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Qihong Xie

Publications and source records attributed to Qihong Xie.

15 recordsLinked to original sources
arXiv2014

Vanishing Theorems on Toric Varieties, with Erratum to a Former Paper

In this paper, we clarify the mistakes made in the former article entitled "Vanishing Theorems on Toric Varieties in Positive Characteristic". On the other hand, we use the positive characteristic method to reprove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties over a field of arbitrary characteristic, where concerned Weil divisors are torus invariant.

math.AG↗
arXiv2014

Cyclic Covers over Strongly Liftable Schemes

A smooth scheme $X$ over a field $k$ of positive characteristic is said to be strongly liftable over $W_2(k)$, if $X$ and all prime divisors on $X$ can be lifted simultaneously over $W_2(k)$. In this paper, we give a criterion for that cyclic covers over strongly liftable schemes are still strongly liftable. As a corollary, cyclic covers over projective spaces of dimension at least three are strongly liftable over $W_2(k)$.

math.AG↗
arXiv2013

Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic III

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable over W_2(k), if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, we first deduce the Kummer covering trick over W_2(k), which can be used to construct a large class of smooth projective varieties liftable over W_2(k), and to give a direct proof of the Kawamata-Viehweg vanishing theorem on strongly liftable schemes. Secondly, we generalize almost all of the results in [Xie10, Xie11] to the case where everything is considered over W(k), the ring of Witt vectors of k.

math.AG↗
arXiv2013

Vanishing Theorems on Toric Varieties in Positive Characteristic

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch-Thomsen-Lauritzen-Mehta, Mustata and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.

math.AG↗
arXiv2011

Decomposition of De Rham Complexes with Smooth Horizontal Coefficients for Semistable Reductions

We generalize Illusie's result to prove the decomposition of the de Rham complex with smooth horizontal coefficients for a semistable $S$-morphism $f:X\ra Y$ which is liftable over $\Z/p^2\Z$. As an application, we prove the Kollár vanishing theorem in positive characteristic for a semistable $S$-morphism $f:X\ra Y$ which is liftable over $\Z/p^2\Z$, where all concerned horizontal divisors are smooth over $Y$.

math.AG↗
arXiv2011

Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic II

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, first we prove that smooth toric varieties are strongly liftable. As a corollary, we obtain the Kawamata-Viehweg vanishing theorem for smooth projective toric varieties. Second, we prove the Kawamata-Viehweg vanishing theorem for normal projective surfaces which are birational to a strongly liftable smooth projective surface. Finally, we deduce the cyclic cover trick over W_2(k), which can be used to construct a large class of liftable smooth projective varieties.

math.AG↗
arXiv2010

Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, we give some concrete examples and properties of strongly liftable schemes. As an application, we prove that the Kawamata-Viehweg vanishing theorem in positive characteristic holds on any normal projective surface which is birational to a strongly liftable surface.

math.AG↗
arXiv2009

Kawamata-Viehweg Vanishing on Rational Surfaces in Positive Characteristic

We prove that the Kawamata-Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W_2(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata-Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.

math.AG↗
arXiv2008

A Note on the Effective Non-vanishing Conjecture

We give a reduction of the irregular case for the effective non-vanishing conjecture by virtue of the Fourier-Mukai transform. As a consequence, we reprove that the effective non-vanishing conjecture holds on algebraic surfaces.

math.AG↗
arXiv2006

Effective Non-vanishing for Algebraic Surfaces in Positive Characteristic

We give a theorem on the effective non-vanishing problem for algebraic surfaces in positive characteristic. For the Kawamata-Viehweg vanishing, the logarithmic Kollar vanishing and the logarithmic semipositivity, we give their counterexamples on ruled surfaces in positive characteristic.

math.AG↗
arXiv2005

On Pseudo-Effectivity of the Second Chern Classes for Terminal Threefolds

We give a reduction of the conjecture that for terminal projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective. On the other hand, some effective non-vanishing results are obtained as applications of the pseudo-effectivity of the second Chern classes.

math.AG↗
arXiv2005

The Nef Curve Cone Theorem Revisited

We revisit the nef curve cone theorem and use it to reprove the pseudo-effectivity of the second Chern classes for terminal weak Q-Fano varieties.

math.AG↗
arXiv2004

On Pseudo-Effectivity of the Second Chern Classes for Smooth Threefolds

We prove that for smooth projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective under a weak assumption. As an application, the pseudo-effectivity of the second Chern classes implies that Kawamata's Effective Non-vanishing Conjecture holds for such threefolds.

math.AG↗