arXiv subjects

Ari Krishna

Publications and source records attributed to Ari Krishna.

6 recordsLinked to original sources

arXiv2026

A negative K\"ahler-Einstein threefold with non-integrable infinitesimal Einstein deformations

We construct a smooth canonically polarized threefold, not biholomorphic to a product of positive-dimensional varieties, whose normalized K\"ahler-Einstein metric admits a non-integrable infinitesimal Einstein deformation. The same tangent direction is non-integrable as an infinitesimal complex deformation. In fact, the space of infinitesimal Einstein deformations in our example has real dimension 8, its integrable directions form a real 6-dimensional subspace, and every direction outside that subspace is obstructed. This answers both parts of a suitably generalized version of a question posed by Dai, Wang, and Wei in real dimension 6.

math.DG↗

arXiv2026

Chow classes of orbit closures of skew-symmetric matrix pencils

Let $V$ be a six-dimensional complex vector space and let $G=\operatorname{Gr}(2,\Lambda^2V^\vee)$ parametrize pencils of skew-symmetric $6\times6$ matrices. The group $\operatorname{PGL}(V)$ has $11$ orbits on $G$, classified by the Jordan-Kronecker canonical form. We compute the Chow class of every orbit closure in the Schubert basis of $\operatorname{CH}^*(G)$.

math.AG↗

arXiv2026

A universal discriminant formula for pencils of quadrics

Let $V$ be a vector space of dimension $n+1$ over an algebraically closed field $\mathbb{k}$ of characteristic zero, and let $G_n = \operatorname{Gr}(2,\operatorname{Sym}^2V^\vee)$ be the Grassmannian parametrizing pencils of quadrics in $\mathbb{P}(V) \cong \mathbb{P}^n$. The determinant of the universal pencil defines a universal binary form of degree $n+1$. We prove that the divisor $\mathcal{D}_n\subseteq G_n$ of pencils whose determinant binary form has a multiple root has Chow class $[\mathcal{D}_n]=n(n+1)\sigma_1\in A^1(G_n),$ where $\sigma_1=c_1(S^\vee)$ and $S$ is the tautological rank-two subbundle on $G_n$. More generally, the higher-contact loci of determinant binary forms are computed by a universal jet formula. We also formulate the determinant-root collision strata as refined pullbacks of the universal collision strata for binary forms. For $n=3$, the main formula recovers the class $12\sigma_1$ for the boundary divisor in $\operatorname{Gr}(2,10)$ that the author established in a prior paper.

math.AG↗

arXiv2026

Pointed Evaluation Fibers of Rational Curves on del Pezzo Manifolds

Let $X$ be a Picard-rank-one del Pezzo manifold of dimension $n\geq 4$ over an algebraically closed field of characteristic zero. Okamura proved that the unpointed Kontsevich spaces $\overline{M}_{0,0}(X,d)$ are irreducible of the expected dimension for every $d\geq 1$. We refine this result by studying pointed evaluation fibers. First, we prove that for every $d\geq 1$, the one-pointed evaluation morphism $\overline{M}_{0,1}(X,d)\to X$ has geometrically irreducible generic fiber. Second, in the very ample cases $H^n=3,4,5$, we prove that for every $d\geq 2$, the two-pointed evaluation morphism $\overline{M}_{0,2}(X,d)\to X\times X$ has geometrically irreducible generic fiber.

math.AG↗

arXiv2026

All Quiet on the Exceptional Locus

We study admissible subcategories of the bounded derived category of a smooth projective surface that are supported on the exceptional locus of a birational morphism. We prove that if $f:X\to Y$ is a birational morphism of smooth projective surfaces, then every admissible subcategory of $D^b(X)$ supported on $\operatorname{Exc}(f)$ is generated by a finite exceptional collection. Moreover, if $K_Y$ is nef, then the same conclusion holds for every admissible subcategory of $D^b(X)$ supported on a proper closed subset of $X$. As a consequence, no nonzero phantom or quasi-phantom subcategory on such a surface can have proper support. The proof combines a splitting lemma for admissible subcategories inside a semiorthogonal decomposition with a single exceptional block, Orlov's blow-up formula, and Pirozhkov's support theorem.

math.AG↗

arXiv2026

On the Codimension-1 $\mathrm{PGL}_4$ Orbit Closures in $\mathrm{Gr}(2,10)$

We study the natural action of $\mathrm{PGL}(V)$ on the Grassmannian $G=\operatorname{Gr}(2,\operatorname{Sym}^2 V^\vee)$, where $\dim V=4$ and points of $G$ are pencils of quadrics in $\mathbb{P}(V)\cong \mathbb{P}^3$. Here $\dim G=16$ while $\dim \mathrm{PGL}(V)=15$, so the generic orbit has codimension one and one expects a one-parameter family of generic orbits. We construct this family via the $j$-invariant of the discriminant binary quartic of a pencil. We then determine the codimension-one orbit closures and compute their Chow classes. The smooth codimension-one orbit closures are the reduced fibers of the $j$-map on the smooth locus, while the unique boundary divisor is the closure of the orbit of a nodal quartic complete intersection of arithmetic genus $1$ and geometric genus $0$. Every divisorial fiber of the rational $j$-map has class $12\sigma_1$ in $A^1(G)$. For the reduced codimension-one orbit closures one has $[\overline{O_a}]=12\sigma_1$ for $a\neq 0,1728,\infty$, $[\overline{O_{1728}}]=6\sigma_1$, $[\overline{O_0}]=4\sigma_1$, and $[T]=12\sigma_1$.

math.AG↗