Writings of Kai Behrend

Here is a selection of some of the mathematics I have written over the years. See also ArXiv and MathSciNet.

•  German Diplom Thesis
My Diplom thesis was written in Bonn (Germany) under the direction of Prof. G. Harder. It is about moduli spaces of vector bundles with level structures on curves. It has not been published, and it does not exist in electronic form. Here is a summary.  pdf

•  Berkeley PhD Thesis
I wrote my PhD thesis under the direction of Prof. A. Ogus in Berkeley, California. It is about the Lefschetz trace formula for the Frobenius on algebraic stacks in general, and the stack of G-bundles over a curve (G a reductive group scheme) in particular. Only parts of the thesis have been published. Here is the thesis in its entirety.  pdf

Preprints: Algebraic Stacks

•  Derived l-adic categories for algebraic stacks
This is work I did while I was a postdoc at MIT. I construct l-adic derived categories for algebraic stacks defined over a finite field. I define pushforward and pullback functors between these categories. The key notion is that of convergent complex. In the end I arrive at a proof of the trace formula for the Frobenius on any algebraic stack. (I conjectured this formula in 1993.) This work eventually appeared in Mem. AMS 774.  pdf

•  Cohomology of Stacks
These are notes for lectures I gave in 2002 in Trieste at the ICTP. They cover de Rham cohomomlogy of differentiable stacks as well as de Rham cohomology with compact supports. I prove Poincaré duality for proper differentiable Deligne-Mumford stacks. In the second part of these notes, I cover singular homology and cohomology of topological stacks. Examples and applications are included.  pdf

• (with Behrang Noohi) Uniformization of Deligne-Mumford curves
We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary Deligne-Mumford curve as a quotient stack. Along the way, we compute the automorphism 2-groups of weighted projective stacks. We also discuss connections with the theory of F-groups, 2-groups, and Bass-Serre theory of graphs of groups. Appeared in Crelle  pdf

•  On the de Rham cohomology of differential and algebraic stacks
We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to etale equivalence. We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E1-term consists entirely of vector bundle valued cohomology groups. Our theory works for differentiable, holomorphic and algebraic stacks. This work appeared in the Artin Festschrift.  pdf

• (with Ajneet Dhillon) On the motive of the stack of bundles
In connection with my work in Bonn on my Diplom Thesis, Harder explained to me the strange connection between two numbers which are 1: The homotopy type of the space of connections and the Tamagawa number. In this paper we make a precise mathematical conjecture which attempts to explain this strange observation. In fact, we give a conjectural formula for the motive of the stack of G-bundles on a curve in terms of special values of the motivic zeta function of the curve.  pdf

Preprints: Gromov-Witten Theory

• (with Y. Manin) Stacks of stable maps and Gromov-Witten invariants
Treats stacks of stable maps, axioms for virtual fundamental classes and genus zero Gromov-Witten invariants for convex varieties. Special emphasis on functorial nature of both stacks of stable maps and Gromov-Witten invariants. Appeared in Duke Math. Journal.  pdf

• (with B. Fantechi) The intrinsic normal cone
Shows how to construct virtual fundamental classes, using the intrinsic normal cone. Appeared in Invent. Math.  pdf

•  Gromov-Witten invariants in algebraic geometry
We apply the construction of virtual fundamental classes to stacks of stable maps to get Gromov-Witten invariants in the general case. Appeared in Invent. Math.  pdf

•  Algebraic Gromov-Witten invariants
An introduction to the theory developed in the above three papers. Apeared in the proceedings of the Warwick conference 1996.  pdf

•  The product formula for Gromov-Witten invariants
We prove that the Gromov-Witten CohFT of a product is the tensor product of the Gromov-Witten CohFT's of the factors. Appeared in JAG.  pdf

•  Localization and Gromov-Witten invariants
These are notes of lectures I gave in the summer of 1997 in Cetraro, Italy, at the CIME session on Quantum cohomology. The first lecture is an introduction to stacks. The second lecture is about equivariant intersection theory. Finally, in the third lecture I prove the localization formula for Gromov-Witten invariants of projective space. My approach is different from the approach of Graber-Pandharipande (Invent. Math. 1999). It avoids equivariant virtual fundamental classes. These notes appeared in LNM 1776.  pdf

• (with A. O'Halloran) On the Cohomology of Stable Map Spaces
We describe an approach to calculating the cohomology rings of stable map spaces of genus zero curves without marks to projective space. Our method uses an equivariant vector field on the big Bialynicki-Birula cell and is based on work of Akildiz and Carrell. Our main result is a complete determination of the stable cohomology as the dimension of the target goes to infinity of the space of degree 3 maps in terms of generators and relations. Appeared in Invent. Math.  pdf

Preprints: Derived Geometry

•  Differential Graded Schemes I   Perfect Resolving Algebras
I introduce perfect resolving algebras and study their fundamental properties. These algebras are basic for any theory of differential graded schemes, as they give rise to affine differential graded schemes. I also introduce étale morphisms. The purpose for studying these, is that they will be used to glue differential graded schemes from affine ones with respect to an étale topology. This has not been published yet.  pdf

•  Differential Graded Schemes II   The 2-category of differential graded schemes
I construct a 2-category of differential graded schemes. The local affine models in this theory are differential graded algebras, which are graded commutative with unit over a field of characteristic zero, are concentrated in non-positive degrees and have perfect cotangent complex. Quasi-isomorphic differential graded algebras give rise to 2-isomorphic differential graded schemes and a differential graded algebra can be recovered up to quasi-isomorphism from the differential graded scheme it defines. Differential graded schemes can be glued with respect to an étale topology and fibered products of differential graded schemes correspond on the algebra level to derived tensor products. While this theory is without internal contradictions, it does not give rise to "correct" derived geometry: 2-categories are not sufficient for that purpose. Many results in this preprint will hopefully still be useful in derived geometry, for example, a theory based on simplicial presheaves.  pdf

Preprints: Donaldson-Thomas Theory

•  Donaldson-Thomas Type Invariants via Microlocal Geometry
This is the paper where I introduce the Behrend function. I prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. I also introduce new invariants generalizing Donaldson-Thomas type invariants to mod- uli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications. Appeared in Annals of Mathematics.  pdf

• (with B. Fantechi) Donaldson-Thomas Type Invariants via Microlocal Geometry
We evaluate the Behrend function for a singularity which is an isolated point of a C*-action and which admits a symmetric obstruction theory compatible with the C*-action. The answer is (-1)^d, where d is the dimension of the Zariski tangent space. We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of n points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi-Yau threefold, we deduce that the Donaldson-Thomas invariant of the Hilbert scheme of n points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik-Nekrasov-Okounkov-Pandharipande.  pdf

• (with J. Bryan) Super-rigid Donaldson-Thomas Invariants
We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson- Thomas correspondence of [MNOP] for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for the quintic threefold in degrees one and two.  pdf

• (with B. Fantechi) Gerstenhaber and Batalin-Vilkovisky structures on Lagrangian intersections
Let M and N be Lagrangian submanifolds of a complex symplectic manifold S . We construct a Gerstenhaber algebra structure on Tor^S(O_M,O_N) and a compatible Batalin-Vilkovisky module structure on Ext_S(O_M,O_N). This gives rise to a de Rham type cohomology theory for Lagrangian intersections. It is an attempt to categorify Lagrangian intersection numbers. This work has appeared in the Manin Festschrift.  pdf

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Last updated March 10, 2010.