Featured Researches

Algebraic Geometry

A closed model structure for n -categories, internal Hom , n -stacks and generalized Seifert-Van Kampen

We define a closed model category containing the n -nerves defined by Tamsamani, and admitting internal Hom . This allows us to construct the n+1 -category nCAT by taking the internal Hom for fibrant objects. We prove a generalized Seifert-Van Kampen theorem for Tamsamani's Poincaré n -groupoid of a topological space. We give a still-speculative discussion of n -stacks, and similarly of comparison with other possible definitions of n -category.

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Algebraic Geometry

A conjectural generating function for numbers of curves on surfaces

I give a conjectural generating function for the numbers of δ -nodal curves in a linear system of dimension δ on an algebraic surface. It reproduces the results of Vainsencher for the case δ≤6 and Kleiman-Piene for the case δ≤8 . The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of Yau-Zaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of Caporaso-Harris for the Severi degrees in ¶ 2 . We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points.

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Algebraic Geometry

A connectivity lemma for the Albanese map

I prove that for any complex projective variety X and a sufficiently large integer N all the fibers of Albanese map of the N -th configuration space of X are dominated by smooth connected projective varieties with vanishing H 1 .

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Algebraic Geometry

A construction of separated quotient space and boundaries of symmetric spaces

We propose a construction of separated quotient space and apply this construction for simple description of Study-Semple-Satake-Furstenberg-De Concini-Procesi-Oshima boundary of symmetric space

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Algebraic Geometry

A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics

We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics $\Q{n}$ which are not of general type, for n=5 and n≥7 . We prove a similar statement also for the case of higher codimension. The case n=6 has been recently settled by Fania-Ottaviani. Keywords: Codimension two, Grassmannians, Lifting, Low codimension, Not of General Type, Polynomial Bound, Quadrics

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Algebraic Geometry

A generalization of curve genus for ample vector bundles, I

A new genus $g=g(X,\ce)$ is defined for the pairs $(X,\ce)$ that consist of n -dimensional compact complex manifolds X and ample vector bundles $\ce$ of rank r less than n on X . In case r=n−1 , g is equal to curve genus. Above pairs $(X,\ce)$ with g less than two are classified. For spanned $\ce$ it is shown that g is greater than or equal to the irregularity of X , and its equality condition is given.

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Algebraic Geometry

A generalization of curve genus for ample vector bundles, II

Let X be a compact complex manifold of dimension n≥2 and $\ce$ an ample vector bundle of rank r<n on X . As the continuation of Part I, we further study the properties of $g(X,\ce)$ that is an invariant for pairs $(X,\ce)$ and is equal to curve genus when r=n−1 . Main results are the classifications of $(X,\ce)$ with $g(X,\ce)=2$ (resp. 3) when $\ce$ has a regular section (resp. $\ce$ is ample and spanned) and 1<r<n−1 .

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Algebraic Geometry

A geometric approach to the fundamental lemma for unitary groups

We consider from a geometric point of view the conjectural fundamental lemma of Langlands and Shelstad for unitary groups over a local field of positive characteristic. We introduce projective algebraic varieties over the finite residue field k and interpret the conjecture in this case as a remarkable identity between the number of k -rational points of them. We prove the corresponding identity for the numbers of k f -rational points, for any extension of even degree f of k . The proof uses the local intersection theory on a regular surface and Deligne's theory of intersection multiplicities with weights. We also discuss a possible descent argument that uses ℓ -adic cohomology to treat extensions of odd degree as well.

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Algebraic Geometry

A geometric construction of Getzler's relation

A geometric construction of Getzler's cohomological relation in the moduli space of 4 pointed elliptic curves is given by a push-forward of a natural rational equivalence in a space of admissible covers. In particular, Getzler's relation is shown to be a rational equivalence. The recursion for the elliptic Gromov-Witten invariants of P^2 predicted by Eguchi, Hori, and Xiong from the Virasoro conjecture is proven via Getzler's equation and the WDVV-equations.

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Algebraic Geometry

A high fibered power of a family of varieties of general type dominates a variety of general type

We prove the following theorem: Fibered Power Theorem: Let $X\rar B$ be a smooth family of positive dimensional varieties of general type, with B irreducible. Then there exists an integer n>0 , a positive dimensional variety of general type W n , and a dominant rational map $X^n_B \das W_n$.

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