34 research outputs found
Topological properties of punctual Hilbert schemes of almost-complex fourfolds (I)
In this article, we study topological properties of Voisin's punctual Hilbert
schemes of an almost-complex fourfold . In this setting, we compute their
Betti numbers and construct Nakajima operators. We also define tautological
bundles associated with any complex bundle on , which are shown to be
canonical in -theory
Surfaces containing a family of plane curves not forming a fibration
We complete the classification of smooth surfaces swept out by a
1-dimensional family of plane curves that do not form a fibration. As a
consequence, we characterize manifolds swept out by a 1-dimensional family of
hypersurfaces that do not form a fibration.Comment: Author's post-print, final version published online in Collect. Mat
The Abelian/Nonabelian Correspondence and Frobenius Manifolds
We propose an approach via Frobenius manifolds to the study (began in
math.AG/0407254) of the relation between rational Gromov-Witten invariants of
nonabelian quotients X//G and those of the corresponding ``abelianized''
quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses
the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove
this conjecture when the nonabelian quotients are partial flag manifolds.Comment: 35 pages, no figure
Enumerative geometry of Calabi-Yau 4-folds
Gromov-Witten theory is used to define an enumerative geometry of curves in
Calabi-Yau 4-folds. The main technique is to find exact solutions to moving
multiple cover integrals. The resulting invariants are analogous to the BPS
counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold
invariants to be integers and expect a sheaf theoretic explanation.
Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including
the sextic Calabi-Yau in CP5, are also studied. A complete solution of the
Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic
anomaly equation.Comment: 44 page
The moduli space of hyper-K{\"a}hler four-fold compactifications
I discuss some aspects of the moduli space of hyper-K{\"a}hler four-fold
compactifications of type II and - theories. The dimension of the
moduli space of these theories is strictly bounded from above. As an example I
study Hilb and the generalized Kummer variety . In both cases
RR-flux (or -flux in -theory) must be turned on, and we show that
they give rise to vacua with or supersymmetry upon
turning on appropriate fluxes. An interesting subtlety involving the symmetric
product limit is pointed out.Comment: 42 pages, discussion of supersymmetry preserving fluxes
added, acknowledegement adde
Equivariant Riemann-Roch theorems for curves over perfect fields
We prove an equivariant Riemann-Roch formula for divisors on algebraic curves over perfect fields. By reduction to the known case of curves over algebraically closed fields, we first show a preliminary formula with coefficients in Q. We then prove and shed some further light on a divisibility result that yields a formula with integral coefficients. Moreover, we give variants of the main theorem for equivariant locally free sheaves of higher rank