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Here are few notes on not necessarily normal toric varieties and resolution by toric blow-up. These notes are independent of, but in the same spirit as the earlier preprint arXiv:math.AG/0306221. That is, they focus on the fact that toric varieties are locally given by monoid algebras.
These notes have two parts. The first is a study of Nekrasov's deformed partition functions $Z(\ve_1,\ve_2,\vec{a};\q,\vecτ)$ of N=2 SUSY Yang-Mills theories, which are generating functions of the integration in the equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$. The second is review of geometry of the Seiberg-Witten curves and the geometric engineering of the gauge theory, which are physical backgrounds of Nekrasov's partition functions. The first part is continuation of math.AG/0306198, where we identified the Seiberg-Witten prepotential with $Z(0,0,\vec{a};\q,0)$. We put higher Casimir operators to the partition function and clarify their relation to the Seiberg-Witten $u$-plane. We also determine the coefficients of $\ve_1\ve_2$ and $(\ve_1^2+\ve_2^2)/3$ (the genus 1 part) of the partition function, which coincide with two measure factors $A$, $B$ appeared in the $u$-plane integral. The proof is based on the blowup equation which we derived in the previous paper.
This is a supplement to the paper "Goncharov's Relations in Bloch's higher Chow Group CH^3(F,5)" submitted as math.AG/0105084. We check the admissiblility of all the cycles appearing in the main article.
Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.
This is a write-up of my talk at the Conference on algebraic structures in Montreal, July 2003. I try to give a brief informal introduction to the proof of Y. Ruan's conjecture on orbifold cohomology multiplication for symplectic quotient singularities given in V. Ginzburg and D. Kaledin, math.AG/0212279. Version 2: minor changes, added some references.
In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in math.AG/0002083. The result can be applied to show the rigidity of all open swallowtails of dimension greater or equal than two. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show that lagrangian complete intersection in dimension greater than two cannot be regular in codimension one.
In this article we classify quadruple Galois canonical covers $ϕ$ of singular surfaces of minimal degree. This complements the work done in math.AG/0302045, so the main output of both papers is the complete classification of quadruple Galois canonical covers of surfaces of minimal degree, both singular and smooth. Our results show that the covers $X$ studied in this article are all regular surfaces and form a bounded family in terms of geometric genus $p_g$. In fact, the geometric genus of $X$ is bounded by 4. Together with the results of Horikawa and Konno for double and triple covers, a striking numerology emerges that motivates some general questions on the existence of higher degree canonical covers. In this article, we also answer some of these questions. The arguments to prove our results include a delicate analysis of the discrepancies of divisors in connection with the ramification and inertia groups of $ϕ$.
This is a sequel to the paper "Frobenius amplitude and strong vanishing theorems for vector bundles" (math.AG/0202129). We introduce a more elementary variant of the notion of F-amplitude from the earlier paper which we call amplitude. This provides another measure of positivity of a vector bundle which is related to a number of preexisting positivity notions such as k-ampleness or q-convexity. We use this to refine the estimates of F-amplitude from the first paper, and to deduce some further vanishing theorems as a consequence. We also give some new proofs of some known vanishing theorems for Abelian and toric varieties by analogous methods. For technical reasons, we need to develop a theory of partial Castelnuovo-Mumford regularity which provides a rough measure of the cohomological complexity of a sheaf. Since this material may have independent interest, it is contained in a section which can be read on its own.
For a smooth Del Pezzo surface the direct sum of global sections of all isomorphism classes of invertible sheaves on it can be almost canonically endowed with a ring structure, called the Cox ring. We show that in characteristic 0 this ring is Gorenstein and its spectrum has rational singularities. Also a gap in math.AG/0309111 is pointed out.
This text is an introduction to math.AG/0110051 (to appear in Ann. Inst. Fourier), and describes a canonical decomposition of compact Kähler manifolds $X$ first by means of their "core", the unique fibration on $X$ with fibres special, and orbifold base of general type. Being special means that there does not exist a fibration onto an orbifold of general type. The core is next decomposed into a tower of fibrations with all orbifold fibres either with $κ=0$, or rationally generated (a weak version of rationally connected).
