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41

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Introduction

## Publications

Publications (41)

Assuming the Kunneth decomposition of the Chow groups of products of general Kummer surfaces, we prove that the Hodge-${\mathcal D}$-conjecture fails for the real regulator $r_{k,1}$ on a product of $n$ general elliptic curves for $2n\ge 3k-1\ge 8$.

We study the algebraic hyperbolicity of the complement of very general degree $2n$ hypersurfaces in P^n. We prove the Algebraic Green-Griffiths-Lang Conjecture for these complements, and in the case of the complement of a quartic plane curve, we completely characterize the exceptional locus as the union of the flex and bitangent lines.

We prove that if X is a complex projective K3 surface and $g>0$ , then there exist infinitely many families of curves of geometric genus g on X with maximal, i.e., g -dimensional, variation in moduli. In particular, every K3 surface contains a curve of geometric genus 1 which moves in a nonisotrivial family. This implies a conjecture of Huybrechts...

We prove a conjecture of Voisin that any two distinct points on a very general hypersurface of degree 2n+2 in Pn+1 are rationally inequivalent.

We prove that for every $g\geq 0$ and every complex projective K3 surface $X$, there exist infinitely many families of integral curves of genus $g$ on $X$ which deform with maximal moduli. In particular every K3 surface contains a curve of geometric genus 1 which moves in a non-isotrivial family. This implies a conjecture of Huybrechts on constant...

We study the relations between the finite generation of Cox ring, the rationality of Euler–Chow series and Poincaré series and Zariski’s conjecture on dimensions of linear systems. We prove that if the Cox ring of a smooth projective variety is finitely generated, then all Poincaré series of the variety are rational. We also prove that the multi-va...

We show that every projective K3 surface over an algebraically closed field of characteristic zero contains infinitely many rational curves. For this, we introduce two new techniques in the deformation theory of curves on K3 surfaces. Regeneration, a process opposite to specialisation, which preserves the geometric genus and does not require the cl...

Fix a K3 lattice $\Lambda$ of rank two and $L\in\Lambda$ a big and nef divisor that is positive enough. We prove that the generic $\Lambda$-polarised K3 surface has an integral nodal rational curve in the linear system $|L|$, in particular strengthening previous work of the first named author. The technique is by degeneration, and also works for ma...

We study the relations between the finite generation of Cox ring, the rationality of Euler-Chow series and Poincar\'e series and Zariski's conjecture on dimensions of linear systems. We prove that if the Cox ring of a smooth projective variety is finitely generated, then all Poincar\'e series of the variety are rational. We also prove that the mult...

We prove a conjecture of Voisin that no two distinct points on a very general hypersurface of degree $2n$ in ${\mathbb P}^n$ are rationally equivalent.

In this paper, we study the Severi variety $V_{L,g}$ of genus $g$ curves in $|L|$ on a general polarized K3 surface $(X,L)$. We show that the closure of every component of $V_{L,g}$ contains a component of $V_{L,g-1}$. As a consequence, we see that the general members of every component of $V_{L,g}$ are nodal.

We generalize the results of Clemens, Ein, and Voisin regarding rational curves and zero cycles on generic projective complete intersections to the logarithmic setup.

In this paper, we study $\mathbb{A}^1$ curves on log K3 surfaces. We classify all genuine log K3 surfaces of type II which admits countably infinite $\mathbb{A}^1$ curves.

We prove that the automorphism group of a general complete intersection $X$
in $\PP^n$ 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.

A canonically fibered surface is a surface whose canonical series maps it to
a curve. Using Miyaoka-Yau inequality, A. Beauville proved that a canonically
fibered surface has relative genus at most 5 when its geometric genus is
sufficiently large. G. Xiao further conjectured that the relative genus cannot
exceed 4. We give a proof of this conjectur...

We prove some general density statements about the subgroup of
invertible points on intermediate jacobians; namely those points in the
Abel-Jacobi image of nullhomologous algebraic cycles on projective
algebraic manifolds.

In this paper we work with a series whose coefficients are the Euler
characteristic of Chow varieties of a given projective variety. For varieties
where the Cox ring is defined, it is easy to see that in this case the ring
associated to the series is the Cox ring. If this ring is noetherian then the
series is rational. It is an open question whethe...

