20 citations · 82 across the 25 of their papers we have counts for
24 papers · 1 filter
Stable maps of curves and algebraic equivalence of 1-cycles
János Kollár, Zhiyu Tian
We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern , the group of 1-cy…
Local-global principle and integral Tate conjecture for certain varieties
Zhiyu Tian
We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimens…
Space of one cycles and coniveau filtrations
Zhiyu Tian
We prove a structural result about the space of one cycles of a separably rationally connected variety or a separably rationally connected fibration over a curve, either as a topol…
Zero cycles on rationally connected varieties over Laurent fields
Zhiyu Tian
We study zero cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces…
Separable rational connectedness and weak approximation in positive characteristic
Jason Michael Starr, Zhiyu Tian
In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a…
Weak approximation for Fano complete intersections in positive characteristic
Jason Michael Starr, Zhiyu Tian, Runhong Zong
For a smooth curve over an algebraically closed field , for every -flat complete intersection in of type …
Motivic multiplicative McKay correspondence for surfaces
Lie Fu, Zhiyu Tian
We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicati…
Motivic HyperKähler Resolution Conjecture : I. Generalized Kummer varieties
Lie Fu, Zhiyu Tian, Charles Vial
Given a smooth projective variety endowed with a faithful action of a finite group , following Jarvis-Kaufmann-Kimura and Fantechi-Göttsche, we define the orbifold motive (o…
Irregular varieites with geometric genus one, theta divisors, and fake tori
Jungkai Chen, Zhi Jiang, Zhiyu Tian
We study the Albanese image of a compact Kähler manifold whose geometric genus is one. We prove that if the Albanese map is not surjective, then the manifold maps surjectively onto…
Weak Approximation for Cubic Hypersurfaces and Degree 4 del Pezzo Surfaces
Letao Zhang, Zhiyu Tian
In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic…
Finiteness of fundamental groups
Zhiyu Tian, Chenyang Xu
We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of klt singu…
Hasse principle for three classes of varieties over global function fields
Zhiyu Tian
We give a geometric proof that Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces in odd characteristic, smooth cub…
Picard groups on moduli of K3 surfaces with Mukai models
Francois Greer, Zhiyuan Li, Zhiyu Tian
We discuss the Picard group of moduli space of quasi-polarized K3 surfaces of genus and . In this range, is unirational and a g…
Integral Hodge classes on fourfolds fiberd by quadric bundles
Zhiyuan Li, Zhiyu Tian
We discuss the space of sections and certain bisections on a quadric surfaces bundle over a smooth curve. The Abel-Jacobi from these spaces to the intermediate Jacobian will be…
Separable rational connectedness and stability
Zhiyu Tian
In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability of the tangent sheaf (if there are no othe…
R-equivalence on del Pezzo surfaces of degree 4 and cubic surfaces
Zhiyu Tian
We prove that there is a unique -equivalence class on every del Pezzo surface of degree defined over the Laurent field in one variable over an algebraically close…
Moduli space of quasi-polarized K3 surfaces of degree 6 and 8
Zhiyuan Li, Zhiyu Tian
In this paper, we study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are comp…
Weak approximation for cubic hypersurfaces
Zhiyu Tian
We prove weak approximation for smooth cubic hypersurfaces of dimension at least 2 defined over the function field of a complex curve.
One Cycles on Rationally Connected Varieties
Zhiyu Tian, Hong R. Zong
All curves on a separably rationally connected variety are rationally equivalent to a (non-effective) integral sum of rational curves, hence the first Chow group is generated by ra…
Towards the symplectic Graber-Harris-Starr theorems
Zhiyu Tian
A theorem of Graber, Harris, and Starr states that a rationally connected fibration over a curve has a section. We study an analogous question in symplectic geometry. Namely, given…
Symplectic geometry and rationally connected 4-folds
Zhiyu Tian
We study some symplectic geometric aspects of rationally connected 4-folds. As a corollary, we prove that any smooth projective 4-fold symplectic deformation equivalent to a Fano 4…
Some examples of symplectic rationally connected 4-folds
Zhiyu Tian
A symplectic manifold is called symplectic rationally connected if there is a non-zero genus zero Gromov-Witten invariant with two point insertions. It is conjectured that every sm…
Symplectic geometry of rationally connected threefolds
Zhiyu Tian
We study symplectic geometry of rationally connected -folds. The first result shows that rationally connectedness is a symplectic deformation invariant in dimension . If a ra…
Weak Approximation for Isotrivial Families
Zhiyu Tian, Hong R. Zong
We prove weak approximation for isotrivial families of rationally connected varieties defined over the function field of a smooth projective complex curve.