3,966 research outputs found
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
Energy Conservation and Hawking Radiation
The conservation of energy implies that an isolated radiating black hole
cannot have an emission spectrum that is precisely thermal. Moreover, the
no-hair theorem is only approximately applicable. We consider the implications
for the black hole information puzzle.Comment: 6 pages, LaTex; v2: references adde
The Volume of Black Holes
We propose a definition of volume for stationary spacetimes. The proposed
volume is independent of the choice of stationary time-slicing, and applies
even though the Killing vector may not be globally timelike. Moreover, it is
constant in time, as well as simple: the volume of a spherical black hole in
four dimensions turns out to be just . We then consider
whether it is possible to construct spacetimes that have finite horizon area
but infinite volume, by sending the radius to infinity while making discrete
identifications to preserve the horizon area. We show that, in three or four
dimensions, no such solutions exist that are not inconsistent in some way. We
discuss the implications for the interpretation of the Bekenstein-Hawking
entropy.Comment: 8 pages, revte
Rindler-AdS/CFT
In anti-de Sitter space a highly accelerating observer perceives a Rindler
horizon. The two Rindler wedges in AdS_{d+1} are holographically dual to an
entangled conformal field theory that lives on two boundaries with geometry R x
H_{d-1}. For AdS_3, the holographic duality is especially tractable, allowing
quantum-gravitational aspects of Rindler horizons to be probed. We recover the
thermodynamics of Rindler-AdS space directly from the boundary conformal field
theory. We derive the temperature from the two-point function and obtain the
Rindler entropy density precisely, including numerical factors, using the Cardy
formula. We also probe the causal structure of the spacetime, and find from the
behavior of the one-point function that the CFT "knows" when a source has
fallen across the Rindler horizon. This is so even though, from the bulk point
of view, there are no local signifiers of the presence of the horizon. Finally,
we discuss an alternate foliation of Rindler-AdS which is dual to a CFT living
in de Sitter space.Comment: 29 Pages, 4 Figures, citations adde
Lehn's formula in Chow and Conjectures of Beauville and Voisin
The Beauville-Voisin conjecture for a hyperk\"ahler manifold X states that
the subring of the Chow ring A^*(X) generated by divisor classes and Chern
characters of the tangent bundle injects into the cohomology ring of X. We
prove a weak version of this conjecture when X is the Hilbert scheme of points
on a K3 surface, for the subring generated by divisor classes and tautological
classes. This in particular implies the weak splitting conjecture of Beauville
for these geometries. In the process, we extend Lehn's formula and the
Li-Qin-Wang W_{1+infinity} algebra action from cohomology to Chow groups, for
the Hilbert scheme of an arbitrary smooth projective surface
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