Tagged Questions
4
votes
0answers
54 views
Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem
From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
4
votes
1answer
89 views
differential operator on manifold
I am currently trying to understand the local expression of a (pseudo)differential operator
$$
\int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi
$$
on a manifold $M$ (compact and boundaryless, ...
8
votes
2answers
188 views
Are there n-th roots of differential operators?
In analogy to a Dirac operator, it seems to me that formally, the equation
$$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$
is solved by
$$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$
Is there a ...
5
votes
1answer
129 views
How to prove (0,1) form is not $\overline\partial$-exact
On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
1
vote
2answers
122 views
What does ad$f$ mean, for $f$ a smooth function?
I am currently reading Nicole Berline "Heat Kernels and Dirac Operators". On page 64 Differential Operators are introduced that are generalized from operators acting on scalar functions to vector ...
