Algebraic Geometry

The result in the title is proven, using the Selberg estimate on the leading eigenvalue of the non-Euclidean Laplacian, and the method of conformal volumes of Li and Yau.

Let X be a smooth projective surface over the complex number field and let L be a nef-big divisor on X . Here we consider the following conjecture; If the Kodaira dimension κ(X)≥0 , then K X L≥2q(X)−4 , where q(X) is the irregularity of X . In this paper, we prove that this conjecture is true if (1) the case in which κ(X)=0 or 1, (2) the case in which κ(X)=2 and h 0 (L)≥2 , or (3) the case in which κ(X)=2 , X is minimal, h 0 (L)=1 , and L satisfies some conditions.

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions. Revision 03.03.97: we correct an error in Introduction.

Let f:X−−>Y be a map of algebraic varieties. Barthel, Brasselet, Fieseler, Gabber and Kaup have shown that there exists a homomorphism of intersection homology groups f ∗ :I H ∗ (Y)−−>I H ∗ (X) compatible with the induced homomorphism on cohomology. The crucial point in the argument is reduction to the finite characteristic. We give an alternative and short proof of the existence of a homomorphism f ∗ . Our construction is an easy application of the Decomposition Theorem.

P-resolutions of a cyclic quotient singularity are known to be in one-to-one correspondence with the components of the base space of its semi-universal deformation. Stevens and Christophersen have shown that P-resolutions are parametrized by so-called chains representing zero or, equivalently, certain subdivisions of polygons. I give here a purely combinatorial proof of the correspondence between subdivisions of polygons and P-resolutions.

Let S d be the vector space of monomials of degree d in the variables x 1 ,..., x s . For a subspace $V \sus S_d$ which is in general coordinates, consider the subspace $\gin V \sus S_d$ generated by initial monomials of polynomials in V for the revlex order. We address the question of what properties of V may be deduced from $\gin V$. % This is an approach for understanding what algebraic or geometric properties of a homogeneous ideal $I \sus k[x_1, ..., x_s]$ that may be deduced from its generic initial ideal $\gin I$.

In this paper we resolve the arithmetic analogues of standard conjectures for an arithmetic variety, which are proposed by Gillet and Soule, into original standard conjectures and similar conjectures for two cycle class groups. One is the homologically trivial cycles. This is well-known as conjectures about the height pairing raised by Beilinson. The other is the cycle class groups consisting of vertical cycles. We show that the conjectures for the above two cycle class groups with Grothendieck's standard conjectures imply their arithmetic analogues.

If S is a scheme of finite type over $k=\cc$, let $\Xx /S$ denote the big etale site of schemes over S . We introduce {\em presentable group sheaves}, a full subcategory of the category of sheaves of groups on $\Xx /S$ which is closed under kernel, quotient, and extension. Group sheaves which are representable by group schemes of finite type over S are presentable; pullback and finite direct image preserve the notions of presentable group sheaves; over S=Spec(k) then presentable group sheaves are just group schemes of finite type over Spec(k) ; there is a notion of connectedness extending the usual notion over Spec(k) ; and a presentable group sheaf G has a Lie algebra object Lie(G . If G is a connected presentable group sheaf then G/Z(G) is determined up to isomorphism by the Lie algebra sheaf Lie(G) . We envision the category of presentable group sheaves as a generalisation relative to an arbitrary base scheme S , of the category of algebraic Lie groups over Spec(k) . The notion of presentable group sheaf is used in order to define {\em presentable n -stacks} over $\Xx$. Roughly, an n -stack is presentable if there is a surjection from a scheme of finite type to its π 0 (the actual condition on π 0 is slightly more subtle), and if its π i (which are sheaves on various $\Xx /S$) are presentable group sheaves. The notion of presentable n -stack is closed under homotopy fiber product and truncation. We propose the notion of presentable n -stack as an answer in characteristic zero for A. Grothendieck's search for what he called ``schematization of homotopy types''.

We propose an algebraic Riemann-Roch formula for moving flat bundles on contant families in characteristic zero with values in the ring of algebraic differential characters. This formula lifts Grothendieck-Riemann-Roch formula in the Chow groups and Bismut-Lott formula in the odd dimensional Betti cohomology with $\C/\Z$ coefficients over the field of complex numbers.

This is a footnote of a recent interesting work of Cohen, Manin and Zagier, where they, among other things, produce a natural isomorphism between the sheaf of (n-1)-th order jets of the n-th tensor power of the tangent bundle of a Riemann surface equipped with a projective structure and the sheaf of differential operators of order n (on the trivial bundle) with vanishing 0-th order part. We give a different proof of this result without using the coordinates, and following the idea of this proof we prove: Take a line bundle L with L 2 =T on a Riemann surface equipped with a projective structure. Then the jet bundle J n ( L n ) has a natural flat connection with J n ( L n )= S n ( J 1 (L)) . For any m>n the obvious surjection J m ( L n )→ J n ( L n ) has a canonical splitting. In particular, taking m=n+1 , one gets a natural differential operator of order n+1 from L n to L −n−2 .