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K. V. Shuddhodan

Publications and source records attributed to K. V. Shuddhodan.

11 recordsLinked to original sources
arXiv2026

Uniform stratified vanishing and equidistribution on $\mathbb{G}_m^d$

We prove a uniform stratified generic vanishing theorem for perverse sheaves on $\mathbb{G}_m^d$ over finite fields, with constants depending only on the dimension and on the complexity à la Sawin. As a corollary, we prove an equidistribution theorem for $\mathbb{G}_m^d$ extending Katz's theorem for $\mathbb{G}_m$. The corollary applies to sequences of perverse sheaves of bounded complexity with common tannakian monodromy group, over finite fields of varying characteristic. As the cardinalities of the fields tend to infinity, the Frobenius conjugacy classes attached to the multiplicative characters equidistribute in the space of conjugacy classes of a maximal compact subgroup.

math.AG↗
arXiv2026

Néron--Severi groups of proper schemes over finite fields

Let $X$ be a proper reduced scheme over a finite field $k$, let $\ell$ be a prime different from $\operatorname{char} k$, and write $\overline{X}=X\times_{k}\overline{k}$ for its base change to an algebraic closure $\overline{k}$ of $k$. Call a class in $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(1))$ Zariski-locally trivial if it vanishes on a Zariski-open cover of $\overline{X}$. We prove that the first Chern class map identifies $\operatorname{NS}(\overline{X})\otimes\mathbb{Z}_{\ell}$ with the group of Zariski-locally trivial classes whose image in $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Q}_{\ell}(1))$ has weight zero. This is the finite-field analogue of a theorem of Barbieri-Viale--Rosenschon--Srinivas for proper seminormal complex varieties. In the finite-field setting neither seminormality nor irreducibility is needed.

math.AG↗
arXiv2026

Generic vanishing on homogeneous spaces in arbitrary characteristic

Let $X$ be a proper homogeneous space for a connected algebraic group $G$ over an algebraically closed field. For locally closed smooth affine subvarieties $W,Z\subset X$, we show that \[ (-1)^{\dim X-\dim W+\dim Z}χ(gW\cap Z)\geq 0 \] for generic $g\in G$. This extends the characteristic-zero theorem of Schürmann--Simpson--Wang. Over finite fields, our methods give a trace-function identity on a dense open subset of $G$ and a Lang--Weil estimate for the non-generic locus.

math.AG↗
arXiv2023

Perverse Filtrations via Brylinski-Radon transformations

In this article, we prove the $t$-exactness of a Brylinski-Radon transformation taking values in sheaves on flag varieties. This implies several weak Lefschetz type results for cohomology. In particular, we obtain de Cataldo-Migliorini's P=Dec(F) and Beilinson's basic lemma, the latter was an important ingredient in their proof of P=Dec(F). Our methods also allow the sharpening of Esnault-Katz's cohomological divisibility theorem and estimates for the Hodge level. Finally, we upgrade P=Dec(F) to an equivalence of functors which is also valid over a base.

math.AG↗
arXiv2023

Brylinski-Radon transformation in characteristic $p>0$

In this article, we characterize the image of the Brylinski-Radon transform in characteristic $p>0$ via Beilinson's theory of singular supports. We also provide an alternate proof of Brylinski's results over $\mathbb{C}$, which also works for sheaves with finite coefficients. Along the way, we also obtain a microlocal criterion for the descent of perverse sheaves which could be of independent interest.

math.AG↗
arXiv2021

Algebraic entropy for smooth projective varieties

We show that the spectral radius for the action of a self map $f$ of a smooth projective variety (over an arbitrary base field) on its $\ell$-adic cohomology is achieved on the $f^*$-stable sub-algebra generated by any ample class. This generalizes a result of Esnault-Srinivas who had obtained an analogous result for automorphisms of surfaces. Over $\mathbb{C}$ we also show that this sub-algebra is naturally an irreducible representation of a Looijenga-Lunts-Verbitsky type Lie algebra acting on the cohomology of a smooth projective variety.

math.AG↗
arXiv2021

The Hrushovski-Lang-Weil estimates

In this work we give a geometric proof of Hrushovski's generalization of the Lang-Weil estimates on the number of points in the intersection of a correspondence with the graph of Frobenius.

math.AG↗
arXiv2013

Using Partial Transpose and Realignment to generate Local Unitary Invariants

Motivated by link transformations of lattice gauge theory, a method for generating local unitary invariants, especially for a system of qubits, has been pointed out in an earlier work [M. S. Williamson {\it et. al.}, Phys. Rev. A {\bf 83}, 062308 (2011)]. This paper first points the equivalence of the so constructed transformations to the combined operations of partial transpose and realignment. This allows construction of local unitary invariants of any system, with subsystems of arbitrary dimensions. Some properties of the resulting operators and consequences for pure tripartite higher dimensional states are briefly discussed.

quant-ph↗
arXiv2011

Entanglement optimizing mixtures of two-qubit states

Entanglement in incoherent mixtures of pure states of two qubits is considered via the concurrence measure. A set of pure states is optimal if the concurrence for any mixture of them is the weighted sum of the concurrences of the generating states. When two or three pure real states are mixed it is shown that 28.5% and 5.12% of the cases respectively, are optimal. Conditions that are obeyed by the pure states generating such optimally entangled mixtures are derived. For four or more pure states it is shown that there are no such sets of real states. The implications of these on superposition of two or more dimerized states is discussed. A corollary of these results also show in how many cases rebit concurrence can be the same as that of qubit concurrence.

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