arXiv subjects

Hao Max Sun

Publications and source records attributed to Hao Max Sun.

10 recordsLinked to original sources

arXiv2026

Frobenius type positivity of Hodge bundles and applications

We define two new Frobenius type positivity notions, $F^t$-ampleness and $F^tGG$-ampleness, for coherent sheaves, show that they are stronger than ordinary ampleness and establish some of their basic properties. The analogous Frobenius type semipositivity notions have also been established. We prove the $F^1GG$-semipositivity of Hodge bundles for semistable morphisms of varieties. This strengthens and generalizes various previous semipositivity results of Fujita, Kawamata, Koll\'ar, Viehweg, Fujino and others. As a consequence we obtain the positivity of top Chern characters of Hodge bundles and Kawamata-Viehweg-Koll\'ar type vanishing theorems for some semistable morphisms. Our results are also applied to study the images of Fano varieties under semistable morphisms and the slope inequality.

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arXiv2025

Bogomolov type inequalities and Frobenius semipositivity

We prove Bogomolov type inequalities for high Chern characters of semistable sheaves satisfying certain Frobenius semipositivity. The key ingredients in the proof are a high rank generalization of the asymptotic Riemann-Roch theorem and Langer's estimation theorem of the global sections of torsion free sheaves. These results give some Bogomolov type inequalities for semistable sheaves with vanishing low Chern characters. Our results are also applied to obtain inequalities of Chern characters of threefolds and varieties of small codimension in projective spaces and abelian varieties.

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arXiv2023

Counting equivariant sheaves on K3 surfaces

We study the equivariant sheaf counting theory on K3 surfaces with finite group actions. Let $\sS=[S/G]$ be a global quotient stack, where $S$ is a K3 surface and $G$ is a finite group acting as symplectic homomorphisms on $S$. We show that the Joyce invariants counting Gieseker semistable sheaves on $\sS$ are independent on the Bridgeland stability conditions. As an application we prove the multiple cover formula of Y. Toda for the counting invariants for semistable sheaves on local K3 surfaces with a symplectic finite group action.

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arXiv2022

Bogomolov-Gieseker type inequalities on ruled threefolds

We strengthen a conjecture by the author. This conjecture is a Bogomolov-Gieseker type inequality involving the third Chern character of mixed tilt-stable complexes on fibred threefolds. We extend it from complexes of mixed tilt-slope zero to arbitrary relative tilt-slope. We show that this stronger conjecture implies the support property of Bridgeland stability conditions, and the existence of explicit stability conditions. We prove our conjecture for ruled threefolds, hence improving a previous result by the author.

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arXiv2022

Stability conditions on fibred threefolds

We give a conjectural construction of Bridgeland stability conditions on the derived category of fibred threefolds. The construction depends on a conjectural Bogomolov-Gieseker type inequality for certain stable complexes. It can be considered as a relative version of the construction of Bayer, Macrì and Toda. We prove the conjectural Bogomolov-Gieseker type inequality in the case of relative projective planes over curves. This gives the the existence of Bridgeland stability conditions on such threefolds.

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arXiv2021

On the Bogomolov-Gieseker inequality for tame Deligne-Mumford surfaces

We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer's method to smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne-Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface $S$. In a joint appendix with Hao Max Sun, we generalize the Bogomolov inequality formula to Simpson Higgs sheaves on tame Deligne-Mumford stacks.

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arXiv2021

Arithmetic genus of integral space curves

We give an estimation for the arithmetic genus of an integral space curve, which are not contained in a surface of degree $k-1$. Our main technique is the Bogomolov-Gieseker type inequality for $\mathbb{P}^3$ proved by Macri.

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arXiv2021

Tilt-stability, vanishing theorems and Bogomolov-Gieseker type inequalities

We investigate the tilt-stability of stable sheaves on projective varieties with respect to certain tilt-stability conditions depends on two parameters constructed by Bridgeland. For a stable sheaf, we give effective bounds of these parameters such that the stable sheaf is tilt-stable. These allow us to prove new vanishing theorems for stable sheaves and an effective Serre vanishing theorem for torsion free sheaves. Using these results, we also prove Bogomolov-Gieseker type inequalities for the third Chern character of a stable sheaf on $\mathbb{P}^3$.

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arXiv2021

Bogomolov's inequality for product type varieties in positive characteristic

We prove Bogomolov's inequality for semistable sheaves on product type varieties in arbitrary characteristic. This gives the first examples of varieties with positive Kodaira dimension in positive characteristic on which Bogomolov's inequality holds for semistable sheaves of any rank. The key ingredient in the proof is a high rank generalization of the slope inequality established by Xiao and Cornalba-Harris. This Bogomolov's inequality is applied to study the positivity of linear systems and semistable sheaves and construct Bridgeland stability conditions on product type surfaces in positive characteristic. We also give some new counterexamples to Bogomolov's inequality and pose some open questions.

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arXiv2021

Stability conditions on threefolds with vanishing Chern classes

We prove the Bogomolov-Gieseker type inequality conjectured by Bayer, Macri and Toda for threefolds with semistable tangent bundles and vanishing Chern classes in any characteristic, which was originally proved by Bayer, Macri and Stellari in characteristic zero. This gives the existence of Bridgeland stability conditions on such threefolds. As applications, we obtain Reider type theorem and confirm Fujita's conjecture for such threefolds in any characteristic.

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