相关论文: Real Regulator on K_1 of elliptic surfaces

I construct regulator indecomposable higher Chow cycles in elliptic surfaces satisfying certain conditions. As an application I give an alternative proof of a theorem of Gordon and Lewis, which asserts that there is a real regulator…

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…

We present the topological classification of real parts of real regular elliptic surfaces with a real section.

We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient…

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…

We study the structure of complex points on real surfaces, embedded into complex Elliptic surfaces. We show, for example, that any compact surface has a totally real embedding into a blow-up of a K3 surface. We also exhibit smooth disc…

We pose the problem to determine explicit defining equations of various elliptic fibrations on a given $K3$ surface, and study the case of the Kummer surfaces of the product of two elliptic curves.

We consider an elliptic optimal control problem where the objective functional contains an integral along a surface of codimension 1, also known as a hypersurface. In particular, we use a fidelity term that encourages the state to take…

We survey some results on real rational surfaces focused on their topology and their birational geometry.

We report on our project to find explicit examples of $K3$ surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for…

On a real regular elliptic surface without multiple fiber, the Betti number $h_1$ and the Hodge number $h^{1,1}$ are related by $h_1\leq h^{1,1}$. We prove that it's always possible to deform such algebraic surface to obtain $h_1=h^{1,1}$.…

We determine obstructedness or unobstructedness of (holomorphic) Poisson deformations of ruled surfaces over an elliptic curve.

We give cohomological criteria for logarithmic good reduction of elliptic surfaces up to modification. Along the way, we prove several more general results about such surfaces in positive characteristic, as well as about log smooth…

We construct explicit examples of $K3$ surfaces over ${\mathbb Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces…

We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is…

We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an…

This paper further studies the matroid realization space of a specific deformation of the regular $n$-gon with its lines of symmetry. Recently, we obtained that these particular realization spaces are birational to the elliptic modular…

Following a restriction argument in the Euclidean space, we derive a geometric invariant formula for a possible viscosity operator for an incompressible fluid flow on an ellipsoid embedded in $\mathbb R^3$. We also give an asymptotic…

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…

The ruled surface is a typical modeling surface in computer aided geometric design. It is usually given in the standard parametric form. However, it can also be in the forms than the standard one. For these forms, it is necessary to…