements & differences and rational sphere maps

We consider proper holomorphic maps of ball complements and differences in complex euclidean spaces of dimension at least two. Such maps are always rational, which naturally leads to a related problem of classifying rational maps taking concentric spheres to concentric spheres, what we call $m$-fold sphere maps; a proper map of the difference of concentric balls is a $2$-fold sphere map. We prove that proper maps of ball complements are in one to one correspondence with polynomial proper maps of balls taking infinity to infinity. We show that rational $m$-fold sphere maps of degree less than $m$ (or polynomial maps of degree $m$ or less) must take all concentric spheres to concentric spheres and we provide a complete classification of them. We prove that these degree bounds are sharp.

math.CV↗

arXiv2024

On perturbations of singular complex analytic curves

Suppose $V$ is a singular complex analytic curve inside $\mathbb{C}^{2}$. We investigate when a singular or non-singular complex analytic curve $W$ inside $\mathbb{C}^{2}$ with sufficiently small Hausdorff distance $d_{H}(V, W)$ from $V$ must intersect $V$. We obtain a sufficient condition on $W$ which when satisfied gives an affirmative answer to our question. More precisely, we show the intersection is non-empty for any such $W$ that admits at most one non-normal crossing type discriminant point associated with some proper projection. As an application, we prove a special case of the higher-dimensional analog, and also a holomorphic multifunction analog of a result by Lyubich-Peters.

math.CV↗