We study constraints on the Chern classes of a vector bundle on a singular
variety. We use this constraint to study a variety which carries a Hodge cycle
that are not a linear combination of Chern classes of vector bundles on it.

In [C-L3] it is shown that the real regulator for a general self-product of a K3 surface is nontrivial. In this note, we prove a theorem which says that the real regulator for a general self-product of a surface of higher order (in a suitable sense), is essentially trivial. 1. Statement of the theorem Let Γ be a smooth projective curve, {Zt}t∈Γ a f...

Using Gauss-Manin derivatives of normal functions, we arrive at some
remarkable results on the non-triviality of the transcendental regulator for
$K_m$ of a very general projective algebraic manifold. Our strongest results
are for the transcendental regulator for $K_1$ of a very general $K3$ surface.
We also construct an explicit family of $K_1$ cy...

We prove that a very general projective K3 surface does not admit a dominant self rational map of degree at least two. Comment: 34 pages

We proved that the union of rational curves is dense on a very general K3
surface and the union of elliptic curves is dense in the 1st jet space of a
very general K3 surface, both in the strong topology.

Let X/C be a projective algebraic manifold, and MX∗ be the sheaf of nonvanishing meromorphic functions on X in the analytic topology. We prove a number of nonvanishing results for H•(X,MX∗). In particular, MX∗ is acyclic iff dimX=1.

Based on a novel application of an archimedean type pairing to the geometry and deformation theory of $K3$ surfaces, we construct a regulator indecomposable $K_1$-class on a self-product of a $K3$ surface. In the Appendix, we explain how this pairing is a special instance of a general pairing on precycles in the equivalence relation defining Bloch'...

I gave a geometric proof of Vojta's 1 + epsilon conjecture.
Some gaps in the published paper were spotted and kindly pointed out to me by
Paul Vojta. These were addressed in "Erratum".

Support Vector Machines (SVMs) have been very successful in text
classification. However, the intrinsic geometric structure of text data
has been ignored by standard kernels commonly used in SVMs. It is natural
to assume that the documents are on the multinomial manifold, which is
the simplex of multinomial models furnished with the Riemannian stru...

Let X be a projective algebraic manifold, and CHk (X, 1) the higher Chow group, with corresponding real regulator r(k,1) circle times R : CHk (X, 1) circle times R --> H-D(2k-1)(X, R(k)). If X is a general K3 surface or Abelian surface, and k = 2, we prove the Hodge-D-conjecture, i.e. the surjectivity of r(2,1) circle times R. Since the Hodge-D-con...

We first give an elementary new proof of the vanishing of the regulator on K-1(Z) where Z subset of P-3 be a general surface of degree d greater than or equal to 5, using a Lefschetz pencil argument. By a similar argument we then show the triviality of the regulator for K-1 of a general product of two curves.

We prove the Hodge-D-conjecture for general K3 and Abelian surfaces. Some consequences of this result, e.g., on the levels of higher Chow groups of products of elliptic curves, are discussed.

Let Z be a general surface in P^3 of degree at least 5. Using a Lefschetz pencil argument, we give an elementary new proof of the vanishing of a regulator on K_1(Z).

Fix a very general hypersurface D in P^n of degree at least 2n + 1 and we show that the complement P^n - D does not contain any algebraic torus C^*.

We call a log variety (X, D) algebraically hyperbolic if there exists a positive number e such that 2g(C) - 2 + i(C, D) >= e deg(C) for all curves C on X, where i(C, D) is the number of the intersections between D and the normalization of C. Among other things, we proved that (P^2, D) is algebraically hyperbolic for a very general curve D of degree...

Continuing the work of Chiantini and Ciliberto (1999) on the Severi varieties of curves on surfaces in ℙ3, we complete the proof of the existence of regular components for such varieties.

We give a simple proof of the statement that every rational curve in the primitive class of a general K3 surface is nodal.

We study the following question: fix a sufficient general curve D of degree d in P^2, what is the least number of intersections between D and an irreducible curve of degree m? G. Xu proved this number i(d, m) is at least d - 2 for all m. This problem can be regarded as the algebraic part of Kobayashi conjecture on the hyperbolicity of P^2 D. We fir...

We proved the existence of rational curves in every linear system on a general K3 surface and that all rational curves in the hyperplane class are nodal on a general K3 surface of small genus.

We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.