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Xuanyu Pan

Publications and source records attributed to Xuanyu Pan.

13 recordsLinked to original sources
arXiv2026

Asymptotic Betti bounds for hypersurfaces in a singular variety

We show that for any degree $d$ hypersurface $Y \subset X$ in a possibly singular projective variety $X \subset \mathbf{P}^N$, the total Betti number of $Y$ is bounded by $3\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$ for some explicit constant $C > 0$ independent of $d$ and $Y$. When $X$ is a local complete intersection, the bound improves to $\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$. In this case, the bound is asymptotically sharp. Similar bounds are also established for general constructible sheaves.

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arXiv2025

Automorphism and Cohomology II: Complete intersections

We prove that the automorphism group of a general complete intersection $X$ in a projective space is trivial with a few well-understood exceptions. We also prove that the automorphism group of a complete intersection $X$ acts on the cohomology of $X$ faithfully with a few well-understood exceptions.

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arXiv2021

Remarks on Automorphism and Cohomology of Cyclic Coverings

For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that the group of automorphisms acts faithfully on the cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and automorphism groups for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristics, a lifting criterion of automorphisms reduces the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge-de Rham spectral sequences for a family of smooth finite cyclic coverings, and the infinitesimal Torelli theorem for finite cyclic coverings defined over an arbitrary field.

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arXiv2020

Logarithmic de Rham comparison for open rigid spaces

In this note, we prove the logarithmic $p$-adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(π,1)$ (in a certain sense) with respect to $\mathbb{F}_p$-local systems and ramified coverings along the divisor. We follow Scholze's method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.

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arXiv2017

1-Cycles on Fano varieties

We prove results about 1-cycles on certain Fano varieties using techniques that rely on rational curves. Firstly, we show that Fano weighted complete intersections with index bigger than half their dimension have trivial first Griffiths group. Secondly, we prove that the first Chow group of most $2$-Fano weighted complete intersections, and of $2$-Fano conic-connected varieties in $\mathbb{P}^n$ of high enough index (with $3$ obvious exceptions), are generated by lines. Furthermore, if the Fano variety of lines is irreducible, the first Chow group is isomorphic to $\mathbb{Z}$.

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arXiv2017

p-adic deformation of graph cycles

In this paper, we show that the infinitesimal Torelli theorem implies the existence of deformations of automorphisms. In the first part, we use Hodge theory and deformation theory to study the deformations of automorphisms of complex projective manifolds. In the second part, we use crystalline cohomology to explore the p-adic analogues of the first part, which generalizes a result of Berthelot and Ogus. The study of the deformations of automorphisms also provides criterions characterizing when the action of the automorphism group of a variety on its cohomology is faithful.

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arXiv2017

Geometry of Rational Curves on Low degree Complete Intersections

For a smooth complete intersection X, we consider a general fiber \mathbb{F} of the evaluation map ev of Kontsevich moduli space \bar{M}_{0,m}(X,m)\rightarrow X^m and the forgetful functor F : \mathbb{F} \rightarrow \bar{M}_{0,m}. We prove that a general fiber of the map $F$ is a smooth complete intersection if $X$ is of low degree. The result sheds some light on the arithmetic and geometry of Fano complete intersections.

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arXiv2017

Spaces of conics on low degree complete intersections

Let X be a smooth complete intersection. Suppose p and q are general points of X, we consider conics in X passing through p and q. We show the moduli space of these conics is a smooth complete intersection. The main ingredients of the proof are a criterion for characterizing when a smooth projective variety is a complete intersection, the Grothendieck-Riemann-Roch theorem and the geometry of the spaces of conics.

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arXiv2016

Automorphism and Cohomology I: Fano variety of lines and Cubic

In this paper, we prove a general principle of lifting an automorphism from positive characteristic to zero characteristic. We based on the principle to prove the automorphism group of Fano variety of cubic threefold (fourfold) acts on its cohomology group faithfully. We also prove the same result for a singular cubic threefold (fourfold) with one ordinary double singularity. As an application, we classify the automorphism groups of cubic fourfolds with small middle Picard number over complex numbers.

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arXiv2016

2-Cycles on Higher Fano Hypersurfaces

Let F(X_d) be a smooth Fano variety of lines of a hypersurface X_d of degree d. In this paper, we prove the Griffiths group Griff_1(F(X_d)) is trivial if the hypersurface X_d is of 2-Fano type. As a result, we give a positive answer to a question of Professor Voisin about the first Griffiths groups of Fano varieties in some cases. Base on this result, we prove that CH_2(X_d)=\mathbb{Z} for a complex smooth $3$-Fano hypersurface X_d whose Fano variety of lines is smooth.

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arXiv2014

Triviality and Split of Vector Bundles on Rationally Connected Varieties

In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert lines are trivial. Using this result and Chern classes of vector bundles, we give a general criterion of a uniform vector bundle on a homogenous space to be splitting. As an application, we prove a uniform vector bundle on classical Grassmannians and quadrics of low rank is splitting.

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arXiv2011

Some examples on moduli spaces of low degree rational curves on low degree complete intersections

This paper is to give some concrete examples of the general fibers of the evaluation map of some Kontsevich mapping spaces parametrize low degree rational curves on low degree complete intersection varieties. We prove these examples are Fano or Calabi-Yau vareities of low dimension, e.g, surface and threefold. In usual, the "moduli" fibers are constructed in an abstract way, but here we give a down-to-earth interpretation to some of these fibers from the perspective of projective geometry.

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