Relating the second and fourth order terms in the asymptotic expansion of the heat equation

Leta _n be the coefficients in the asymptotic expansion of the heat equation. In this paper, we study the relationship betweena ₂ ² anda ₄ in the context of both Riemannian and affine geometry.Research partially supported by the DFG project on Affine differential geometryResearch partially supported by the NSF (USA) and MSRI (USA)     

Relative equidimensionality and stability of actions¶of a reductive algebraic group

In order to inquire into invariants of non-semisimple groups, we introduce and study relative versions of equidimensionality and stabilty, which are called relative quasi-equidimensionality and relative stability, of actions of affine algebraic groups, especially of reductive groups, on affine varieties. As an application of our results, for complex reductive groups of semisimple rank one, we characterize, respectively, relatively stable representations and relatively equidimensional representations and, consequently, show that every equidimensional representation is cofree. Received: 23 October 1998     

Motives of some Fano varieties

We study the Fano varieties of projective k-planes lying in hypersurfaces and investigate the associated motives. The first author is partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The second author is partially supported by TüBİTAK-BDP funds and Bilkent University research development funds.     

Weights of mixed tilting sheaves and geometric Ringel duality

We describe several general methods for calculating weights of mixed tilting sheaves. We introduce a notion called “non-cancellation property” which implies a strong uniqueness of mixed tilting sheaves and enables one to calculate their weights effectively. When we have a certain Radon transform, we prove a geometric analogue of Ringel duality which sends tilting objects to projective objects. We apply these methods to (partial) flag varieties and affine (partial) flag varieties and show that the weight polynomials of mixed tilting sheaves on flag and affine flag varieties are essentially given by Kazhdan-Lusztig polynomials. This verifies a mixed geometric analogue of a conjecture by W. Soergel in [10].      

Tropical representation of Weyl groups associated with certain rational varieties

Starting from certain rational varieties blown-up from N(P¹), we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo-isomorphisms of the varieties. We develop an algebro-geometric framework of τ-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields a class of (higher order) q-difference Painlevé equations and its algebraic degree grows quadratically.     

On a map between two K3 surfaces associated to a net of quadrics

We study the geometry of the birational map between an intersection of a net of quadrics in that contains a line and the double sextic branched along the discriminant of the net. We show that the branch locus of a smooth double sextic S ₆ is discriminant of a net of quadrics in such that S ₆ is isomorphic to the intersection of this net iff a certain configuration of rational curves on S₆ is weakly even. Received: 14 September 2005 Suported by the DFG Schwerpunktprogramm ‘Global methods in complex geometry’. The first named author is partially supported by the KBN Grant No. 1 P03A 008 28. The second named author is partially supported by the KBN Grant No. 2 P03A 016 25.     

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined overQ there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G, PO(3)) = Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties. This research was partially supported by NSF grant DMS-96-26633 at University of Utah. This research was partially supported by NSF grant DMS-95-04193 at University of Maryland.     

Generic vanishing and minimal cohomology classes on abelian varieties

We establish a—and conjecture further—relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of the cohomology of twisted ideal sheaves. The main ingredient is the Generic Vanishing criterion established in Pareschi G. and Popa M. (GV-sheaves, Fourier–Mukai transform, and Generic Vanishing. Preprint math.AG/0608127), based on the Fourier–Mukai transform. MP was partially supported by the NSF grant DMS 0500985 and by an AMS Centennial Fellowship.     

On the cancellation problem

Let k be an algebraically closed field. For every n ≥ 8 we give examples of Zariski open, dense, affine subsets of the affine space A ⁿ (k) which do not have the cancellation property. Dedicated to Professor Mikhail Zaidenberg. The author was partially supported by the grant of Polish Ministry of Science, 2006–2009.     

Singular Poisson-Kähler geometry of Scorza varieties and their secant varieties

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.     

Hamiltonian / Gauge theoretical Gromov-Witten invariants of toric varieties

We present a purely algebraic approach to the Hamiltonian / Gauge theoretical invariants associated to torus actions on affine spaces. Secondly, we address the issue of computing the invariants: a localization and a genus recursion formula are deduced. Partially supported by: EAGER - European Algebraic Geometry Research Training Network, contract No. HPRN-CT-2000-00099 (BBW).     

Equivariant deformations of the affine multicone over a flag variety

We prove that the invariant Hilbert scheme parameterising the equivariant deformations of the affine multicone over a flag variety is, under certain hypotheses, an affine space. More specifically, we obtain that the isomorphism classes of equivariant deformations of such a multicone are in correspondence with the orbits of a well-determined wonderful variety.     

Rigidity and moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ⁻¹ is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular. The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006 (HPRN–CT–00271).     

Simple immersions of wonderful varieties

Let G be a semisimple connected linear algebraic group over , and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions. We also prove that any ample line bundle on a wonderful variety is very ample. Research supported by European Research Training Network LIEGRITS (MRTN-CT 2003- 505078), in contract with CNRS DR17, No 2.     

Quantization of curvature for compact surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n⋆</Emphasis></Superscript>

For minimal surfaces in spheres, there is a well known conjecture about the quantization of intrinsic curvature which has been solved only in special cases so far. We recall an intrinsic and an extrinsic version for the known results and extend them to compact non-minimal surfaces in spheres. In particular we discuss special classes like Willmore surfaces and surfaces with parallel mean curvature vector. Mathematics Subject Classification (2000):53C42, 53A10.H.Li is partially supported by a research fellowship of the Alexander von Humboldt Stiftung 2001/2002 and the Zhongdian grant of NSFC. U. Simon is partially supported by DFG 163/Si-7-2 and a Chinese–German research cooperation of NSFC and DFG.     

Modular curves and Poncelet polygons

Parts of the results and the essential techniques of this note are taken from the Erlangen thesis (1991) of the second author. They were circulated as Nr. 122 of Schriftenreihe Komplexe Mannigfaltigkeiten. Our research was supported by DFG grant Ba 423/3-3 and the European Science Project Geometry of Algebraic Varieties SCI-0398-C(A)     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.