Upper bounds for regularized determinants

Let $E$ be a holomorphic vector bundle on a compact K\"ahler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one defines its regularized determinant $det'(\Delta)$. We conjectured elsewhere that, when $h$ varies, this determinant $det'(\Delta)$ remains bounded from above. In this paper we prove this in two special cases. The first case is when $X$ is a Riemann surface, $E$ is a line bundle and $dim(H^0 (X,E)) + dim(H^1 (X,E)) \leq 2$, and the second case is when $X$ is the projective line, $E$ is a line bundle, and all metrics under consideration are invariant under rotation around a fixed axis.Comment: 22 pages, plain Te

An Explicit Proof of the Generalized Gauss-Bonnet Formula

In this paper we construct an explicit representative for the Grothendieck fundamental class [Z] of a complex submanifold Z of a complex manifold X, under the assumption that Z is the zero locus of a real analytic section of a holomorphic vector bundle E. To this data we associate a super-connection A on the exterior algebra of E, which gives a "twisted resolution" of the structure sheaf of Z. The "generalized super-trace" of A^{2r}/r!, where r is the rank of E, is an explicit map of complexes from the twisted resolution to the Dolbeault complex of X, which represents [Z]. One may then read off the Gauss-Bonnet formula from this map of complexes.Comment: 21 pages. Paper reorganized to improve exposition. To appear in Asterisqu

On the arithmetic Chern character

We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern character class of the sequence and its Chern character with supports on the finite fibers. Next, we compute these classes in the situation encountered by the second author when proving a "Kodaira vanishing theorem" for arithmetic surfaces

Rational points of varieties with ample cotangent bundle over function fields of positive characteristic

Let $K$ be the function field of a smooth curve over an algebraically closed field $k$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $\Omega_{X/K}$ is ample. Let $R:={\rm Zar}(X)(K)\cap X)$ be the Zariski closure of the set of all $K$-rational points of $X$, endowed with its reduced induced structure. We prove that there is a projective variety $X_0$ over $k$ and a finite and surjective $K^{\rm sep}$-morphism $X_{0,K^{\rm sep}}\to R_{K^{\rm sep}}$, which is birational when ${\rm char}(K)=0$.Comment: Final version; to appear in Mathematische Annale

The structure of radiative shock waves. III. The model grid for partially ionized hydrogen gas

The grid of the models of radiative shock waves propagating through partially ionized hydrogen gas with temperature 3000K <= T_1 <= 8000K and density 10^{-12} gm/cm^3 <= \rho_1 <= 10^{-9}gm/cm^3 is computed for shock velocities 20 km/s <= U_1 <= 90 km/s. The fraction of the total energy of the shock wave irreversibly lost due to radiation flux ranges from 0.3 to 0.8 for 20 km/s <= U_1 <= 70 km/s. The postshock gas is compressed mostly due to radiative cooling in the hydrogen recombination zone and final compression ratios are within 1 <\rho_N/\rho_1 \lesssim 10^2, depending mostly on the shock velocity U_1. The preshock gas temperature affects the shock wave structure due to the equilibrium ionization of the unperturbed hydrogen gas, since the rates of postshock relaxation processes are very sensitive to the number density of hydrogen ions ahead the discontinuous jump. Both the increase of the preshock gas temperature and the decrease of the preshock gas density lead to lower postshock compression ratios. The width of the shock wave decreases with increasing upstream velocity while the postshock gas is still partially ionized and increases as soon as the hydrogen is fully ionized. All shock wave models exhibit stronger upstream radiation flux emerging from the preshock outer boundary in comparison with downstream radiation flux emerging in the opposite direction from the postshock outer boundary. The difference between these fluxes depends on the shock velocity and ranges from 1% to 16% for 20 km/s <= U_1 <= 60 km/s. The monochromatic radiation flux transported in hydrogen lines significantly exceeds the flux of the background continuum and all shock wave models demonstrate the hydrogen lines in emission.Comment: 11 pages, 11 figures, LaTeX, to appear in A