An orbifold version of Bogomolov decomposition theorem is established for compact Kähler spaces with quotient singularities and first Chern class zero.The proof is a direct adaptation of the classical smooth case, using Ricci-flat Kähler metrics, and Cheeger-Gromoll splitting theorem. It implies that normal K3 surfaces are uniformised in the orbifold sense either by normal K3 surfaces or by a complex torus, as conjectured by D.Q. Zhang. It also implies some conjectures on "special" threefolds raised in math.AG/0110051
In this paper, we shall prove that any two (projective) symplectic resolutions of a nilpotent orbit closure in a classical simple Lie algebra are connected by a finite sequence of diagrams which are locally trivial families of Mukai flops of type A or of type D. A Mukai flop of type A is the same as a stratified Mukai flop in our previous paper "Mukai flops and derived categories II". On the other hand, a Mukai flop of type D comes from a nilpotent orbit in so(2k). Let G^+ and G^- be two connected components of the orthogonal Grassmannian G_{iso}(k,2k). Then the closure of this orbit has two Springer resolutions via the cotanegent bundles T^*G^+ and T^*G^-, which is the Mukai flop of type D. Our result would clarify the geometric meaning of the results of Spaltenstein and Hesselink. To illustrate this, three examples will be given. Another purpose of this paper is to give an affimative answer to a conjecture in the paper "Uniqueness of crepant resolutions and symplectic singularities" math.AG/0306091, for nilpotent orbit cases. A key idea is the notion of a "Dixmier sheet" studied by Borho and Kraft.
The existence of bivariant Chern classes was conjectured by W.Fulton and R.MacPherson and proved by J.P.Brasselet for cellular morphisms of analytic varieties. In this paper we show that restricted to morphisms whose target varieties are possible singular but (rational) homology manifolds, the bivariant Chern classes (with rational coefficients) are uniquely determined. In the final sections we construct a unique bivariant Chern class satisfying a suitable normalization condition. In fact, it will be a special case of a general construction of unique Grothendieck transformations, which in a sense gives a positive answer to a corresponding question posed by Fulton and MacPherson. This construction generalizes and unifies previous attemps, in particular the main result of the paper math.AG/0209299.
This paper has been superseded by math.AG/0510287 and withdrawn by the author.
The paper is a short supplement of the longer paper "The Algebraic Proof of the Universality Theorem", preprint math.AG/0402045. In this short note, we outline the geometric meaning of Universality theorem (conjecture by Gottsche) as a non-linear extension of surface Riemann-Roch Theorem, inspired by the string theory argument of Yau-Zaslow to probe non-linear information from linear systems of algebraic surfaces. The universality theorem is an existence result which reflects the topological nature of the Riemann-Roch problem. We also outline the crucial role that Yau-Zaslow formula has played in our theory. At the end, we list a few open problems related to the algebraic solution of the problem.
Let X be a Kahler manifold that is presented as a Kahler quotient of C^n by the linear action of a compact group G. We define the hyperkahler analogue M of X as a hyperkahler quotient of the cotangent bundle T^*C^n by the induced G-action. Special instances of this construction include hypertoric varieties and quiver varieties. Our aim is to provide a unified treatment of these two previously studied examples, with specific attention to the geometry and topology of the circle action on M that descends from the scalar action on the fibers of the cotangent bundle. We provide a detailed study of this action in the cases where M is a hypertoric variety or a hyperpolygon space. Most of this document consists of material from the papers math.DG/0207012, math.AG/0308218, and math.SG/0310141. Sections 2.2 and 3.5 contain previously unannounced results.
This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-Borisov construction for complete intersections in toric varieties. Given a pair of reflexive polytopes with nef decompositions dual to each other, and given polarizations on the corresponding toric varieties, we construct examples of toric degenerations and their dual intersection complexes, which are affine manifolds with singularities. We show these affine manifolds are related by the discrete Legendre transform, thus showing that the Batyrev-Borisov construction is a special case of our more general construction. The description of the dual intersection complexes in terms of the combinatoricsof the setup generalises work of Haase and Zharkov in the toric hypersurface case, and similar work of Ruan. In particular, this gives a topological description of the Strominger-Yau-Zaslow fibrations on complete intersections in toric varieties.
Let X be a minuscule Schubert variety and $α$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $α$ from a rational curve to X. The irreducible components are described in the following way : the class $α$ can be seen as an element of $Pic(X)^*$ the dual of the Picard group. Because any Weil-divisor need not to be a Cartier-divisor, there is (only) a surjective map $s:A^1(X)^*\to Pic(X)^*$ from the dual of the group of codimension 1 cycles to the dual of the Picard group. The irreducible components are given by the effective elements $β$ in $A^1(X)^*$ such that $s(β)=α$. The proof of the result uses the Bott-Samelson resolution Y of X. We prove that any curve on X can be lifted in Y (after deformation). This is because any divisor on minuscule Schubert variety is a moving one. Then we prove that any curve coming from X can be deformed so that it does not meet the contracted divisor of $Y\to X$. This is possible because for minuscule Schubert variety there are lines in the projectivised tangent space to a singularity. It is now sufficient to deal with the case of the orbit of $Stab(X)$ the stabiliser of X and we can apply results of our previous paper math.AG/0003199.