1,951 research outputs found
Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
Let X -> Y be a fibration whose fibers are complete intersections of two
quadrics. We develop new categorical and algebraic tools---a theory of relative
homological projective duality and the Morita invariance of the even Clifford
algebra under quadric reduction by hyperbolic splitting---to study
semiorthogonal decompositions of the bounded derived category of X. Together
with new results in the theory of quadratic forms, we apply these tools in the
case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers
are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds,
respectively. In the latter two cases, if Y is the projective line over an
algebraically closed field of characteristic zero, we relate rationality
questions to categorical representability of X.Comment: 43 pages, changes made and some material added and corrected in
sections 1, 4, and 5; this is the final version accepted for publication at
Journal de Math\'ematiques Pures et Appliqu\'ee
Supersingular K3 surfaces for large primes
Given a K3 surface X over a field of characteristic p, Artin conjectured that
if X is supersingular (meaning infinite height) then its Picard rank is 22.
Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture
for K3 surfaces over finite fields with p \geq 5. We prove Artin's conjecture
under the additional assumption that X has a polarization of degree 2d with p >
2d+4. Assuming semistable reduction for surfaces in characteristic p, we can
improve the main result to K3 surfaces which admit a polarization of degree
prime-to-p when p \geq 5.
The argument uses Borcherds' construction of automorphic forms on O(2,n) to
construct ample divisors on the moduli space. We also establish
finite-characteristic versions of the positivity of the Hodge bundle and the
Kulikov-Pinkham-Persson classification of K3 degenerations. In the appendix by
A. Snowden, a compatibility statement is proven between Clifford constructions
and integral p-adic comparison functors.Comment: Some minor edits made; German error fixed; comments still welcom
Complex valued Ray-Singer torsion
In the spirit of Ray and Singer we define a complex valued analytic torsion
using non-selfadjoint Laplacians. We establish an anomaly formula which permits
to turn this into a topological invariant. Conjecturally this analytically
defined invariant computes the complex valued Reidemeister torsion, including
its phase. We establish this conjecture in some non-trivial situations.Comment: Fixed two sign mistakes and added a few more details here and ther
T-Duality from super Lie n-algebra cocycles for super p-branes
We compute the -theoretic dimensional reduction of the
F1/D-brane super -cocycles with coefficients in rationalized
twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to
9d. We show that the two resulting coefficient -algebras are
naturally related by an -isomorphism which we find to act on the
super -brane cocycles by the infinitesimal version of the rules of
topological T-duality and inducing an isomorphism between and ,
rationally. In particular this is a derivation of the Buscher rules for
RR-fields (Hori's formula) from first principles. Moreover, we show that these
-algebras are the homotopy quotients of the RR-charge coefficients by
the "T-duality Lie 2-algebra". We find that the induced -extension is
a gerby extension of a 9+(1+1) dimensional (i.e. "doubled") T-duality
correspondence super-spacetime, which serves as a local model for T-folds. We
observe that this still extends, via the D0-brane cocycle of its type IIA
factor, to a 10+(1+1)-dimensional super Lie algebra. Finally we observe that
this satisfies expected properties of a local model space for F-theory elliptic
fibrations.Comment: 44 pages; v2: added more discussion of double dimensional reduction
via cyclic L-infinity cohomology; v3: added derivation of Buscher rules for
RR-fields, expanded on role of curved L-infinity algebras in double
dimensional reductio
Topological modular forms and conformal nets
We describe the role conformal nets, a mathematical model for conformal field
theory, could play in a geometric definition of the generalized cohomology
theory TMF of topological modular forms. Inspired by work of Segal and
Stolz-Teichner, we speculate that bundles of boundary conditions for the net of
free fermions will be the basic underlying objects representing TMF-cohomology
classes. String structures, which are the fundamental orientations for
TMF-cohomology, can be encoded by defects between free fermions, and we
construct the bundle of fermionic boundary conditions for the TMF-Euler class
of a string vector bundle. We conjecture that the free fermion net exhibits an
algebraic periodicity corresponding to the 576-fold cohomological periodicity
of TMF; using a homotopy-theoretic invariant of invertible conformal nets, we
establish a lower bound of 24 on this periodicity of the free fermions
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