Searcharxiv⌕ Search

arXiv subjects

Omprokash Das

Publications and source records attributed to Omprokash Das.

At least 19 recordsLinked to original sources
arXiv2026

MMP for Generalized Pairs on Kähler 3-folds

In this article we define generalized pairs $(X, B+\boldsymbolβ)$ where $X$ is an analytic variety and $\boldsymbolβ$ is a b-(1,1) current. We then prove that almost all standard results of the MMP hold in this generality for compact Kähler varieties of dim $X\leq 3$. More specifically, we prove the cone theorem, existence of flips, existence of log terminal models, log canonical models and Mori fiber spaces, the geography of log canonical and log terminal models, etc.

math.AG↗
arXiv2026

Transcendental Minimal Model Program for Projective Varieties

In this article we prove that if $(X,B+β)$ is a projective generalized klt pair such that $B+β$ is big, then $(X,B+β)$ admits a good Minimal Model or Mori fiber space. In particular, this implies Tossati's transcendental base-point-free conjecture for projective manifolds.

math.AG↗
arXiv2025

On the Log Abundance for Compact {K{ä}hler} threefolds II

In this article we show that if $(X, Δ)$ is a log canonical compact Kähler threefold pair such that $K_X+Δ$ is nef and the numerical dimension $ν(X, K_X+Δ)=2$, then $K_X+Δ$ is semi-ample. This result combined with our previous work in arXiv:2201.01202 shows that the log abundance holds for log canonical compact Kähler threefold pairs.

math.AG↗
arXiv2024

On the Minimal Model Program for Kähler 3-folds

In this article we prove the existence of pl-flipping and divisorial contractions and pl flips in dimension $n$ for compact Kähler varieties, assuming results of the minimal model program in dimension $n-1$. We also give a self contained proof of the cone theorem, the existence of flipping and divisorial contractions, of flips and minimal models in dimension 3.

math.AG↗
arXiv2024

The log minimal model program for Kähler $3$-folds

In this article we show that the Log Minimal Model Program for $\mathbb{Q}$-factorial dlt pairs $(X, B)$ on a compact Kähler $3$-fold holds. More specifically, we show that after finitely many divisorial contractions and flips we obtain either a (log) minimal model or a Mori fiber space. We also prove a base point free theorem Kähler $3$-folds.

math.AG↗
arXiv2024

On the $4$-dimensional minimal model program for Kähler varieties

In this article we establish the following results: Let $(X, B)$ be a dlt pair, where $X$ is a $\mathbb Q$-factorial Kähler $4$-fold -- (i) if $X$ is compact and $K_X+B\sim_{\mathbb Q} D\geq 0$ for some effective $\mathbb Q$-divisor, then $(X, B)$ has a log minimal model, (ii) if $(X/T, B)$ is a semi-stable klt pair, $W\subset T$ a compact subset and $K_X+B$ is effective over $W$ (resp. not effective over $W$), then we can run a $(K_X+B)$-MMP over $T$ (in a neighborhood of $W$) which ends with a minimal model over $T$ (resp. a Mori fiber space over $T$). We also give a proof of the existence of flips for analytic varieties in all dimensions and the relative MMP for projective morphisms between analytic varieties.

math.AG↗
arXiv2023

A basepoint free theorem for algebraically integrable foliations

We show that if $\mathcal{F}$ is an algebraically integrable foliation on a $\mathbb{Q}$-factorial normal projective variety $X$, $ A, B \geq 0$ are $\mathbb{Q}$-divisors on $X$ with $A$ ample such that $(\mathcal{F}, B)$ is foliated dlt and $K_{\mathcal{F}}+ A+B$ is nef, then $K_{\mathcal{F}}+A+B$ is semiample. We also provide some applications of this and related results such as contraction theorem for F-dlt pairs and a special case of the b-semiampleness conjecture.

math.AG↗
arXiv2022

On the log minimal model program for $3$-folds over imperfect fields of characteristic $p>5$

We prove that many of the results of the LMMP hold for $3$-folds over fields of characteristic $p>5$ which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal rays, and the existence of log minimal models. As well as pertaining to the geometry of fibrations of relative dimension $3$ over algebraically closed fields, they have applications to tight closure in dimension $4$.

math.AG↗
arXiv2020

Boundedness of log-pluricanonical maps for surfaces of log-general types in positive characteristic

In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, Δ)$ satisfying the following properties: (i) $X$ is a projective surface defined over an algebraically closed field, (ii) $(X, Δ)$ is log canonical and the coefficients of $Δ$ are in $I$, and (iii) $K_X+Δ$ is big. Then there is a positive integer $N=N(I)$ depending only on the set $I$ such that the linear system $|\lceil m(K_X+Δ)\rceil|$ defines a birational map onto its image for all $m\geq N$ and $(X, Δ)\in\mathfrak{D}$

math.AG↗
arXiv2019

On the boundedness of anti-canonical volumes of singular Fano $3$-folds in characteristic $p>5$

In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt pairs $(X, Δ)$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $Δ$ is an $\mathbb{R}$-divisor with coefficients in $I$, (3) $K_X+Δ\equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\mbox{vol}_X(-K_X) \ | \ (X, Δ)\in\mathfrak{D}\mbox{ for some }Δ\}$ is bounded from above.

math.AG↗
arXiv2018

Finiteness of Log Minimal Models and Nef curves on $3$-folds in characteristic $p>5$

In this article we prove a finiteness result on the number of log minimal models for $3$-folds in char $p>5$. We then use this result to prove a version of Batyrev's conjecture on the structure of nef cone of curves on $3$-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic $0$. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_X$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.

math.AG↗
arXiv2018

Bridgeland Stability on Blow Ups and Counterexamples

We give further counterexamples to the conjectural construction of Bridgeland stability on threefolds due to Bayer, Macrì, and Toda. This includes smooth projective threefolds containing a divisor that contracts to a point, and Weierstraß elliptic Calabi-Yau threefolds. Furthermore, we show that if the original conjecture, or a minor modification of it, holds on a smooth projective threefold, then the space of stability conditions is non-empty on the blow up at an arbitrary point. More precisely, there are stability conditions on the blow up for which all skyscraper sheaves are semistable.

math.AG↗
arXiv2016

On the Adjunction Formula for $3$-folds in characteristic $p>5$

In this article we prove a relative Kawamata-Viehweg vanishing-type theorem for PLT $3$-folds in characteristic $p>5$. We use this to prove the normality of minimal log canonical centers and the adjunction formula for codimension $2$ subvarieties on $\mathbb{Q}$-factorial $3$-folds in characteristic $p>5$.

math.AG↗